3.11.100 \(\int \frac {A+B x}{(d+e x)^{5/2} (b x+c x^2)^2} \, dx\)
Optimal. Leaf size=344 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-5 A b e-4 A c d+2 b B d)}{b^3 d^{7/2}}-\frac {e \left (b^2 (-e) (2 B d-5 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}+\frac {c^{5/2} \left (9 A b c e-4 A c^2 d-7 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}-\frac {e \left (b^3 e^2 (2 B d-5 A e)-b^2 c d e (6 B d-11 A e)-b c^2 d^2 (3 A e+B d)+2 A c^3 d^3\right )}{b^2 d^3 \sqrt {d+e x} (c d-b e)^3} \]
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Rubi [A] time = 0.92, antiderivative size = 344, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{822, 828, 826, 1166, 208} \begin {gather*} -\frac {e \left (-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (3 A e+B d)+2 A c^3 d^3\right )}{b^2 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {e \left (b^2 (-e) (2 B d-5 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}+\frac {c^{5/2} \left (9 A b c e-4 A c^2 d-7 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-5 A b e-4 A c d+2 b B d)}{b^3 d^{7/2}}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]
[Out]
-(e*(6*A*c^2*d^2 - b^2*e*(2*B*d - 5*A*e) - 3*b*c*d*(B*d + 2*A*e)))/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) -
(e*(2*A*c^3*d^3 - b^2*c*d*e*(6*B*d - 11*A*e) + b^3*e^2*(2*B*d - 5*A*e) - b*c^2*d^2*(B*d + 3*A*e)))/(b^2*d^3*(
c*d - b*e)^3*Sqrt[d + e*x]) - (A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(
3/2)*(b*x + c*x^2)) - ((2*b*B*d - 4*A*c*d - 5*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(7/2)) + (c^(5/2)*
(2*b*B*c*d - 4*A*c^2*d - 7*b^2*B*e + 9*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d -
b*e)^(7/2))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 822
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 826
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 828
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]
Rule 1166
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx &=-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} (c d-b e) (2 b B d-4 A c d-5 A b e)-\frac {5}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {e \left (6 A c^2 d^2-b^2 e (2 B d-5 A e)-3 b c d (B d+2 A e)\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} (c d-b e)^2 (2 b B d-4 A c d-5 A b e)+\frac {1}{2} c e \left (6 A c^2 d^2-b^2 e (2 B d-5 A e)-3 b c d (B d+2 A e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac {e \left (6 A c^2 d^2-b^2 e (2 B d-5 A e)-3 b c d (B d+2 A e)\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} (c d-b e)^3 (2 b B d-4 A c d-5 A b e)+\frac {1}{2} c e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^3 (c d-b e)^3}\\ &=-\frac {e \left (6 A c^2 d^2-b^2 e (2 B d-5 A e)-3 b c d (B d+2 A e)\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {-\frac {1}{2} e (c d-b e)^3 (2 b B d-4 A c d-5 A b e)-\frac {1}{2} c d e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right )+\frac {1}{2} c e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^2 d^3 (c d-b e)^3}\\ &=-\frac {e \left (6 A c^2 d^2-b^2 e (2 B d-5 A e)-3 b c d (B d+2 A e)\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {(c (2 b B d-4 A c d-5 A b e)) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3 d^3}-\frac {\left (c^3 \left (2 b B c d-4 A c^2 d-7 b^2 B e+9 A b c e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3 (c d-b e)^3}\\ &=-\frac {e \left (6 A c^2 d^2-b^2 e (2 B d-5 A e)-3 b c d (B d+2 A e)\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {(2 b B d-4 A c d-5 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {c^{5/2} \left (4 A c^2 d+7 b^2 B e-b c (2 B d+9 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 194, normalized size = 0.56 \begin {gather*} \frac {-x (b+c x) \left (c d^2 \left (b c (9 A e+2 B d)-4 A c^2 d-7 b^2 B e\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )+(c d-b e)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e x}{d}+1\right ) (5 A b e+4 A c d-2 b B d)\right )-3 A b^2 d (c d-b e)^2-3 b c d x (b e-c d) (A b e-2 A c d+b B d)}{3 b^3 d^2 x (b+c x) (d+e x)^{3/2} (c d-b e)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]
[Out]
(-3*A*b^2*d*(c*d - b*e)^2 - 3*b*c*d*(-(c*d) + b*e)*(b*B*d - 2*A*c*d + A*b*e)*x - x*(b + c*x)*(c*d^2*(-4*A*c^2*
d - 7*b^2*B*e + b*c*(2*B*d + 9*A*e))*Hypergeometric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] + (c*d - b*e)
^2*(-2*b*B*d + 4*A*c*d + 5*A*b*e)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (e*x)/d]))/(3*b^3*d^2*(c*d - b*e)^2*x*(
b + c*x)*(d + e*x)^(3/2))
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IntegrateAlgebraic [A] time = 1.05, size = 672, normalized size = 1.95 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (5 A b e+4 A c d-2 b B d)}{b^3 d^{7/2}}+\frac {\left (9 A b c^{7/2} e-4 A c^{9/2} d-7 b^2 B c^{5/2} e+2 b B c^{7/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 (c d-b e)^3 \sqrt {b e-c d}}-\frac {-2 A b^4 d^2 e^4-10 A b^4 d e^4 (d+e x)+15 A b^4 e^4 (d+e x)^2+4 A b^3 c d^3 e^3+30 A b^3 c d^2 e^3 (d+e x)-58 A b^3 c d e^3 (d+e x)^2+15 A b^3 c e^3 (d+e x)^3-2 A b^2 c^2 d^4 e^2-20 A b^2 c^2 d^3 e^2 (d+e x)+64 A b^2 c^2 d^2 e^2 (d+e x)^2-33 A b^2 c^2 d e^2 (d+e x)^3-12 A b c^3 d^3 e (d+e x)^2+9 A b c^3 d^2 e (d+e x)^3+6 A c^4 d^4 (d+e x)^2-6 A c^4 d^3 (d+e x)^3+2 b^4 B d^3 e^3+4 b^4 B d^2 e^3 (d+e x)-6 b^4 B d e^3 (d+e x)^2-4 b^3 B c d^4 e^2-18 b^3 B c d^3 e^2 (d+e x)+28 b^3 B c d^2 e^2 (d+e x)^2-6 b^3 B c d e^2 (d+e x)^3+2 b^2 B c^2 d^5 e+14 b^2 B c^2 d^4 e (d+e x)-34 b^2 B c^2 d^3 e (d+e x)^2+18 b^2 B c^2 d^2 e (d+e x)^3-3 b B c^3 d^4 (d+e x)^2+3 b B c^3 d^3 (d+e x)^3}{3 b^2 d^3 x (d+e x)^{3/2} (b e-c d)^3 (b e+c (d+e x)-c d)} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]
[Out]
-1/3*(2*b^2*B*c^2*d^5*e - 4*b^3*B*c*d^4*e^2 - 2*A*b^2*c^2*d^4*e^2 + 2*b^4*B*d^3*e^3 + 4*A*b^3*c*d^3*e^3 - 2*A*
b^4*d^2*e^4 + 14*b^2*B*c^2*d^4*e*(d + e*x) - 18*b^3*B*c*d^3*e^2*(d + e*x) - 20*A*b^2*c^2*d^3*e^2*(d + e*x) + 4
*b^4*B*d^2*e^3*(d + e*x) + 30*A*b^3*c*d^2*e^3*(d + e*x) - 10*A*b^4*d*e^4*(d + e*x) - 3*b*B*c^3*d^4*(d + e*x)^2
+ 6*A*c^4*d^4*(d + e*x)^2 - 34*b^2*B*c^2*d^3*e*(d + e*x)^2 - 12*A*b*c^3*d^3*e*(d + e*x)^2 + 28*b^3*B*c*d^2*e^
2*(d + e*x)^2 + 64*A*b^2*c^2*d^2*e^2*(d + e*x)^2 - 6*b^4*B*d*e^3*(d + e*x)^2 - 58*A*b^3*c*d*e^3*(d + e*x)^2 +
15*A*b^4*e^4*(d + e*x)^2 + 3*b*B*c^3*d^3*(d + e*x)^3 - 6*A*c^4*d^3*(d + e*x)^3 + 18*b^2*B*c^2*d^2*e*(d + e*x)^
3 + 9*A*b*c^3*d^2*e*(d + e*x)^3 - 6*b^3*B*c*d*e^2*(d + e*x)^3 - 33*A*b^2*c^2*d*e^2*(d + e*x)^3 + 15*A*b^3*c*e^
3*(d + e*x)^3)/(b^2*d^3*(-(c*d) + b*e)^3*x*(d + e*x)^(3/2)*(-(c*d) + b*e + c*(d + e*x))) + ((2*b*B*c^(7/2)*d -
4*A*c^(9/2)*d - 7*b^2*B*c^(5/2)*e + 9*A*b*c^(7/2)*e)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d -
b*e)])/(b^3*(c*d - b*e)^3*Sqrt[-(c*d) + b*e]) + ((-2*b*B*d + 4*A*c*d + 5*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]
])/(b^3*d^(7/2))
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fricas [B] time = 49.60, size = 5665, normalized size = 16.47
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
[-1/6*(3*((2*(B*b*c^4 - 2*A*c^5)*d^5*e^2 - (7*B*b^2*c^3 - 9*A*b*c^4)*d^4*e^3)*x^4 + (4*(B*b*c^4 - 2*A*c^5)*d^6
*e - 2*(6*B*b^2*c^3 - 7*A*b*c^4)*d^5*e^2 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^4*e^3)*x^3 + (2*(B*b*c^4 - 2*A*c^5)*d
^7 - (3*B*b^2*c^3 - A*b*c^4)*d^6*e - 2*(7*B*b^3*c^2 - 9*A*b^2*c^3)*d^5*e^2)*x^2 + (2*(B*b^2*c^3 - 2*A*b*c^4)*d
^7 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^6*e)*x)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e
*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 3*((5*A*b^4*c*e^6 + 2*(B*b*c^4 - 2*A*c^5)*d^4*e^2 - (6*B*b^2*c^3 - 7
*A*b*c^4)*d^3*e^3 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^2*e^4 - (2*B*b^4*c + 11*A*b^3*c^2)*d*e^5)*x^4 + (5*A*b^5*e^6
+ 4*(B*b*c^4 - 2*A*c^5)*d^5*e - 10*(B*b^2*c^3 - A*b*c^4)*d^4*e^2 + (6*B*b^3*c^2 + 13*A*b^2*c^3)*d^3*e^3 + (2*
B*b^4*c - 19*A*b^3*c^2)*d^2*e^4 - (2*B*b^5 + A*b^4*c)*d*e^5)*x^3 + (10*A*b^5*d*e^5 + 2*(B*b*c^4 - 2*A*c^5)*d^6
- (2*B*b^2*c^3 + A*b*c^4)*d^5*e - (6*B*b^3*c^2 - 17*A*b^2*c^3)*d^4*e^2 + 5*(2*B*b^4*c - A*b^3*c^2)*d^3*e^3 -
(4*B*b^5 + 17*A*b^4*c)*d^2*e^4)*x^2 + (5*A*b^5*d^2*e^4 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^6 - (6*B*b^3*c^2 - 7*A*b^
2*c^3)*d^5*e + 3*(2*B*b^4*c + A*b^3*c^2)*d^4*e^2 - (2*B*b^5 + 11*A*b^4*c)*d^3*e^3)*x)*sqrt(d)*log((e*x + 2*sqr
t(e*x + d)*sqrt(d) + 2*d)/x) + 2*(3*A*b^2*c^3*d^6 - 9*A*b^3*c^2*d^5*e + 9*A*b^4*c*d^4*e^2 - 3*A*b^5*d^3*e^3 -
3*(5*A*b^4*c*d*e^5 + (B*b^2*c^3 - 2*A*b*c^4)*d^4*e^2 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^3*e^3 - (2*B*b^4*c + 11*A
*b^3*c^2)*d^2*e^4)*x^3 - (15*A*b^5*d*e^5 + 6*(B*b^2*c^3 - 2*A*b*c^4)*d^5*e + 5*(4*B*b^3*c^2 + 3*A*b^2*c^3)*d^4
*e^2 + 5*(2*B*b^4*c - 7*A*b^3*c^2)*d^3*e^3 - (6*B*b^5 + 13*A*b^4*c)*d^2*e^4)*x^2 - (3*A*b^2*c^3*d^5*e + 20*A*b
^5*d^2*e^4 + 3*(B*b^2*c^3 - 2*A*b*c^4)*d^6 + (20*B*b^4*c + 9*A*b^3*c^2)*d^4*e^2 - (8*B*b^5 + 41*A*b^4*c)*d^3*e
^3)*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4*e^5)*x^4 + (2*b^3*
c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*d^9 - b^4*c^3*
d^8*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2*d^8*e + 3*b^6*c*d^
7*e^2 - b^7*d^6*e^3)*x), 1/6*(6*((2*(B*b*c^4 - 2*A*c^5)*d^5*e^2 - (7*B*b^2*c^3 - 9*A*b*c^4)*d^4*e^3)*x^4 + (4*
(B*b*c^4 - 2*A*c^5)*d^6*e - 2*(6*B*b^2*c^3 - 7*A*b*c^4)*d^5*e^2 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^4*e^3)*x^3 + (
2*(B*b*c^4 - 2*A*c^5)*d^7 - (3*B*b^2*c^3 - A*b*c^4)*d^6*e - 2*(7*B*b^3*c^2 - 9*A*b^2*c^3)*d^5*e^2)*x^2 + (2*(B
*b^2*c^3 - 2*A*b*c^4)*d^7 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^6*e)*x)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqr
t(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 3*((5*A*b^4*c*e^6 + 2*(B*b*c^4 - 2*A*c^5)*d^4*e^2 - (6*B*b^2*
c^3 - 7*A*b*c^4)*d^3*e^3 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^2*e^4 - (2*B*b^4*c + 11*A*b^3*c^2)*d*e^5)*x^4 + (5*A*
b^5*e^6 + 4*(B*b*c^4 - 2*A*c^5)*d^5*e - 10*(B*b^2*c^3 - A*b*c^4)*d^4*e^2 + (6*B*b^3*c^2 + 13*A*b^2*c^3)*d^3*e^
3 + (2*B*b^4*c - 19*A*b^3*c^2)*d^2*e^4 - (2*B*b^5 + A*b^4*c)*d*e^5)*x^3 + (10*A*b^5*d*e^5 + 2*(B*b*c^4 - 2*A*c
^5)*d^6 - (2*B*b^2*c^3 + A*b*c^4)*d^5*e - (6*B*b^3*c^2 - 17*A*b^2*c^3)*d^4*e^2 + 5*(2*B*b^4*c - A*b^3*c^2)*d^3
*e^3 - (4*B*b^5 + 17*A*b^4*c)*d^2*e^4)*x^2 + (5*A*b^5*d^2*e^4 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^6 - (6*B*b^3*c^2 -
7*A*b^2*c^3)*d^5*e + 3*(2*B*b^4*c + A*b^3*c^2)*d^4*e^2 - (2*B*b^5 + 11*A*b^4*c)*d^3*e^3)*x)*sqrt(d)*log((e*x
+ 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(3*A*b^2*c^3*d^6 - 9*A*b^3*c^2*d^5*e + 9*A*b^4*c*d^4*e^2 - 3*A*b^5*d^3
*e^3 - 3*(5*A*b^4*c*d*e^5 + (B*b^2*c^3 - 2*A*b*c^4)*d^4*e^2 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^3*e^3 - (2*B*b^4*c
+ 11*A*b^3*c^2)*d^2*e^4)*x^3 - (15*A*b^5*d*e^5 + 6*(B*b^2*c^3 - 2*A*b*c^4)*d^5*e + 5*(4*B*b^3*c^2 + 3*A*b^2*c
^3)*d^4*e^2 + 5*(2*B*b^4*c - 7*A*b^3*c^2)*d^3*e^3 - (6*B*b^5 + 13*A*b^4*c)*d^2*e^4)*x^2 - (3*A*b^2*c^3*d^5*e +
20*A*b^5*d^2*e^4 + 3*(B*b^2*c^3 - 2*A*b*c^4)*d^6 + (20*B*b^4*c + 9*A*b^3*c^2)*d^4*e^2 - (8*B*b^5 + 41*A*b^4*c
)*d^3*e^3)*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4*e^5)*x^4 +
(2*b^3*c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*d^9 - b
^4*c^3*d^8*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2*d^8*e + 3*b
^6*c*d^7*e^2 - b^7*d^6*e^3)*x), 1/6*(6*((5*A*b^4*c*e^6 + 2*(B*b*c^4 - 2*A*c^5)*d^4*e^2 - (6*B*b^2*c^3 - 7*A*b*
c^4)*d^3*e^3 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^2*e^4 - (2*B*b^4*c + 11*A*b^3*c^2)*d*e^5)*x^4 + (5*A*b^5*e^6 + 4*
(B*b*c^4 - 2*A*c^5)*d^5*e - 10*(B*b^2*c^3 - A*b*c^4)*d^4*e^2 + (6*B*b^3*c^2 + 13*A*b^2*c^3)*d^3*e^3 + (2*B*b^4
*c - 19*A*b^3*c^2)*d^2*e^4 - (2*B*b^5 + A*b^4*c)*d*e^5)*x^3 + (10*A*b^5*d*e^5 + 2*(B*b*c^4 - 2*A*c^5)*d^6 - (2
*B*b^2*c^3 + A*b*c^4)*d^5*e - (6*B*b^3*c^2 - 17*A*b^2*c^3)*d^4*e^2 + 5*(2*B*b^4*c - A*b^3*c^2)*d^3*e^3 - (4*B*
b^5 + 17*A*b^4*c)*d^2*e^4)*x^2 + (5*A*b^5*d^2*e^4 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^6 - (6*B*b^3*c^2 - 7*A*b^2*c^3
)*d^5*e + 3*(2*B*b^4*c + A*b^3*c^2)*d^4*e^2 - (2*B*b^5 + 11*A*b^4*c)*d^3*e^3)*x)*sqrt(-d)*arctan(sqrt(e*x + d)
*sqrt(-d)/d) - 3*((2*(B*b*c^4 - 2*A*c^5)*d^5*e^2 - (7*B*b^2*c^3 - 9*A*b*c^4)*d^4*e^3)*x^4 + (4*(B*b*c^4 - 2*A*
c^5)*d^6*e - 2*(6*B*b^2*c^3 - 7*A*b*c^4)*d^5*e^2 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^4*e^3)*x^3 + (2*(B*b*c^4 - 2*
A*c^5)*d^7 - (3*B*b^2*c^3 - A*b*c^4)*d^6*e - 2*(7*B*b^3*c^2 - 9*A*b^2*c^3)*d^5*e^2)*x^2 + (2*(B*b^2*c^3 - 2*A*
b*c^4)*d^7 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^6*e)*x)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e
)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 2*(3*A*b^2*c^3*d^6 - 9*A*b^3*c^2*d^5*e + 9*A*b^4*c*d^4*e^2 -
3*A*b^5*d^3*e^3 - 3*(5*A*b^4*c*d*e^5 + (B*b^2*c^3 - 2*A*b*c^4)*d^4*e^2 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^3*e^3
- (2*B*b^4*c + 11*A*b^3*c^2)*d^2*e^4)*x^3 - (15*A*b^5*d*e^5 + 6*(B*b^2*c^3 - 2*A*b*c^4)*d^5*e + 5*(4*B*b^3*c^2
+ 3*A*b^2*c^3)*d^4*e^2 + 5*(2*B*b^4*c - 7*A*b^3*c^2)*d^3*e^3 - (6*B*b^5 + 13*A*b^4*c)*d^2*e^4)*x^2 - (3*A*b^2
*c^3*d^5*e + 20*A*b^5*d^2*e^4 + 3*(B*b^2*c^3 - 2*A*b*c^4)*d^6 + (20*B*b^4*c + 9*A*b^3*c^2)*d^4*e^2 - (8*B*b^5
+ 41*A*b^4*c)*d^3*e^3)*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4
*e^5)*x^4 + (2*b^3*c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3
*c^4*d^9 - b^4*c^3*d^8*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2
*d^8*e + 3*b^6*c*d^7*e^2 - b^7*d^6*e^3)*x), 1/3*(3*((2*(B*b*c^4 - 2*A*c^5)*d^5*e^2 - (7*B*b^2*c^3 - 9*A*b*c^4)
*d^4*e^3)*x^4 + (4*(B*b*c^4 - 2*A*c^5)*d^6*e - 2*(6*B*b^2*c^3 - 7*A*b*c^4)*d^5*e^2 - (7*B*b^3*c^2 - 9*A*b^2*c^
3)*d^4*e^3)*x^3 + (2*(B*b*c^4 - 2*A*c^5)*d^7 - (3*B*b^2*c^3 - A*b*c^4)*d^6*e - 2*(7*B*b^3*c^2 - 9*A*b^2*c^3)*d
^5*e^2)*x^2 + (2*(B*b^2*c^3 - 2*A*b*c^4)*d^7 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^6*e)*x)*sqrt(-c/(c*d - b*e))*arct
an(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + 3*((5*A*b^4*c*e^6 + 2*(B*b*c^4 - 2*A*c^5)*
d^4*e^2 - (6*B*b^2*c^3 - 7*A*b*c^4)*d^3*e^3 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^2*e^4 - (2*B*b^4*c + 11*A*b^3*c^2)
*d*e^5)*x^4 + (5*A*b^5*e^6 + 4*(B*b*c^4 - 2*A*c^5)*d^5*e - 10*(B*b^2*c^3 - A*b*c^4)*d^4*e^2 + (6*B*b^3*c^2 + 1
3*A*b^2*c^3)*d^3*e^3 + (2*B*b^4*c - 19*A*b^3*c^2)*d^2*e^4 - (2*B*b^5 + A*b^4*c)*d*e^5)*x^3 + (10*A*b^5*d*e^5 +
2*(B*b*c^4 - 2*A*c^5)*d^6 - (2*B*b^2*c^3 + A*b*c^4)*d^5*e - (6*B*b^3*c^2 - 17*A*b^2*c^3)*d^4*e^2 + 5*(2*B*b^4
*c - A*b^3*c^2)*d^3*e^3 - (4*B*b^5 + 17*A*b^4*c)*d^2*e^4)*x^2 + (5*A*b^5*d^2*e^4 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d
^6 - (6*B*b^3*c^2 - 7*A*b^2*c^3)*d^5*e + 3*(2*B*b^4*c + A*b^3*c^2)*d^4*e^2 - (2*B*b^5 + 11*A*b^4*c)*d^3*e^3)*x
)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (3*A*b^2*c^3*d^6 - 9*A*b^3*c^2*d^5*e + 9*A*b^4*c*d^4*e^2 - 3*A*b
^5*d^3*e^3 - 3*(5*A*b^4*c*d*e^5 + (B*b^2*c^3 - 2*A*b*c^4)*d^4*e^2 + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^3*e^3 - (2*B
*b^4*c + 11*A*b^3*c^2)*d^2*e^4)*x^3 - (15*A*b^5*d*e^5 + 6*(B*b^2*c^3 - 2*A*b*c^4)*d^5*e + 5*(4*B*b^3*c^2 + 3*A
*b^2*c^3)*d^4*e^2 + 5*(2*B*b^4*c - 7*A*b^3*c^2)*d^3*e^3 - (6*B*b^5 + 13*A*b^4*c)*d^2*e^4)*x^2 - (3*A*b^2*c^3*d
^5*e + 20*A*b^5*d^2*e^4 + 3*(B*b^2*c^3 - 2*A*b*c^4)*d^6 + (20*B*b^4*c + 9*A*b^3*c^2)*d^4*e^2 - (8*B*b^5 + 41*A
*b^4*c)*d^3*e^3)*x)*sqrt(e*x + d))/((b^3*c^4*d^7*e^2 - 3*b^4*c^3*d^6*e^3 + 3*b^5*c^2*d^5*e^4 - b^6*c*d^4*e^5)*
x^4 + (2*b^3*c^4*d^8*e - 5*b^4*c^3*d^7*e^2 + 3*b^5*c^2*d^6*e^3 + b^6*c*d^5*e^4 - b^7*d^4*e^5)*x^3 + (b^3*c^4*d
^9 - b^4*c^3*d^8*e - 3*b^5*c^2*d^7*e^2 + 5*b^6*c*d^6*e^3 - 2*b^7*d^5*e^4)*x^2 + (b^4*c^3*d^9 - 3*b^5*c^2*d^8*e
+ 3*b^6*c*d^7*e^2 - b^7*d^6*e^3)*x)]
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giac [A] time = 0.32, size = 603, normalized size = 1.75 \begin {gather*} -\frac {{\left (2 \, B b c^{4} d - 4 \, A c^{5} d - 7 \, B b^{2} c^{3} e + 9 \, A b c^{4} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}\right )} \sqrt {-c^{2} d + b c e}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c^{3} d^{3} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{4} d^{3} e - \sqrt {x e + d} B b c^{3} d^{4} e + 2 \, \sqrt {x e + d} A c^{4} d^{4} e + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{3} d^{2} e^{2} - 4 \, \sqrt {x e + d} A b c^{3} d^{3} e^{2} - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{2} d e^{3} + 6 \, \sqrt {x e + d} A b^{2} c^{2} d^{2} e^{3} + {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c e^{4} - 4 \, \sqrt {x e + d} A b^{3} c d e^{4} + \sqrt {x e + d} A b^{4} e^{5}}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}} + \frac {2 \, {\left (9 \, {\left (x e + d\right )} B c d^{2} e^{2} + B c d^{3} e^{2} - 3 \, {\left (x e + d\right )} B b d e^{3} - 12 \, {\left (x e + d\right )} A c d e^{3} - B b d^{2} e^{3} - A c d^{2} e^{3} + 6 \, {\left (x e + d\right )} A b e^{4} + A b d e^{4}\right )}}{3 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {{\left (2 \, B b d - 4 \, A c d - 5 \, A b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
-(2*B*b*c^4*d - 4*A*c^5*d - 7*B*b^2*c^3*e + 9*A*b*c^4*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^
3*d^3 - 3*b^4*c^2*d^2*e + 3*b^5*c*d*e^2 - b^6*e^3)*sqrt(-c^2*d + b*c*e)) + ((x*e + d)^(3/2)*B*b*c^3*d^3*e - 2*
(x*e + d)^(3/2)*A*c^4*d^3*e - sqrt(x*e + d)*B*b*c^3*d^4*e + 2*sqrt(x*e + d)*A*c^4*d^4*e + 3*(x*e + d)^(3/2)*A*
b*c^3*d^2*e^2 - 4*sqrt(x*e + d)*A*b*c^3*d^3*e^2 - 3*(x*e + d)^(3/2)*A*b^2*c^2*d*e^3 + 6*sqrt(x*e + d)*A*b^2*c^
2*d^2*e^3 + (x*e + d)^(3/2)*A*b^3*c*e^4 - 4*sqrt(x*e + d)*A*b^3*c*d*e^4 + sqrt(x*e + d)*A*b^4*e^5)/((b^2*c^3*d
^6 - 3*b^3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*d^3*e^3)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e
- b*d*e)) + 2/3*(9*(x*e + d)*B*c*d^2*e^2 + B*c*d^3*e^2 - 3*(x*e + d)*B*b*d*e^3 - 12*(x*e + d)*A*c*d*e^3 - B*b
*d^2*e^3 - A*c*d^2*e^3 + 6*(x*e + d)*A*b*e^4 + A*b*d*e^4)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^
3*e^3)*(x*e + d)^(3/2)) + (2*B*b*d - 4*A*c*d - 5*A*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^3)
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maple [A] time = 0.08, size = 535, normalized size = 1.56 \begin {gather*} \frac {9 A \,c^{4} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {4 A \,c^{5} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{3}}-\frac {7 B \,c^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b}+\frac {2 B \,c^{4} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {\sqrt {e x +d}\, A \,c^{4} e}{\left (b e -c d \right )^{3} \left (c e x +b e \right ) b^{2}}-\frac {\sqrt {e x +d}\, B \,c^{3} e}{\left (b e -c d \right )^{3} \left (c e x +b e \right ) b}-\frac {4 A b \,e^{4}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{3}}+\frac {8 A c \,e^{3}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{2}}+\frac {2 B b \,e^{3}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{2}}-\frac {6 B c \,e^{2}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d}-\frac {2 A \,e^{3}}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} d^{2}}+\frac {2 B \,e^{2}}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} d}+\frac {5 A e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {7}{2}}}+\frac {4 A c \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} d^{\frac {5}{2}}}-\frac {2 B \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {5}{2}}}-\frac {\sqrt {e x +d}\, A}{b^{2} d^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x)
[Out]
e*c^4/(b*e-c*d)^3/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A-e*c^3/(b*e-c*d)^3/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B+9*e*c^4/(b*e
-c*d)^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A-4*c^5/(b*e-c*d)^3/b^3/((b*e-c*d)
*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d-7*e*c^3/(b*e-c*d)^3/b/((b*e-c*d)*c)^(1/2)*arctan((e*
x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+2*c^4/(b*e-c*d)^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*
c)^(1/2)*c)*B*d-1/b^2/d^3*A*(e*x+d)^(1/2)/x+5*e/b^2/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+4/b^3/d^(5/2)*arc
tanh((e*x+d)^(1/2)/d^(1/2))*A*c-2/b^2/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B-2/3*e^3/(b*e-c*d)^2/d^2/(e*x+d)
^(3/2)*A+2/3*e^2/(b*e-c*d)^2/d/(e*x+d)^(3/2)*B-4*e^4/(b*e-c*d)^3/d^3/(e*x+d)^(1/2)*A*b+8*e^3/(b*e-c*d)^3/d^2/(
e*x+d)^(1/2)*A*c+2*e^3/(b*e-c*d)^3/d^2/(e*x+d)^(1/2)*B*b-6*e^2/(b*e-c*d)^3/d/(e*x+d)^(1/2)*B*c
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?
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mupad [B] time = 6.10, size = 18450, normalized size = 53.63
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((b*x + c*x^2)^2*(d + e*x)^(5/2)),x)
[Out]
atan((A^2*c^13*d^12*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*
b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c
^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5
- 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*32i - b^6*c^11*d^17*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b
^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d
*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 -
35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*2i + b^17*d^6*e^11*(-(16*A
^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^
6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e -
21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(
d + e*x)^(1/2)*1i + B^2*b^2*c^11*d^12*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*
c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)
/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 2
1*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*8i - b^8*c^9*d^15*e^2*(-(16*A^2*c^9*d^2 + 81*A^2*b
^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*
d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 +
35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*100i +
b^9*c^8*d^14*e^3*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^
3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7
*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 -
7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*285i - b^10*c^7*d^13*e^4*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^
2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^
8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^
3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*540i + b^11*c^6*d^12*e^
5*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B
^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6
*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6)
)^(3/2)*(d + e*x)^(1/2)*714i - b^12*c^5*d^11*e^6*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 +
49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b
^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*
d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*672i + b^13*c^4*d^10*e^7*(-(16*A^2*c^9*
d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e -
16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*
c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x
)^(1/2)*450i - b^14*c^3*d^9*e^8*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^
2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13
*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11
*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*210i + b^15*c^2*d^8*e^9*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c
^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2
- 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b
^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*65i + A^2*
b^12*c*e^12*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*
e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 +
7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^1
2*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*25i + b^7*c^10*d^16*e*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7
*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 9
2*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^
10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d + e*x)^(1/2)*21i - b^16*c*d^7*e^10*(-(16*A^2*
c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d
*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*
b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(3/2)*(d +
e*x)^(1/2)*12i - A^2*b*c^12*d^11*e*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^
5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(
b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*
b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*256i - A^2*b^11*c^2*d*e^11*(-(16*A^2*c^9*d^2 + 81*A^
2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c
^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2
+ 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*210
i - B^2*b^3*c^10*d^11*e*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*
A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b
^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2
*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*84i + B^2*b^12*c*d^2*e^10*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2
+ 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A
^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4
*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*4i - A*B*b*c^12*
d^12*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 2
8*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*
c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e
^6))^(1/2)*(d + e*x)^(1/2)*32i + A^2*b^2*c^11*d^10*e^2*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*
d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92
*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^1
0*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*810i - A^2*b^3*c^10*d^9*e^3*(-(16
*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*
c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e
- 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)
*(d + e*x)^(1/2)*1190i + A^2*b^4*c^9*d^8*e^4*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B
^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c
^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*
e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*475i + A^2*b^5*c^8*d^7*e^5*(-(16*A^2*c^9*d^
2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 1
6*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^
5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^
(1/2)*972i - A^2*b^6*c^7*d^6*e^6*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e
^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^1
3*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^1
1*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*1389i + A^2*b^7*c^6*d^5*e^7*(-(16*A^2*c^9*d^2 + 81*A^2*
b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8
*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 +
35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*180i
+ A^2*b^8*c^5*d^4*e^8*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*
B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6
*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e
^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*1170i - A^2*b^9*c^4*d^3*e^9*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2
+ 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A
^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4
*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*1360i + A^2*b^10
*c^3*d^2*e^10*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^
6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7
+ 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b
^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*741i + B^2*b^4*c^9*d^10*e^2*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2
*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8
*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3
- 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*365i - B^2*b^5*c^8*d^9*e
^3*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*
B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^
6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6
))^(1/2)*(d + e*x)^(1/2)*860i + B^2*b^6*c^7*d^8*e^4*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2
+ 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*
B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c
^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*1250i - B^2*b^7*c^6*d^7*e^5*(-(16*A^
2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6
*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 2
1*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d
+ e*x)^(1/2)*1232i + B^2*b^8*c^5*d^6*e^6*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*
b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*
d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4
+ 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*889i - B^2*b^9*c^4*d^5*e^7*(-(16*A^2*c^9*d^2 +
81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A
*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d
^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/
2)*480i + B^2*b^10*c^3*d^4*e^8*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2
- 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*
e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*
c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*180i - B^2*b^11*c^2*d^3*e^9*(-(16*A^2*c^9*d^2 + 81*A^2*b^
2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d
^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 3
5*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*40i - A
*B*b^12*c*d*e^11*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3
*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*
d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 -
7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*20i + A*B*b^2*c^11*d^11*e*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^
2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^
8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^
3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*296i - A*B*b^3*c^10*d^1
0*e^2*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 -
28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7
*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*
e^6))^(1/2)*(d + e*x)^(1/2)*1110i + A*B*b^4*c^9*d^9*e^3*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7
*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 9
2*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^
10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*2140i - A*B*b^5*c^8*d^8*e^4*(-(1
6*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3
*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e
- 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2
)*(d + e*x)^(1/2)*2100i + A*B*b^6*c^7*d^7*e^5*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*
B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*
c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3
*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*428i + A*B*b^7*c^6*d^6*e^6*(-(16*A^2*c^9*d
^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e -
16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c
^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)
^(1/2)*1554i - A*B*b^8*c^5*d^5*e^7*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5
*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b
^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b
^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*2280i + A*B*b^9*c^4*d^4*e^8*(-(16*A^2*c^9*d^2 + 81*A^
2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c
^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2
+ 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*168
0i - A*B*b^10*c^3*d^3*e^9*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 12
6*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 -
b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d
^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*740i + A*B*b^11*c^2*d^2*e^10*(-(16*A^2*c^9*d^2 + 81*A^2*b^2*c^
7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^8*d^2 -
72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e^2 + 35*b^
9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6))^(1/2)*(d + e*x)^(1/2)*184i)/(225*
A^3*b^7*c^4*e^10 + 420*A^3*c^11*d^7*e^3 + 3591*A^3*b^2*c^9*d^5*e^5 - 1113*A^3*b^3*c^8*d^4*e^6 - 2367*A^3*b^4*c
^7*d^3*e^7 + 2889*A^3*b^5*c^6*d^2*e^8 - 70*B^3*b^2*c^9*d^8*e^2 + 525*B^3*b^3*c^8*d^7*e^3 - 1260*B^3*b^4*c^7*d^
6*e^4 + 1148*B^3*b^5*c^6*d^5*e^5 - 644*B^3*b^6*c^5*d^4*e^6 + 204*B^3*b^7*c^4*d^3*e^7 - 28*B^3*b^8*c^3*d^2*e^8
- 175*A^2*B*b^8*c^3*e^10 - 280*A^2*B*c^11*d^8*e^2 - 2205*A^3*b*c^10*d^6*e^4 - 1315*A^3*b^6*c^5*d*e^9 - 1645*A*
B^2*b^2*c^9*d^7*e^3 + 2520*A*B^2*b^3*c^8*d^6*e^4 + 112*A*B^2*b^4*c^7*d^5*e^5 - 2296*A*B^2*b^5*c^6*d^4*e^6 + 21
36*A*B^2*b^6*c^5*d^3*e^7 - 872*A*B^2*b^7*c^4*d^2*e^8 + 665*A^2*B*b^2*c^9*d^6*e^4 - 5299*A^2*B*b^3*c^8*d^5*e^5
+ 5537*A^2*B*b^4*c^7*d^4*e^6 - 1717*A^2*B*b^5*c^6*d^3*e^7 - 901*A^2*B*b^6*c^5*d^2*e^8 + 280*A*B^2*b*c^10*d^8*e
^2 + 140*A*B^2*b^8*c^3*d*e^9 + 980*A^2*B*b*c^10*d^7*e^3 + 815*A^2*B*b^7*c^4*d*e^9))*(-(16*A^2*c^9*d^2 + 81*A^2
*b^2*c^7*e^2 + 4*B^2*b^2*c^7*d^2 + 49*B^2*b^4*c^5*e^2 - 126*A*B*b^3*c^6*e^2 - 28*B^2*b^3*c^6*d*e - 16*A*B*b*c^
8*d^2 - 72*A^2*b*c^8*d*e + 92*A*B*b^2*c^7*d*e)/(4*(b^13*e^7 - b^6*c^7*d^7 + 7*b^7*c^6*d^6*e - 21*b^8*c^5*d^5*e
^2 + 35*b^9*c^4*d^4*e^3 - 35*b^10*c^3*d^3*e^4 + 21*b^11*c^2*d^2*e^5 - 7*b^12*c*d*e^6)))^(1/2)*2i - log(32*A^3*
b^4*c^20*d^23*e^3 - ((((25*A^2*b^2*e^2)/4 + 4*A^2*c^2*d^2 + B^2*b^2*d^2 - 4*A*B*b*c*d^2 - 5*A*B*b^2*d*e + 10*A
^2*b*c*d*e)/(b^6*d^7))^(1/2)*((d + e*x)^(1/2)*(((25*A^2*b^2*e^2)/4 + 4*A^2*c^2*d^2 + B^2*b^2*d^2 - 4*A*B*b*c*d
^2 - 5*A*B*b^2*d*e + 10*A^2*b*c*d*e)/(b^6*d^7))^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 + 1800*b
^14*c^16*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 104104*b^1
8*c^12*d^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 + 88088*b
^22*c^8*d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 1080*b^26*
c^4*d^17*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18) + 8*A*b^10*c^18*d^28*e^3 - 112*A*b^11*c^17*d^27
*e^4 + 664*A*b^12*c^16*d^26*e^5 - 2080*A*b^13*c^15*d^25*e^6 + 2996*A*b^14*c^14*d^24*e^7 + 2528*A*b^15*c^13*d^2
3*e^8 - 23056*A*b^16*c^12*d^22*e^9 + 59312*A*b^17*c^11*d^21*e^10 - 95700*A*b^18*c^10*d^20*e^11 + 109648*A*b^19
*c^9*d^19*e^12 - 92840*A*b^20*c^8*d^18*e^13 + 58688*A*b^21*c^7*d^17*e^14 - 27476*A*b^22*c^6*d^16*e^15 + 9280*A
*b^23*c^5*d^15*e^16 - 2144*A*b^24*c^4*d^14*e^17 + 304*A*b^25*c^3*d^13*e^18 - 20*A*b^26*c^2*d^12*e^19 - 4*B*b^1
1*c^17*d^28*e^3 + 96*B*b^12*c^16*d^27*e^4 - 872*B*b^13*c^15*d^26*e^5 + 4440*B*b^14*c^14*d^25*e^6 - 14748*B*b^1
5*c^13*d^24*e^7 + 34496*B*b^16*c^12*d^23*e^8 - 59312*B*b^17*c^11*d^22*e^9 + 76824*B*b^18*c^10*d^21*e^10 - 7590
0*B*b^19*c^9*d^20*e^11 + 57376*B*b^20*c^8*d^19*e^12 - 33000*B*b^21*c^7*d^18*e^13 + 14216*B*b^22*c^6*d^17*e^14
- 4452*B*b^23*c^5*d^16*e^15 + 960*B*b^24*c^4*d^15*e^16 - 128*B*b^25*c^3*d^14*e^17 + 8*B*b^26*c^2*d^13*e^18) -
(d + e*x)^(1/2)*(64*A^2*b^6*c^20*d^26*e^2 - 832*A^2*b^7*c^19*d^25*e^3 + 4820*A^2*b^8*c^18*d^24*e^4 - 16240*A^2
*b^9*c^17*d^23*e^5 + 34490*A^2*b^10*c^16*d^22*e^6 - 45430*A^2*b^11*c^15*d^21*e^7 + 29414*A^2*b^12*c^14*d^20*e^
8 + 10670*A^2*b^13*c^13*d^19*e^9 - 39550*A^2*b^14*c^12*d^18*e^10 + 25730*A^2*b^15*c^11*d^17*e^11 + 19048*A^2*b
^16*c^10*d^16*e^12 - 53852*A^2*b^17*c^9*d^15*e^13 + 55510*A^2*b^18*c^8*d^14*e^14 - 35210*A^2*b^19*c^7*d^13*e^1
5 + 14830*A^2*b^20*c^6*d^12*e^16 - 4082*A^2*b^21*c^5*d^11*e^17 + 670*A^2*b^22*c^4*d^10*e^18 - 50*A^2*b^23*c^3*
d^9*e^19 + 16*B^2*b^8*c^18*d^26*e^2 - 248*B^2*b^9*c^17*d^25*e^3 + 1730*B^2*b^10*c^16*d^24*e^4 - 7210*B^2*b^11*
c^15*d^23*e^5 + 20160*B^2*b^12*c^14*d^22*e^6 - 40320*B^2*b^13*c^13*d^21*e^7 + 60116*B^2*b^14*c^12*d^20*e^8 - 6
8820*B^2*b^15*c^11*d^19*e^9 + 61800*B^2*b^16*c^10*d^18*e^10 - 44080*B^2*b^17*c^9*d^17*e^11 + 24962*B^2*b^18*c^
8*d^16*e^12 - 11018*B^2*b^19*c^7*d^15*e^13 + 3640*B^2*b^20*c^6*d^14*e^14 - 840*B^2*b^21*c^5*d^13*e^15 + 120*B^
2*b^22*c^4*d^12*e^16 - 8*B^2*b^23*c^3*d^11*e^17 - 64*A*B*b^7*c^19*d^26*e^2 + 912*A*B*b^8*c^18*d^25*e^3 - 5820*
A*B*b^9*c^17*d^24*e^4 + 21940*A*B*b^10*c^16*d^23*e^5 - 54040*A*B*b^11*c^15*d^22*e^6 + 89880*A*B*b^12*c^14*d^21
*e^7 - 97664*A*B*b^13*c^13*d^20*e^8 + 54080*A*B*b^14*c^12*d^19*e^9 + 23400*A*B*b^15*c^11*d^18*e^10 - 86480*A*B
*b^16*c^10*d^17*e^11 + 101652*A*B*b^17*c^9*d^16*e^12 - 76188*A*B*b^18*c^8*d^15*e^13 + 40040*A*B*b^19*c^7*d^14*
e^14 - 14840*A*B*b^20*c^6*d^13*e^15 + 3720*A*B*b^21*c^5*d^12*e^16 - 568*A*B*b^22*c^4*d^11*e^17 + 40*A*B*b^23*c
^3*d^10*e^18))*(((25*A^2*b^2*e^2)/4 + 4*A^2*c^2*d^2 + B^2*b^2*d^2 - 4*A*B*b*c*d^2 - 5*A*B*b^2*d*e + 10*A^2*b*c
*d*e)/(b^6*d^7))^(1/2) - 368*A^3*b^5*c^19*d^22*e^4 + 2006*A^3*b^6*c^18*d^21*e^5 - 6895*A^3*b^7*c^17*d^20*e^6 +
16250*A^3*b^8*c^16*d^19*e^7 - 25764*A^3*b^9*c^15*d^18*e^8 + 22851*A^3*b^10*c^14*d^17*e^9 + 2958*A^3*b^11*c^13
*d^16*e^10 - 41520*A^3*b^12*c^12*d^15*e^11 + 64900*A^3*b^13*c^11*d^14*e^12 - 57568*A^3*b^14*c^10*d^13*e^13 + 3
2617*A^3*b^15*c^9*d^12*e^14 - 11714*A^3*b^16*c^8*d^11*e^15 + 2440*A^3*b^17*c^7*d^10*e^16 - 225*A^3*b^18*c^6*d^
9*e^17 - 4*B^3*b^7*c^17*d^23*e^3 + 26*B^3*b^8*c^16*d^22*e^4 + 38*B^3*b^9*c^15*d^21*e^5 - 880*B^3*b^10*c^14*d^2
0*e^6 + 3900*B^3*b^11*c^13*d^19*e^7 - 9492*B^3*b^12*c^12*d^18*e^8 + 14868*B^3*b^13*c^11*d^17*e^9 - 15816*B^3*b
^14*c^10*d^16*e^10 + 11580*B^3*b^15*c^9*d^15*e^11 - 5750*B^3*b^16*c^8*d^14*e^12 + 1846*B^3*b^17*c^7*d^13*e^13
- 344*B^3*b^18*c^6*d^12*e^14 + 28*B^3*b^19*c^5*d^11*e^15 + 24*A*B^2*b^6*c^18*d^23*e^3 - 196*A*B^2*b^7*c^17*d^2
2*e^4 + 487*A*B^2*b^8*c^16*d^21*e^5 + 165*A*B^2*b^9*c^15*d^20*e^6 - 2800*A*B^2*b^10*c^14*d^19*e^7 + 3552*A*B^2
*b^11*c^13*d^18*e^8 + 5922*A*B^2*b^12*c^12*d^17*e^9 - 25434*A*B^2*b^13*c^11*d^16*e^10 + 39900*A*B^2*b^14*c^10*
d^15*e^11 - 36600*A*B^2*b^15*c^9*d^14*e^12 + 21199*A*B^2*b^16*c^8*d^13*e^13 - 7651*A*B^2*b^17*c^7*d^12*e^14 +
1572*A*B^2*b^18*c^6*d^11*e^15 - 140*A*B^2*b^19*c^5*d^10*e^16 - 48*A^2*B*b^5*c^19*d^23*e^3 + 472*A^2*B*b^6*c^18
*d^22*e^4 - 2129*A^2*B*b^7*c^17*d^21*e^5 + 6450*A^2*B*b^8*c^16*d^20*e^6 - 16250*A^2*B*b^9*c^15*d^19*e^7 + 3524
6*A^2*B*b^10*c^14*d^18*e^8 - 59679*A^2*B*b^11*c^13*d^17*e^9 + 71028*A^2*B*b^12*c^12*d^16*e^10 - 52860*A^2*B*b^
13*c^11*d^15*e^11 + 16500*A^2*B*b^14*c^10*d^14*e^12 + 9377*A^2*B*b^15*c^9*d^13*e^13 - 13318*A^2*B*b^16*c^8*d^1
2*e^14 + 6726*A^2*B*b^17*c^7*d^11*e^15 - 1690*A^2*B*b^18*c^6*d^10*e^16 + 175*A^2*B*b^19*c^5*d^9*e^17)*(((25*A^
2*b^2*e^2)/4 + 4*A^2*c^2*d^2 + B^2*b^2*d^2 - 4*A*B*b*c*d^2 - 5*A*B*b^2*d*e + 10*A^2*b*c*d*e)/(b^6*d^7))^(1/2)
+ log((((25*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*d^2 - 16*A*B*b*c*d^2 - 20*A*B*b^2*d*e + 40*A^2*b*c*d*e)/(
4*b^6*d^7))^(1/2)*((d + e*x)^(1/2)*((25*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*d^2 - 16*A*B*b*c*d^2 - 20*A*B
*b^2*d*e + 40*A^2*b*c*d*e)/(4*b^6*d^7))^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 + 1800*b^14*c^16
*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 104104*b^18*c^12*d
^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 + 88088*b^22*c^8*
d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 1080*b^26*c^4*d^17
*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18) - 8*A*b^10*c^18*d^28*e^3 + 112*A*b^11*c^17*d^27*e^4 - 6
64*A*b^12*c^16*d^26*e^5 + 2080*A*b^13*c^15*d^25*e^6 - 2996*A*b^14*c^14*d^24*e^7 - 2528*A*b^15*c^13*d^23*e^8 +
23056*A*b^16*c^12*d^22*e^9 - 59312*A*b^17*c^11*d^21*e^10 + 95700*A*b^18*c^10*d^20*e^11 - 109648*A*b^19*c^9*d^1
9*e^12 + 92840*A*b^20*c^8*d^18*e^13 - 58688*A*b^21*c^7*d^17*e^14 + 27476*A*b^22*c^6*d^16*e^15 - 9280*A*b^23*c^
5*d^15*e^16 + 2144*A*b^24*c^4*d^14*e^17 - 304*A*b^25*c^3*d^13*e^18 + 20*A*b^26*c^2*d^12*e^19 + 4*B*b^11*c^17*d
^28*e^3 - 96*B*b^12*c^16*d^27*e^4 + 872*B*b^13*c^15*d^26*e^5 - 4440*B*b^14*c^14*d^25*e^6 + 14748*B*b^15*c^13*d
^24*e^7 - 34496*B*b^16*c^12*d^23*e^8 + 59312*B*b^17*c^11*d^22*e^9 - 76824*B*b^18*c^10*d^21*e^10 + 75900*B*b^19
*c^9*d^20*e^11 - 57376*B*b^20*c^8*d^19*e^12 + 33000*B*b^21*c^7*d^18*e^13 - 14216*B*b^22*c^6*d^17*e^14 + 4452*B
*b^23*c^5*d^16*e^15 - 960*B*b^24*c^4*d^15*e^16 + 128*B*b^25*c^3*d^14*e^17 - 8*B*b^26*c^2*d^13*e^18) - (d + e*x
)^(1/2)*(64*A^2*b^6*c^20*d^26*e^2 - 832*A^2*b^7*c^19*d^25*e^3 + 4820*A^2*b^8*c^18*d^24*e^4 - 16240*A^2*b^9*c^1
7*d^23*e^5 + 34490*A^2*b^10*c^16*d^22*e^6 - 45430*A^2*b^11*c^15*d^21*e^7 + 29414*A^2*b^12*c^14*d^20*e^8 + 1067
0*A^2*b^13*c^13*d^19*e^9 - 39550*A^2*b^14*c^12*d^18*e^10 + 25730*A^2*b^15*c^11*d^17*e^11 + 19048*A^2*b^16*c^10
*d^16*e^12 - 53852*A^2*b^17*c^9*d^15*e^13 + 55510*A^2*b^18*c^8*d^14*e^14 - 35210*A^2*b^19*c^7*d^13*e^15 + 1483
0*A^2*b^20*c^6*d^12*e^16 - 4082*A^2*b^21*c^5*d^11*e^17 + 670*A^2*b^22*c^4*d^10*e^18 - 50*A^2*b^23*c^3*d^9*e^19
+ 16*B^2*b^8*c^18*d^26*e^2 - 248*B^2*b^9*c^17*d^25*e^3 + 1730*B^2*b^10*c^16*d^24*e^4 - 7210*B^2*b^11*c^15*d^2
3*e^5 + 20160*B^2*b^12*c^14*d^22*e^6 - 40320*B^2*b^13*c^13*d^21*e^7 + 60116*B^2*b^14*c^12*d^20*e^8 - 68820*B^2
*b^15*c^11*d^19*e^9 + 61800*B^2*b^16*c^10*d^18*e^10 - 44080*B^2*b^17*c^9*d^17*e^11 + 24962*B^2*b^18*c^8*d^16*e
^12 - 11018*B^2*b^19*c^7*d^15*e^13 + 3640*B^2*b^20*c^6*d^14*e^14 - 840*B^2*b^21*c^5*d^13*e^15 + 120*B^2*b^22*c
^4*d^12*e^16 - 8*B^2*b^23*c^3*d^11*e^17 - 64*A*B*b^7*c^19*d^26*e^2 + 912*A*B*b^8*c^18*d^25*e^3 - 5820*A*B*b^9*
c^17*d^24*e^4 + 21940*A*B*b^10*c^16*d^23*e^5 - 54040*A*B*b^11*c^15*d^22*e^6 + 89880*A*B*b^12*c^14*d^21*e^7 - 9
7664*A*B*b^13*c^13*d^20*e^8 + 54080*A*B*b^14*c^12*d^19*e^9 + 23400*A*B*b^15*c^11*d^18*e^10 - 86480*A*B*b^16*c^
10*d^17*e^11 + 101652*A*B*b^17*c^9*d^16*e^12 - 76188*A*B*b^18*c^8*d^15*e^13 + 40040*A*B*b^19*c^7*d^14*e^14 - 1
4840*A*B*b^20*c^6*d^13*e^15 + 3720*A*B*b^21*c^5*d^12*e^16 - 568*A*B*b^22*c^4*d^11*e^17 + 40*A*B*b^23*c^3*d^10*
e^18))*((25*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*d^2 - 16*A*B*b*c*d^2 - 20*A*B*b^2*d*e + 40*A^2*b*c*d*e)/(
4*b^6*d^7))^(1/2) + 32*A^3*b^4*c^20*d^23*e^3 - 368*A^3*b^5*c^19*d^22*e^4 + 2006*A^3*b^6*c^18*d^21*e^5 - 6895*A
^3*b^7*c^17*d^20*e^6 + 16250*A^3*b^8*c^16*d^19*e^7 - 25764*A^3*b^9*c^15*d^18*e^8 + 22851*A^3*b^10*c^14*d^17*e^
9 + 2958*A^3*b^11*c^13*d^16*e^10 - 41520*A^3*b^12*c^12*d^15*e^11 + 64900*A^3*b^13*c^11*d^14*e^12 - 57568*A^3*b
^14*c^10*d^13*e^13 + 32617*A^3*b^15*c^9*d^12*e^14 - 11714*A^3*b^16*c^8*d^11*e^15 + 2440*A^3*b^17*c^7*d^10*e^16
- 225*A^3*b^18*c^6*d^9*e^17 - 4*B^3*b^7*c^17*d^23*e^3 + 26*B^3*b^8*c^16*d^22*e^4 + 38*B^3*b^9*c^15*d^21*e^5 -
880*B^3*b^10*c^14*d^20*e^6 + 3900*B^3*b^11*c^13*d^19*e^7 - 9492*B^3*b^12*c^12*d^18*e^8 + 14868*B^3*b^13*c^11*
d^17*e^9 - 15816*B^3*b^14*c^10*d^16*e^10 + 11580*B^3*b^15*c^9*d^15*e^11 - 5750*B^3*b^16*c^8*d^14*e^12 + 1846*B
^3*b^17*c^7*d^13*e^13 - 344*B^3*b^18*c^6*d^12*e^14 + 28*B^3*b^19*c^5*d^11*e^15 + 24*A*B^2*b^6*c^18*d^23*e^3 -
196*A*B^2*b^7*c^17*d^22*e^4 + 487*A*B^2*b^8*c^16*d^21*e^5 + 165*A*B^2*b^9*c^15*d^20*e^6 - 2800*A*B^2*b^10*c^14
*d^19*e^7 + 3552*A*B^2*b^11*c^13*d^18*e^8 + 5922*A*B^2*b^12*c^12*d^17*e^9 - 25434*A*B^2*b^13*c^11*d^16*e^10 +
39900*A*B^2*b^14*c^10*d^15*e^11 - 36600*A*B^2*b^15*c^9*d^14*e^12 + 21199*A*B^2*b^16*c^8*d^13*e^13 - 7651*A*B^2
*b^17*c^7*d^12*e^14 + 1572*A*B^2*b^18*c^6*d^11*e^15 - 140*A*B^2*b^19*c^5*d^10*e^16 - 48*A^2*B*b^5*c^19*d^23*e^
3 + 472*A^2*B*b^6*c^18*d^22*e^4 - 2129*A^2*B*b^7*c^17*d^21*e^5 + 6450*A^2*B*b^8*c^16*d^20*e^6 - 16250*A^2*B*b^
9*c^15*d^19*e^7 + 35246*A^2*B*b^10*c^14*d^18*e^8 - 59679*A^2*B*b^11*c^13*d^17*e^9 + 71028*A^2*B*b^12*c^12*d^16
*e^10 - 52860*A^2*B*b^13*c^11*d^15*e^11 + 16500*A^2*B*b^14*c^10*d^14*e^12 + 9377*A^2*B*b^15*c^9*d^13*e^13 - 13
318*A^2*B*b^16*c^8*d^12*e^14 + 6726*A^2*B*b^17*c^7*d^11*e^15 - 1690*A^2*B*b^18*c^6*d^10*e^16 + 175*A^2*B*b^19*
c^5*d^9*e^17)*((25*A^2*b^2*e^2 + 16*A^2*c^2*d^2 + 4*B^2*b^2*d^2 - 16*A*B*b*c*d^2 - 20*A*B*b^2*d*e + 40*A^2*b*c
*d*e)/(4*b^6*d^7))^(1/2) - ((2*(A*e^3 - B*d*e^2))/(3*(c*d^2 - b*d*e)) - (2*(d + e*x)*(5*A*b*e^4 - 10*A*c*d*e^3
- 2*B*b*d*e^3 + 7*B*c*d^2*e^2))/(3*(c*d^2 - b*d*e)^2) + ((d + e*x)^2*(6*B*b^4*d*e^4 - 6*A*c^4*d^4*e - 15*A*b^
4*e^5 + 12*A*b*c^3*d^3*e^2 - 28*B*b^3*c*d^2*e^3 - 64*A*b^2*c^2*d^2*e^3 + 34*B*b^2*c^2*d^3*e^2 + 58*A*b^3*c*d*e
^4 + 3*B*b*c^3*d^4*e))/(3*b^2*(c*d^2 - b*d*e)^3) - ((d + e*x)^3*(5*A*b^3*c*e^4 - 2*A*c^4*d^3*e + 3*A*b*c^3*d^2
*e^2 - 11*A*b^2*c^2*d*e^3 + 6*B*b^2*c^2*d^2*e^2 + B*b*c^3*d^3*e - 2*B*b^3*c*d*e^3))/(b^2*(c*d^2 - b*d*e)^3))/(
c*(d + e*x)^(7/2) + (c*d^2 - b*d*e)*(d + e*x)^(3/2) + (b*e - 2*c*d)*(d + e*x)^(5/2))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**(5/2)/(c*x**2+b*x)**2,x)
[Out]
Timed out
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