3.11.91 \(\int \frac {A+B x}{(d+e x)^{7/2} (b x+c x^2)} \, dx\)
Optimal. Leaf size=225 \[ \frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {2 c^{5/2} (b B-A c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}+\frac {2 (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}} \]
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Rubi [A] time = 0.45, antiderivative size = 225, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used =
{828, 826, 1166, 208} \begin {gather*} \frac {2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {2 c^{5/2} (b B-A c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{7/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}+\frac {2 (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(7/2)*(b*x + c*x^2)),x]
[Out]
(2*(B*d - A*e))/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) + (2*(B*c*d^2 - A*e*(2*c*d - b*e)))/(3*d^2*(c*d - b*e)^2*(d
+ e*x)^(3/2)) + (2*(B*c^2*d^3 - A*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2)))/(d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (2
*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(7/2)) - (2*c^(5/2)*(b*B - A*c)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c
*d - b*e]])/(b*(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 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)^{7/2} \left (b x+c x^2\right )} \, dx &=\frac {2 (B d-A e)}{5 d (c d-b e) (d+e x)^{5/2}}+\frac {\int \frac {A (c d-b e)+c (B d-A e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{d (c d-b e)}\\ &=\frac {2 (B d-A e)}{5 d (c d-b e) (d+e x)^{5/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {\int \frac {A (c d-b e)^2+c \left (B c d^2-A e (2 c d-b e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{d^2 (c d-b e)^2}\\ &=\frac {2 (B d-A e)}{5 d (c d-b e) (d+e x)^{5/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\int \frac {A (c d-b e)^3+c \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{d^3 (c d-b e)^3}\\ &=\frac {2 (B d-A e)}{5 d (c d-b e) (d+e x)^{5/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {2 \operatorname {Subst}\left (\int \frac {A e (c d-b e)^3-c d \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )+c \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\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 )}{d^3 (c d-b e)^3}\\ &=\frac {2 (B d-A e)}{5 d (c d-b e) (d+e x)^{5/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {(2 A c) \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 d^3}+\frac {\left (2 c^3 (b B-A c)\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 (c d-b e)^3}\\ &=\frac {2 (B d-A e)}{5 d (c d-b e) (d+e x)^{5/2}}+\frac {2 \left (B c d^2-A e (2 c d-b e)\right )}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}}-\frac {2 c^{5/2} (b B-A c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 91, normalized size = 0.40 \begin {gather*} \frac {2 \left (d (b B-A c) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {c (d+e x)}{c d-b e}\right )+A (c d-b e) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {e x}{d}+1\right )\right )}{5 b d (d+e x)^{5/2} (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(7/2)*(b*x + c*x^2)),x]
[Out]
(2*((b*B - A*c)*d*Hypergeometric2F1[-5/2, 1, -3/2, (c*(d + e*x))/(c*d - b*e)] + A*(c*d - b*e)*Hypergeometric2F
1[-5/2, 1, -3/2, 1 + (e*x)/d]))/(5*b*d*(c*d - b*e)*(d + e*x)^(5/2))
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IntegrateAlgebraic [A] time = 0.66, size = 329, normalized size = 1.46 \begin {gather*} \frac {2 \left (-3 A b^2 d^2 e^3-5 A b^2 d e^3 (d+e x)-15 A b^2 e^3 (d+e x)^2+6 A b c d^3 e^2+15 A b c d^2 e^2 (d+e x)+45 A b c d e^2 (d+e x)^2-3 A c^2 d^4 e-10 A c^2 d^3 e (d+e x)-45 A c^2 d^2 e (d+e x)^2+3 b^2 B d^3 e^2-6 b B c d^4 e-5 b B c d^3 e (d+e x)+3 B c^2 d^5+5 B c^2 d^4 (d+e x)+15 B c^2 d^3 (d+e x)^2\right )}{15 d^3 (d+e x)^{5/2} (c d-b e)^3}-\frac {2 \left (A c^{7/2}-b B c^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b (b e-c d)^{7/2}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^(7/2)*(b*x + c*x^2)),x]
[Out]
(2*(3*B*c^2*d^5 - 6*b*B*c*d^4*e - 3*A*c^2*d^4*e + 3*b^2*B*d^3*e^2 + 6*A*b*c*d^3*e^2 - 3*A*b^2*d^2*e^3 + 5*B*c^
2*d^4*(d + e*x) - 5*b*B*c*d^3*e*(d + e*x) - 10*A*c^2*d^3*e*(d + e*x) + 15*A*b*c*d^2*e^2*(d + e*x) - 5*A*b^2*d*
e^3*(d + e*x) + 15*B*c^2*d^3*(d + e*x)^2 - 45*A*c^2*d^2*e*(d + e*x)^2 + 45*A*b*c*d*e^2*(d + e*x)^2 - 15*A*b^2*
e^3*(d + e*x)^2))/(15*d^3*(c*d - b*e)^3*(d + e*x)^(5/2)) - (2*(-(b*B*c^(5/2)) + A*c^(7/2))*ArcTan[(Sqrt[c]*Sqr
t[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(b*(-(c*d) + b*e)^(7/2)) - (2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(
b*d^(7/2))
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fricas [B] time = 7.48, size = 3081, normalized size = 13.69
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x),x, algorithm="fricas")
[Out]
[1/15*(15*((B*b*c^2 - A*c^3)*d^4*e^3*x^3 + 3*(B*b*c^2 - A*c^3)*d^5*e^2*x^2 + 3*(B*b*c^2 - A*c^3)*d^6*e*x + (B*
b*c^2 - A*c^3)*d^7)*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)) + 15*(A*c^3*d^6 - 3*A*b*c^2*d^5*e + 3*A*b^2*c*d^4*e^2 - A*b^3*d^3*e^3 + (A*c^3*d^3*e^3 - 3*A*
b*c^2*d^2*e^4 + 3*A*b^2*c*d*e^5 - A*b^3*e^6)*x^3 + 3*(A*c^3*d^4*e^2 - 3*A*b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e^4 -
A*b^3*d*e^5)*x^2 + 3*(A*c^3*d^5*e - 3*A*b*c^2*d^4*e^2 + 3*A*b^2*c*d^3*e^3 - A*b^3*d^2*e^4)*x)*sqrt(d)*log((e*x
- 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(23*B*b*c^2*d^6 - 23*A*b^3*d^3*e^3 - (11*B*b^2*c + 58*A*b*c^2)*d^5*e
+ 3*(B*b^3 + 22*A*b^2*c)*d^4*e^2 + 15*(B*b*c^2*d^4*e^2 - 3*A*b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e^4 - A*b^3*d*e^5)*
x^2 + 5*(7*B*b*c^2*d^5*e + 21*A*b^2*c*d^3*e^3 - 7*A*b^3*d^2*e^4 - (B*b^2*c + 20*A*b*c^2)*d^4*e^2)*x)*sqrt(e*x
+ d))/(b*c^3*d^10 - 3*b^2*c^2*d^9*e + 3*b^3*c*d^8*e^2 - b^4*d^7*e^3 + (b*c^3*d^7*e^3 - 3*b^2*c^2*d^6*e^4 + 3*b
^3*c*d^5*e^5 - b^4*d^4*e^6)*x^3 + 3*(b*c^3*d^8*e^2 - 3*b^2*c^2*d^7*e^3 + 3*b^3*c*d^6*e^4 - b^4*d^5*e^5)*x^2 +
3*(b*c^3*d^9*e - 3*b^2*c^2*d^8*e^2 + 3*b^3*c*d^7*e^3 - b^4*d^6*e^4)*x), -1/15*(30*((B*b*c^2 - A*c^3)*d^4*e^3*x
^3 + 3*(B*b*c^2 - A*c^3)*d^5*e^2*x^2 + 3*(B*b*c^2 - A*c^3)*d^6*e*x + (B*b*c^2 - A*c^3)*d^7)*sqrt(-c/(c*d - b*e
))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 15*(A*c^3*d^6 - 3*A*b*c^2*d^5*e + 3
*A*b^2*c*d^4*e^2 - A*b^3*d^3*e^3 + (A*c^3*d^3*e^3 - 3*A*b*c^2*d^2*e^4 + 3*A*b^2*c*d*e^5 - A*b^3*e^6)*x^3 + 3*(
A*c^3*d^4*e^2 - 3*A*b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e^4 - A*b^3*d*e^5)*x^2 + 3*(A*c^3*d^5*e - 3*A*b*c^2*d^4*e^2
+ 3*A*b^2*c*d^3*e^3 - A*b^3*d^2*e^4)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(23*B*b*c^2*d
^6 - 23*A*b^3*d^3*e^3 - (11*B*b^2*c + 58*A*b*c^2)*d^5*e + 3*(B*b^3 + 22*A*b^2*c)*d^4*e^2 + 15*(B*b*c^2*d^4*e^2
- 3*A*b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e^4 - A*b^3*d*e^5)*x^2 + 5*(7*B*b*c^2*d^5*e + 21*A*b^2*c*d^3*e^3 - 7*A*b^
3*d^2*e^4 - (B*b^2*c + 20*A*b*c^2)*d^4*e^2)*x)*sqrt(e*x + d))/(b*c^3*d^10 - 3*b^2*c^2*d^9*e + 3*b^3*c*d^8*e^2
- b^4*d^7*e^3 + (b*c^3*d^7*e^3 - 3*b^2*c^2*d^6*e^4 + 3*b^3*c*d^5*e^5 - b^4*d^4*e^6)*x^3 + 3*(b*c^3*d^8*e^2 - 3
*b^2*c^2*d^7*e^3 + 3*b^3*c*d^6*e^4 - b^4*d^5*e^5)*x^2 + 3*(b*c^3*d^9*e - 3*b^2*c^2*d^8*e^2 + 3*b^3*c*d^7*e^3 -
b^4*d^6*e^4)*x), 1/15*(30*(A*c^3*d^6 - 3*A*b*c^2*d^5*e + 3*A*b^2*c*d^4*e^2 - A*b^3*d^3*e^3 + (A*c^3*d^3*e^3 -
3*A*b*c^2*d^2*e^4 + 3*A*b^2*c*d*e^5 - A*b^3*e^6)*x^3 + 3*(A*c^3*d^4*e^2 - 3*A*b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e
^4 - A*b^3*d*e^5)*x^2 + 3*(A*c^3*d^5*e - 3*A*b*c^2*d^4*e^2 + 3*A*b^2*c*d^3*e^3 - A*b^3*d^2*e^4)*x)*sqrt(-d)*ar
ctan(sqrt(e*x + d)*sqrt(-d)/d) + 15*((B*b*c^2 - A*c^3)*d^4*e^3*x^3 + 3*(B*b*c^2 - A*c^3)*d^5*e^2*x^2 + 3*(B*b*
c^2 - A*c^3)*d^6*e*x + (B*b*c^2 - A*c^3)*d^7)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqr
t(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(23*B*b*c^2*d^6 - 23*A*b^3*d^3*e^3 - (11*B*b^2*c + 58*A*b*c^2)*
d^5*e + 3*(B*b^3 + 22*A*b^2*c)*d^4*e^2 + 15*(B*b*c^2*d^4*e^2 - 3*A*b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e^4 - A*b^3*d
*e^5)*x^2 + 5*(7*B*b*c^2*d^5*e + 21*A*b^2*c*d^3*e^3 - 7*A*b^3*d^2*e^4 - (B*b^2*c + 20*A*b*c^2)*d^4*e^2)*x)*sqr
t(e*x + d))/(b*c^3*d^10 - 3*b^2*c^2*d^9*e + 3*b^3*c*d^8*e^2 - b^4*d^7*e^3 + (b*c^3*d^7*e^3 - 3*b^2*c^2*d^6*e^4
+ 3*b^3*c*d^5*e^5 - b^4*d^4*e^6)*x^3 + 3*(b*c^3*d^8*e^2 - 3*b^2*c^2*d^7*e^3 + 3*b^3*c*d^6*e^4 - b^4*d^5*e^5)*
x^2 + 3*(b*c^3*d^9*e - 3*b^2*c^2*d^8*e^2 + 3*b^3*c*d^7*e^3 - b^4*d^6*e^4)*x), -2/15*(15*((B*b*c^2 - A*c^3)*d^4
*e^3*x^3 + 3*(B*b*c^2 - A*c^3)*d^5*e^2*x^2 + 3*(B*b*c^2 - A*c^3)*d^6*e*x + (B*b*c^2 - A*c^3)*d^7)*sqrt(-c/(c*d
- b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 15*(A*c^3*d^6 - 3*A*b*c^2*d^5
*e + 3*A*b^2*c*d^4*e^2 - A*b^3*d^3*e^3 + (A*c^3*d^3*e^3 - 3*A*b*c^2*d^2*e^4 + 3*A*b^2*c*d*e^5 - A*b^3*e^6)*x^3
+ 3*(A*c^3*d^4*e^2 - 3*A*b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e^4 - A*b^3*d*e^5)*x^2 + 3*(A*c^3*d^5*e - 3*A*b*c^2*d^
4*e^2 + 3*A*b^2*c*d^3*e^3 - A*b^3*d^2*e^4)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (23*B*b*c^2*d^6 - 23
*A*b^3*d^3*e^3 - (11*B*b^2*c + 58*A*b*c^2)*d^5*e + 3*(B*b^3 + 22*A*b^2*c)*d^4*e^2 + 15*(B*b*c^2*d^4*e^2 - 3*A*
b*c^2*d^3*e^3 + 3*A*b^2*c*d^2*e^4 - A*b^3*d*e^5)*x^2 + 5*(7*B*b*c^2*d^5*e + 21*A*b^2*c*d^3*e^3 - 7*A*b^3*d^2*e
^4 - (B*b^2*c + 20*A*b*c^2)*d^4*e^2)*x)*sqrt(e*x + d))/(b*c^3*d^10 - 3*b^2*c^2*d^9*e + 3*b^3*c*d^8*e^2 - b^4*d
^7*e^3 + (b*c^3*d^7*e^3 - 3*b^2*c^2*d^6*e^4 + 3*b^3*c*d^5*e^5 - b^4*d^4*e^6)*x^3 + 3*(b*c^3*d^8*e^2 - 3*b^2*c^
2*d^7*e^3 + 3*b^3*c*d^6*e^4 - b^4*d^5*e^5)*x^2 + 3*(b*c^3*d^9*e - 3*b^2*c^2*d^8*e^2 + 3*b^3*c*d^7*e^3 - b^4*d^
6*e^4)*x)]
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giac [A] time = 0.23, size = 386, normalized size = 1.72 \begin {gather*} \frac {2 \, {\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} \sqrt {-c^{2} d + b c e}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} B c^{2} d^{3} + 5 \, {\left (x e + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 45 \, {\left (x e + d\right )}^{2} A c^{2} d^{2} e - 5 \, {\left (x e + d\right )} B b c d^{3} e - 10 \, {\left (x e + d\right )} A c^{2} d^{3} e - 6 \, B b c d^{4} e - 3 \, A c^{2} d^{4} e + 45 \, {\left (x e + d\right )}^{2} A b c d e^{2} + 15 \, {\left (x e + d\right )} A b c d^{2} e^{2} + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} - 15 \, {\left (x e + d\right )}^{2} A b^{2} e^{3} - 5 \, {\left (x e + d\right )} A b^{2} d e^{3} - 3 \, A b^{2} d^{2} e^{3}\right )}}{15 \, {\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 {5}{2}}} + \frac {2 \, A \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x),x, algorithm="giac")
[Out]
2*(B*b*c^3 - A*c^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2
- b^4*e^3)*sqrt(-c^2*d + b*c*e)) + 2/15*(15*(x*e + d)^2*B*c^2*d^3 + 5*(x*e + d)*B*c^2*d^4 + 3*B*c^2*d^5 - 45*
(x*e + d)^2*A*c^2*d^2*e - 5*(x*e + d)*B*b*c*d^3*e - 10*(x*e + d)*A*c^2*d^3*e - 6*B*b*c*d^4*e - 3*A*c^2*d^4*e +
45*(x*e + d)^2*A*b*c*d*e^2 + 15*(x*e + d)*A*b*c*d^2*e^2 + 3*B*b^2*d^3*e^2 + 6*A*b*c*d^3*e^2 - 15*(x*e + d)^2*
A*b^2*e^3 - 5*(x*e + d)*A*b^2*d*e^3 - 3*A*b^2*d^2*e^3)/((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)^(5/2)) + 2*A*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)*d^3)
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maple [A] time = 0.07, size = 350, normalized size = 1.56 \begin {gather*} \frac {2 A \,c^{4} \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^{3} \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}}+\frac {2 A \,b^{2} e^{3}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{3}}-\frac {6 A b c \,e^{2}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{2}}+\frac {6 A \,c^{2} e}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d}-\frac {2 B \,c^{2}}{\left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {2 A b \,e^{2}}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} d^{2}}-\frac {4 A c e}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} d}+\frac {2 B c}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 A e}{5 \left (b e -c d \right ) \left (e x +d \right )^{\frac {5}{2}} d}-\frac {2 B}{5 \left (b e -c d \right ) \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 A \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x),x)
[Out]
2/(b*e-c*d)^3*c^4/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A-2/(b*e-c*d)^3*c^3/((b*e-
c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B-2*A*arctanh((e*x+d)^(1/2)/d^(1/2))/b/d^(7/2)+2/5/(
b*e-c*d)/d/(e*x+d)^(5/2)*A*e-2/5/(b*e-c*d)/(e*x+d)^(5/2)*B+2/3/(b*e-c*d)^2/d^2/(e*x+d)^(3/2)*A*b*e^2-4/3/(b*e-
c*d)^2/d/(e*x+d)^(3/2)*A*c*e+2/3/(b*e-c*d)^2/(e*x+d)^(3/2)*B*c+2/(b*e-c*d)^3/d^3/(e*x+d)^(1/2)*A*b^2*e^3-6/(b*
e-c*d)^3/d^2/(e*x+d)^(1/2)*A*b*c*e^2+6/(b*e-c*d)^3/d/(e*x+d)^(1/2)*A*c^2*e-2/(b*e-c*d)^3/(e*x+d)^(1/2)*B*c^2
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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)^(7/2)/(c*x^2+b*x),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 = 4.72, size = 13404, normalized size = 59.57
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)*(d + e*x)^(7/2)),x)
[Out]
- ((2*(A*e - B*d))/(5*(c*d^2 - b*d*e)) + (2*(d + e*x)^2*(A*b^2*e^3 - B*c^2*d^3 + 3*A*c^2*d^2*e - 3*A*b*c*d*e^2
))/(c*d^2 - b*d*e)^3 - (2*(d + e*x)*(A*b*e^2 + B*c*d^2 - 2*A*c*d*e))/(3*(c*d^2 - b*d*e)^2))/(d + e*x)^(5/2) -
(atan((((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^18*d^24*e^2 + 1128*A^2*b^2*c^16*d^22
*e^4 - 4312*A^2*b^3*c^15*d^21*e^5 + 11928*A^2*b^4*c^14*d^20*e^6 - 25032*A^2*b^5*c^13*d^19*e^7 + 40712*A^2*b^6*
c^12*d^18*e^8 - 51768*A^2*b^7*c^11*d^17*e^9 + 51552*A^2*b^8*c^10*d^16*e^10 - 40048*A^2*b^9*c^9*d^15*e^11 + 240
24*A^2*b^10*c^8*d^14*e^12 - 10920*A^2*b^11*c^7*d^13*e^13 + 3640*A^2*b^12*c^6*d^12*e^14 - 840*A^2*b^13*c^5*d^11
*e^15 + 120*A^2*b^14*c^4*d^10*e^16 - 8*A^2*b^15*c^3*d^9*e^17 + 8*B^2*b^2*c^16*d^24*e^2 - 72*B^2*b^3*c^15*d^23*
e^3 + 288*B^2*b^4*c^14*d^22*e^4 - 672*B^2*b^5*c^13*d^21*e^5 + 1008*B^2*b^6*c^12*d^20*e^6 - 1008*B^2*b^7*c^11*d
^19*e^7 + 672*B^2*b^8*c^10*d^18*e^8 - 288*B^2*b^9*c^9*d^17*e^9 + 72*B^2*b^10*c^8*d^16*e^10 - 8*B^2*b^11*c^7*d^
15*e^11 - 192*A^2*b*c^17*d^23*e^3 - 16*A*B*b*c^17*d^24*e^2 + 144*A*B*b^2*c^16*d^23*e^3 - 576*A*B*b^3*c^15*d^22
*e^4 + 1344*A*B*b^4*c^14*d^21*e^5 - 2016*A*B*b^5*c^13*d^20*e^6 + 2016*A*B*b^6*c^12*d^19*e^7 - 1344*A*B*b^7*c^1
1*d^18*e^8 + 576*A*B*b^8*c^10*d^17*e^9 - 144*A*B*b^9*c^9*d^16*e^10 + 16*A*B*b^10*c^8*d^15*e^11) - ((-c^5*(b*e
- c*d)^7)^(1/2)*(A*c - B*b)*(((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^18*d^31*e^2 - 2
48*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^27*e^6 - 58968*b^7*c
^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*e^10 - 131560*b^11*
c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e^14 - 5320*b^15*c^5
*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b^8*e^7 - b*c^7*d^7 +
7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*
d*e^6) - 32*A*b^2*c^17*d^27*e^3 + 432*A*b^3*c^16*d^26*e^4 - 2720*A*b^4*c^15*d^25*e^5 + 10600*A*b^5*c^14*d^24*e
^6 - 28608*A*b^6*c^13*d^23*e^7 + 56672*A*b^7*c^12*d^22*e^8 - 85184*A*b^8*c^11*d^21*e^9 + 99000*A*b^9*c^10*d^20
*e^10 - 89760*A*b^10*c^9*d^19*e^11 + 63536*A*b^11*c^8*d^18*e^12 - 34848*A*b^12*c^7*d^17*e^13 + 14552*A*b^13*c^
6*d^16*e^14 - 4480*A*b^14*c^5*d^15*e^15 + 960*A*b^15*c^4*d^14*e^16 - 128*A*b^16*c^3*d^13*e^17 + 8*A*b^17*c^2*d
^12*e^18 + 8*B*b^2*c^17*d^28*e^2 - 96*B*b^3*c^16*d^27*e^3 + 528*B*b^4*c^15*d^26*e^4 - 1760*B*b^5*c^14*d^25*e^5
+ 3960*B*b^6*c^13*d^24*e^6 - 6336*B*b^7*c^12*d^23*e^7 + 7392*B*b^8*c^11*d^22*e^8 - 6336*B*b^9*c^10*d^21*e^9 +
3960*B*b^10*c^9*d^20*e^10 - 1760*B*b^11*c^8*d^19*e^11 + 528*B*b^12*c^7*d^18*e^12 - 96*B*b^13*c^6*d^17*e^13 +
8*B*b^14*c^5*d^16*e^14))/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35
*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6))*1i)/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^
5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6) + ((-c^5*(b*e - c*d)
^7)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^18*d^24*e^2 + 1128*A^2*b^2*c^16*d^22*e^4 - 4312*A^2*b^3*c^15*
d^21*e^5 + 11928*A^2*b^4*c^14*d^20*e^6 - 25032*A^2*b^5*c^13*d^19*e^7 + 40712*A^2*b^6*c^12*d^18*e^8 - 51768*A^2
*b^7*c^11*d^17*e^9 + 51552*A^2*b^8*c^10*d^16*e^10 - 40048*A^2*b^9*c^9*d^15*e^11 + 24024*A^2*b^10*c^8*d^14*e^12
- 10920*A^2*b^11*c^7*d^13*e^13 + 3640*A^2*b^12*c^6*d^12*e^14 - 840*A^2*b^13*c^5*d^11*e^15 + 120*A^2*b^14*c^4*
d^10*e^16 - 8*A^2*b^15*c^3*d^9*e^17 + 8*B^2*b^2*c^16*d^24*e^2 - 72*B^2*b^3*c^15*d^23*e^3 + 288*B^2*b^4*c^14*d^
22*e^4 - 672*B^2*b^5*c^13*d^21*e^5 + 1008*B^2*b^6*c^12*d^20*e^6 - 1008*B^2*b^7*c^11*d^19*e^7 + 672*B^2*b^8*c^1
0*d^18*e^8 - 288*B^2*b^9*c^9*d^17*e^9 + 72*B^2*b^10*c^8*d^16*e^10 - 8*B^2*b^11*c^7*d^15*e^11 - 192*A^2*b*c^17*
d^23*e^3 - 16*A*B*b*c^17*d^24*e^2 + 144*A*B*b^2*c^16*d^23*e^3 - 576*A*B*b^3*c^15*d^22*e^4 + 1344*A*B*b^4*c^14*
d^21*e^5 - 2016*A*B*b^5*c^13*d^20*e^6 + 2016*A*B*b^6*c^12*d^19*e^7 - 1344*A*B*b^7*c^11*d^18*e^8 + 576*A*B*b^8*
c^10*d^17*e^9 - 144*A*B*b^9*c^9*d^16*e^10 + 16*A*B*b^10*c^8*d^15*e^11) - ((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*
b)*(((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 18
00*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8
*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^1
2*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^
4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*
c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6) + 32*A*b^2*c^17*d^
27*e^3 - 432*A*b^3*c^16*d^26*e^4 + 2720*A*b^4*c^15*d^25*e^5 - 10600*A*b^5*c^14*d^24*e^6 + 28608*A*b^6*c^13*d^2
3*e^7 - 56672*A*b^7*c^12*d^22*e^8 + 85184*A*b^8*c^11*d^21*e^9 - 99000*A*b^9*c^10*d^20*e^10 + 89760*A*b^10*c^9*
d^19*e^11 - 63536*A*b^11*c^8*d^18*e^12 + 34848*A*b^12*c^7*d^17*e^13 - 14552*A*b^13*c^6*d^16*e^14 + 4480*A*b^14
*c^5*d^15*e^15 - 960*A*b^15*c^4*d^14*e^16 + 128*A*b^16*c^3*d^13*e^17 - 8*A*b^17*c^2*d^12*e^18 - 8*B*b^2*c^17*d
^28*e^2 + 96*B*b^3*c^16*d^27*e^3 - 528*B*b^4*c^15*d^26*e^4 + 1760*B*b^5*c^14*d^25*e^5 - 3960*B*b^6*c^13*d^24*e
^6 + 6336*B*b^7*c^12*d^23*e^7 - 7392*B*b^8*c^11*d^22*e^8 + 6336*B*b^9*c^10*d^21*e^9 - 3960*B*b^10*c^9*d^20*e^1
0 + 1760*B*b^11*c^8*d^19*e^11 - 528*B*b^12*c^7*d^18*e^12 + 96*B*b^13*c^6*d^17*e^13 - 8*B*b^14*c^5*d^16*e^14))/
(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6
*c^2*d^2*e^5 - 7*b^7*c*d*e^6))*1i)/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^
4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6))/(((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*((d
+ e*x)^(1/2)*(16*A^2*c^18*d^24*e^2 + 1128*A^2*b^2*c^16*d^22*e^4 - 4312*A^2*b^3*c^15*d^21*e^5 + 11928*A^2*b^4*
c^14*d^20*e^6 - 25032*A^2*b^5*c^13*d^19*e^7 + 40712*A^2*b^6*c^12*d^18*e^8 - 51768*A^2*b^7*c^11*d^17*e^9 + 5155
2*A^2*b^8*c^10*d^16*e^10 - 40048*A^2*b^9*c^9*d^15*e^11 + 24024*A^2*b^10*c^8*d^14*e^12 - 10920*A^2*b^11*c^7*d^1
3*e^13 + 3640*A^2*b^12*c^6*d^12*e^14 - 840*A^2*b^13*c^5*d^11*e^15 + 120*A^2*b^14*c^4*d^10*e^16 - 8*A^2*b^15*c^
3*d^9*e^17 + 8*B^2*b^2*c^16*d^24*e^2 - 72*B^2*b^3*c^15*d^23*e^3 + 288*B^2*b^4*c^14*d^22*e^4 - 672*B^2*b^5*c^13
*d^21*e^5 + 1008*B^2*b^6*c^12*d^20*e^6 - 1008*B^2*b^7*c^11*d^19*e^7 + 672*B^2*b^8*c^10*d^18*e^8 - 288*B^2*b^9*
c^9*d^17*e^9 + 72*B^2*b^10*c^8*d^16*e^10 - 8*B^2*b^11*c^7*d^15*e^11 - 192*A^2*b*c^17*d^23*e^3 - 16*A*B*b*c^17*
d^24*e^2 + 144*A*B*b^2*c^16*d^23*e^3 - 576*A*B*b^3*c^15*d^22*e^4 + 1344*A*B*b^4*c^14*d^21*e^5 - 2016*A*B*b^5*c
^13*d^20*e^6 + 2016*A*B*b^6*c^12*d^19*e^7 - 1344*A*B*b^7*c^11*d^18*e^8 + 576*A*B*b^8*c^10*d^17*e^9 - 144*A*B*b
^9*c^9*d^16*e^10 + 16*A*B*b^10*c^8*d^15*e^11) - ((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(((-c^5*(b*e - c*d)^7)
^(1/2)*(A*c - B*b)*(d + e*x)^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 81
20*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b
^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b
^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^
3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*
d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6) + 32*A*b^2*c^17*d^27*e^3 - 432*A*b^3*c^16*d
^26*e^4 + 2720*A*b^4*c^15*d^25*e^5 - 10600*A*b^5*c^14*d^24*e^6 + 28608*A*b^6*c^13*d^23*e^7 - 56672*A*b^7*c^12*
d^22*e^8 + 85184*A*b^8*c^11*d^21*e^9 - 99000*A*b^9*c^10*d^20*e^10 + 89760*A*b^10*c^9*d^19*e^11 - 63536*A*b^11*
c^8*d^18*e^12 + 34848*A*b^12*c^7*d^17*e^13 - 14552*A*b^13*c^6*d^16*e^14 + 4480*A*b^14*c^5*d^15*e^15 - 960*A*b^
15*c^4*d^14*e^16 + 128*A*b^16*c^3*d^13*e^17 - 8*A*b^17*c^2*d^12*e^18 - 8*B*b^2*c^17*d^28*e^2 + 96*B*b^3*c^16*d
^27*e^3 - 528*B*b^4*c^15*d^26*e^4 + 1760*B*b^5*c^14*d^25*e^5 - 3960*B*b^6*c^13*d^24*e^6 + 6336*B*b^7*c^12*d^23
*e^7 - 7392*B*b^8*c^11*d^22*e^8 + 6336*B*b^9*c^10*d^21*e^9 - 3960*B*b^10*c^9*d^20*e^10 + 1760*B*b^11*c^8*d^19*
e^11 - 528*B*b^12*c^7*d^18*e^12 + 96*B*b^13*c^6*d^17*e^13 - 8*B*b^14*c^5*d^16*e^14))/(b^8*e^7 - b*c^7*d^7 + 7*
b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*
e^6)))/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 +
21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6) - ((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*((d + e*x)^(1/2)*(16*A^2*c^18*d
^24*e^2 + 1128*A^2*b^2*c^16*d^22*e^4 - 4312*A^2*b^3*c^15*d^21*e^5 + 11928*A^2*b^4*c^14*d^20*e^6 - 25032*A^2*b^
5*c^13*d^19*e^7 + 40712*A^2*b^6*c^12*d^18*e^8 - 51768*A^2*b^7*c^11*d^17*e^9 + 51552*A^2*b^8*c^10*d^16*e^10 - 4
0048*A^2*b^9*c^9*d^15*e^11 + 24024*A^2*b^10*c^8*d^14*e^12 - 10920*A^2*b^11*c^7*d^13*e^13 + 3640*A^2*b^12*c^6*d
^12*e^14 - 840*A^2*b^13*c^5*d^11*e^15 + 120*A^2*b^14*c^4*d^10*e^16 - 8*A^2*b^15*c^3*d^9*e^17 + 8*B^2*b^2*c^16*
d^24*e^2 - 72*B^2*b^3*c^15*d^23*e^3 + 288*B^2*b^4*c^14*d^22*e^4 - 672*B^2*b^5*c^13*d^21*e^5 + 1008*B^2*b^6*c^1
2*d^20*e^6 - 1008*B^2*b^7*c^11*d^19*e^7 + 672*B^2*b^8*c^10*d^18*e^8 - 288*B^2*b^9*c^9*d^17*e^9 + 72*B^2*b^10*c
^8*d^16*e^10 - 8*B^2*b^11*c^7*d^15*e^11 - 192*A^2*b*c^17*d^23*e^3 - 16*A*B*b*c^17*d^24*e^2 + 144*A*B*b^2*c^16*
d^23*e^3 - 576*A*B*b^3*c^15*d^22*e^4 + 1344*A*B*b^4*c^14*d^21*e^5 - 2016*A*B*b^5*c^13*d^20*e^6 + 2016*A*B*b^6*
c^12*d^19*e^7 - 1344*A*B*b^7*c^11*d^18*e^8 + 576*A*B*b^8*c^10*d^17*e^9 - 144*A*B*b^9*c^9*d^16*e^10 + 16*A*B*b^
10*c^8*d^15*e^11) - ((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(((-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*(d + e*x)
^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480
*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^
10*c^10*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^
14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^1
5*e^18))/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4
+ 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6) - 32*A*b^2*c^17*d^27*e^3 + 432*A*b^3*c^16*d^26*e^4 - 2720*A*b^4*c^15*d^
25*e^5 + 10600*A*b^5*c^14*d^24*e^6 - 28608*A*b^6*c^13*d^23*e^7 + 56672*A*b^7*c^12*d^22*e^8 - 85184*A*b^8*c^11*
d^21*e^9 + 99000*A*b^9*c^10*d^20*e^10 - 89760*A*b^10*c^9*d^19*e^11 + 63536*A*b^11*c^8*d^18*e^12 - 34848*A*b^12
*c^7*d^17*e^13 + 14552*A*b^13*c^6*d^16*e^14 - 4480*A*b^14*c^5*d^15*e^15 + 960*A*b^15*c^4*d^14*e^16 - 128*A*b^1
6*c^3*d^13*e^17 + 8*A*b^17*c^2*d^12*e^18 + 8*B*b^2*c^17*d^28*e^2 - 96*B*b^3*c^16*d^27*e^3 + 528*B*b^4*c^15*d^2
6*e^4 - 1760*B*b^5*c^14*d^25*e^5 + 3960*B*b^6*c^13*d^24*e^6 - 6336*B*b^7*c^12*d^23*e^7 + 7392*B*b^8*c^11*d^22*
e^8 - 6336*B*b^9*c^10*d^21*e^9 + 3960*B*b^10*c^9*d^20*e^10 - 1760*B*b^11*c^8*d^19*e^11 + 528*B*b^12*c^7*d^18*e
^12 - 96*B*b^13*c^6*d^17*e^13 + 8*B*b^14*c^5*d^16*e^14))/(b^8*e^7 - b*c^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d
^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*c*d*e^6)))/(b^8*e^7 - b*c^7*d^7
+ 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 - 7*b^7*
c*d*e^6) - 48*A^3*c^17*d^20*e^3 - 2176*A^3*b^2*c^15*d^18*e^5 + 5904*A^3*b^3*c^14*d^17*e^6 - 10656*A^3*b^4*c^13
*d^16*e^7 + 13440*A^3*b^5*c^12*d^15*e^8 - 12096*A^3*b^6*c^11*d^14*e^9 + 7776*A^3*b^7*c^10*d^13*e^10 - 3504*A^3
*b^8*c^9*d^12*e^11 + 1056*A^3*b^9*c^8*d^11*e^12 - 192*A^3*b^10*c^7*d^10*e^13 + 16*A^3*b^11*c^6*d^9*e^14 + 16*A
^2*B*c^17*d^21*e^2 + 480*A^3*b*c^16*d^19*e^4 + 144*A*B^2*b^2*c^15*d^20*e^3 - 576*A*B^2*b^3*c^14*d^19*e^4 + 134
4*A*B^2*b^4*c^13*d^18*e^5 - 2016*A*B^2*b^5*c^12*d^17*e^6 + 2016*A*B^2*b^6*c^11*d^16*e^7 - 1344*A*B^2*b^7*c^10*
d^15*e^8 + 576*A*B^2*b^8*c^9*d^14*e^9 - 144*A*B^2*b^9*c^8*d^13*e^10 + 16*A*B^2*b^10*c^7*d^12*e^11 + 96*A^2*B*b
^2*c^15*d^19*e^4 + 832*A^2*B*b^3*c^14*d^18*e^5 - 3888*A^2*B*b^4*c^13*d^17*e^6 + 8640*A^2*B*b^5*c^12*d^16*e^7 -
12096*A^2*B*b^6*c^11*d^15*e^8 + 11520*A^2*B*b^7*c^10*d^14*e^9 - 7632*A^2*B*b^8*c^9*d^13*e^10 + 3488*A^2*B*b^9
*c^8*d^12*e^11 - 1056*A^2*B*b^10*c^7*d^11*e^12 + 192*A^2*B*b^11*c^6*d^10*e^13 - 16*A^2*B*b^12*c^5*d^9*e^14 - 1
6*A*B^2*b*c^16*d^21*e^2 - 96*A^2*B*b*c^16*d^20*e^3))*(-c^5*(b*e - c*d)^7)^(1/2)*(A*c - B*b)*2i)/(b^8*e^7 - b*c
^7*d^7 + 7*b^2*c^6*d^6*e - 21*b^3*c^5*d^5*e^2 + 35*b^4*c^4*d^4*e^3 - 35*b^5*c^3*d^3*e^4 + 21*b^6*c^2*d^2*e^5 -
7*b^7*c*d*e^6) - (A*atan(((A*((d + e*x)^(1/2)*(16*A^2*c^18*d^24*e^2 + 1128*A^2*b^2*c^16*d^22*e^4 - 4312*A^2*b
^3*c^15*d^21*e^5 + 11928*A^2*b^4*c^14*d^20*e^6 - 25032*A^2*b^5*c^13*d^19*e^7 + 40712*A^2*b^6*c^12*d^18*e^8 - 5
1768*A^2*b^7*c^11*d^17*e^9 + 51552*A^2*b^8*c^10*d^16*e^10 - 40048*A^2*b^9*c^9*d^15*e^11 + 24024*A^2*b^10*c^8*d
^14*e^12 - 10920*A^2*b^11*c^7*d^13*e^13 + 3640*A^2*b^12*c^6*d^12*e^14 - 840*A^2*b^13*c^5*d^11*e^15 + 120*A^2*b
^14*c^4*d^10*e^16 - 8*A^2*b^15*c^3*d^9*e^17 + 8*B^2*b^2*c^16*d^24*e^2 - 72*B^2*b^3*c^15*d^23*e^3 + 288*B^2*b^4
*c^14*d^22*e^4 - 672*B^2*b^5*c^13*d^21*e^5 + 1008*B^2*b^6*c^12*d^20*e^6 - 1008*B^2*b^7*c^11*d^19*e^7 + 672*B^2
*b^8*c^10*d^18*e^8 - 288*B^2*b^9*c^9*d^17*e^9 + 72*B^2*b^10*c^8*d^16*e^10 - 8*B^2*b^11*c^7*d^15*e^11 - 192*A^2
*b*c^17*d^23*e^3 - 16*A*B*b*c^17*d^24*e^2 + 144*A*B*b^2*c^16*d^23*e^3 - 576*A*B*b^3*c^15*d^22*e^4 + 1344*A*B*b
^4*c^14*d^21*e^5 - 2016*A*B*b^5*c^13*d^20*e^6 + 2016*A*B*b^6*c^12*d^19*e^7 - 1344*A*B*b^7*c^11*d^18*e^8 + 576*
A*B*b^8*c^10*d^17*e^9 - 144*A*B*b^9*c^9*d^16*e^10 + 16*A*B*b^10*c^8*d^15*e^11) - (A*((A*(d + e*x)^(1/2)*(16*b^
2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^2
7*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*
e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e
^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b*(
d^7)^(1/2)) - 32*A*b^2*c^17*d^27*e^3 + 432*A*b^3*c^16*d^26*e^4 - 2720*A*b^4*c^15*d^25*e^5 + 10600*A*b^5*c^14*d
^24*e^6 - 28608*A*b^6*c^13*d^23*e^7 + 56672*A*b^7*c^12*d^22*e^8 - 85184*A*b^8*c^11*d^21*e^9 + 99000*A*b^9*c^10
*d^20*e^10 - 89760*A*b^10*c^9*d^19*e^11 + 63536*A*b^11*c^8*d^18*e^12 - 34848*A*b^12*c^7*d^17*e^13 + 14552*A*b^
13*c^6*d^16*e^14 - 4480*A*b^14*c^5*d^15*e^15 + 960*A*b^15*c^4*d^14*e^16 - 128*A*b^16*c^3*d^13*e^17 + 8*A*b^17*
c^2*d^12*e^18 + 8*B*b^2*c^17*d^28*e^2 - 96*B*b^3*c^16*d^27*e^3 + 528*B*b^4*c^15*d^26*e^4 - 1760*B*b^5*c^14*d^2
5*e^5 + 3960*B*b^6*c^13*d^24*e^6 - 6336*B*b^7*c^12*d^23*e^7 + 7392*B*b^8*c^11*d^22*e^8 - 6336*B*b^9*c^10*d^21*
e^9 + 3960*B*b^10*c^9*d^20*e^10 - 1760*B*b^11*c^8*d^19*e^11 + 528*B*b^12*c^7*d^18*e^12 - 96*B*b^13*c^6*d^17*e^
13 + 8*B*b^14*c^5*d^16*e^14))/(b*(d^7)^(1/2)))*1i)/(b*(d^7)^(1/2)) + (A*((d + e*x)^(1/2)*(16*A^2*c^18*d^24*e^2
+ 1128*A^2*b^2*c^16*d^22*e^4 - 4312*A^2*b^3*c^15*d^21*e^5 + 11928*A^2*b^4*c^14*d^20*e^6 - 25032*A^2*b^5*c^13*
d^19*e^7 + 40712*A^2*b^6*c^12*d^18*e^8 - 51768*A^2*b^7*c^11*d^17*e^9 + 51552*A^2*b^8*c^10*d^16*e^10 - 40048*A^
2*b^9*c^9*d^15*e^11 + 24024*A^2*b^10*c^8*d^14*e^12 - 10920*A^2*b^11*c^7*d^13*e^13 + 3640*A^2*b^12*c^6*d^12*e^1
4 - 840*A^2*b^13*c^5*d^11*e^15 + 120*A^2*b^14*c^4*d^10*e^16 - 8*A^2*b^15*c^3*d^9*e^17 + 8*B^2*b^2*c^16*d^24*e^
2 - 72*B^2*b^3*c^15*d^23*e^3 + 288*B^2*b^4*c^14*d^22*e^4 - 672*B^2*b^5*c^13*d^21*e^5 + 1008*B^2*b^6*c^12*d^20*
e^6 - 1008*B^2*b^7*c^11*d^19*e^7 + 672*B^2*b^8*c^10*d^18*e^8 - 288*B^2*b^9*c^9*d^17*e^9 + 72*B^2*b^10*c^8*d^16
*e^10 - 8*B^2*b^11*c^7*d^15*e^11 - 192*A^2*b*c^17*d^23*e^3 - 16*A*B*b*c^17*d^24*e^2 + 144*A*B*b^2*c^16*d^23*e^
3 - 576*A*B*b^3*c^15*d^22*e^4 + 1344*A*B*b^4*c^14*d^21*e^5 - 2016*A*B*b^5*c^13*d^20*e^6 + 2016*A*B*b^6*c^12*d^
19*e^7 - 1344*A*B*b^7*c^11*d^18*e^8 + 576*A*B*b^8*c^10*d^17*e^9 - 144*A*B*b^9*c^9*d^16*e^10 + 16*A*B*b^10*c^8*
d^15*e^11) - (A*((A*(d + e*x)^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8
120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*
b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*
b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c
^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b*(d^7)^(1/2)) + 32*A*b^2*c^17*d^27*e^3 - 432*A*b^3*c^16*d^26*e^4 + 272
0*A*b^4*c^15*d^25*e^5 - 10600*A*b^5*c^14*d^24*e^6 + 28608*A*b^6*c^13*d^23*e^7 - 56672*A*b^7*c^12*d^22*e^8 + 85
184*A*b^8*c^11*d^21*e^9 - 99000*A*b^9*c^10*d^20*e^10 + 89760*A*b^10*c^9*d^19*e^11 - 63536*A*b^11*c^8*d^18*e^12
+ 34848*A*b^12*c^7*d^17*e^13 - 14552*A*b^13*c^6*d^16*e^14 + 4480*A*b^14*c^5*d^15*e^15 - 960*A*b^15*c^4*d^14*e
^16 + 128*A*b^16*c^3*d^13*e^17 - 8*A*b^17*c^2*d^12*e^18 - 8*B*b^2*c^17*d^28*e^2 + 96*B*b^3*c^16*d^27*e^3 - 528
*B*b^4*c^15*d^26*e^4 + 1760*B*b^5*c^14*d^25*e^5 - 3960*B*b^6*c^13*d^24*e^6 + 6336*B*b^7*c^12*d^23*e^7 - 7392*B
*b^8*c^11*d^22*e^8 + 6336*B*b^9*c^10*d^21*e^9 - 3960*B*b^10*c^9*d^20*e^10 + 1760*B*b^11*c^8*d^19*e^11 - 528*B*
b^12*c^7*d^18*e^12 + 96*B*b^13*c^6*d^17*e^13 - 8*B*b^14*c^5*d^16*e^14))/(b*(d^7)^(1/2)))*1i)/(b*(d^7)^(1/2)))/
((A*((d + e*x)^(1/2)*(16*A^2*c^18*d^24*e^2 + 1128*A^2*b^2*c^16*d^22*e^4 - 4312*A^2*b^3*c^15*d^21*e^5 + 11928*A
^2*b^4*c^14*d^20*e^6 - 25032*A^2*b^5*c^13*d^19*e^7 + 40712*A^2*b^6*c^12*d^18*e^8 - 51768*A^2*b^7*c^11*d^17*e^9
+ 51552*A^2*b^8*c^10*d^16*e^10 - 40048*A^2*b^9*c^9*d^15*e^11 + 24024*A^2*b^10*c^8*d^14*e^12 - 10920*A^2*b^11*
c^7*d^13*e^13 + 3640*A^2*b^12*c^6*d^12*e^14 - 840*A^2*b^13*c^5*d^11*e^15 + 120*A^2*b^14*c^4*d^10*e^16 - 8*A^2*
b^15*c^3*d^9*e^17 + 8*B^2*b^2*c^16*d^24*e^2 - 72*B^2*b^3*c^15*d^23*e^3 + 288*B^2*b^4*c^14*d^22*e^4 - 672*B^2*b
^5*c^13*d^21*e^5 + 1008*B^2*b^6*c^12*d^20*e^6 - 1008*B^2*b^7*c^11*d^19*e^7 + 672*B^2*b^8*c^10*d^18*e^8 - 288*B
^2*b^9*c^9*d^17*e^9 + 72*B^2*b^10*c^8*d^16*e^10 - 8*B^2*b^11*c^7*d^15*e^11 - 192*A^2*b*c^17*d^23*e^3 - 16*A*B*
b*c^17*d^24*e^2 + 144*A*B*b^2*c^16*d^23*e^3 - 576*A*B*b^3*c^15*d^22*e^4 + 1344*A*B*b^4*c^14*d^21*e^5 - 2016*A*
B*b^5*c^13*d^20*e^6 + 2016*A*B*b^6*c^12*d^19*e^7 - 1344*A*B*b^7*c^11*d^18*e^8 + 576*A*B*b^8*c^10*d^17*e^9 - 14
4*A*B*b^9*c^9*d^16*e^10 + 16*A*B*b^10*c^8*d^15*e^11) - (A*((A*(d + e*x)^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*
c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^2
6*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*e^10 - 131560*b^11*c^9*d^2
2*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e
^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b*(d^7)^(1/2)) + 32*A*b^2*c^1
7*d^27*e^3 - 432*A*b^3*c^16*d^26*e^4 + 2720*A*b^4*c^15*d^25*e^5 - 10600*A*b^5*c^14*d^24*e^6 + 28608*A*b^6*c^13
*d^23*e^7 - 56672*A*b^7*c^12*d^22*e^8 + 85184*A*b^8*c^11*d^21*e^9 - 99000*A*b^9*c^10*d^20*e^10 + 89760*A*b^10*
c^9*d^19*e^11 - 63536*A*b^11*c^8*d^18*e^12 + 34848*A*b^12*c^7*d^17*e^13 - 14552*A*b^13*c^6*d^16*e^14 + 4480*A*
b^14*c^5*d^15*e^15 - 960*A*b^15*c^4*d^14*e^16 + 128*A*b^16*c^3*d^13*e^17 - 8*A*b^17*c^2*d^12*e^18 - 8*B*b^2*c^
17*d^28*e^2 + 96*B*b^3*c^16*d^27*e^3 - 528*B*b^4*c^15*d^26*e^4 + 1760*B*b^5*c^14*d^25*e^5 - 3960*B*b^6*c^13*d^
24*e^6 + 6336*B*b^7*c^12*d^23*e^7 - 7392*B*b^8*c^11*d^22*e^8 + 6336*B*b^9*c^10*d^21*e^9 - 3960*B*b^10*c^9*d^20
*e^10 + 1760*B*b^11*c^8*d^19*e^11 - 528*B*b^12*c^7*d^18*e^12 + 96*B*b^13*c^6*d^17*e^13 - 8*B*b^14*c^5*d^16*e^1
4))/(b*(d^7)^(1/2))))/(b*(d^7)^(1/2)) - (A*((d + e*x)^(1/2)*(16*A^2*c^18*d^24*e^2 + 1128*A^2*b^2*c^16*d^22*e^4
- 4312*A^2*b^3*c^15*d^21*e^5 + 11928*A^2*b^4*c^14*d^20*e^6 - 25032*A^2*b^5*c^13*d^19*e^7 + 40712*A^2*b^6*c^12
*d^18*e^8 - 51768*A^2*b^7*c^11*d^17*e^9 + 51552*A^2*b^8*c^10*d^16*e^10 - 40048*A^2*b^9*c^9*d^15*e^11 + 24024*A
^2*b^10*c^8*d^14*e^12 - 10920*A^2*b^11*c^7*d^13*e^13 + 3640*A^2*b^12*c^6*d^12*e^14 - 840*A^2*b^13*c^5*d^11*e^1
5 + 120*A^2*b^14*c^4*d^10*e^16 - 8*A^2*b^15*c^3*d^9*e^17 + 8*B^2*b^2*c^16*d^24*e^2 - 72*B^2*b^3*c^15*d^23*e^3
+ 288*B^2*b^4*c^14*d^22*e^4 - 672*B^2*b^5*c^13*d^21*e^5 + 1008*B^2*b^6*c^12*d^20*e^6 - 1008*B^2*b^7*c^11*d^19*
e^7 + 672*B^2*b^8*c^10*d^18*e^8 - 288*B^2*b^9*c^9*d^17*e^9 + 72*B^2*b^10*c^8*d^16*e^10 - 8*B^2*b^11*c^7*d^15*e
^11 - 192*A^2*b*c^17*d^23*e^3 - 16*A*B*b*c^17*d^24*e^2 + 144*A*B*b^2*c^16*d^23*e^3 - 576*A*B*b^3*c^15*d^22*e^4
+ 1344*A*B*b^4*c^14*d^21*e^5 - 2016*A*B*b^5*c^13*d^20*e^6 + 2016*A*B*b^6*c^12*d^19*e^7 - 1344*A*B*b^7*c^11*d^
18*e^8 + 576*A*B*b^8*c^10*d^17*e^9 - 144*A*B*b^9*c^9*d^16*e^10 + 16*A*B*b^10*c^8*d^15*e^11) - (A*((A*(d + e*x)
^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480
*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^
10*c^10*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^
14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^1
5*e^18))/(b*(d^7)^(1/2)) - 32*A*b^2*c^17*d^27*e^3 + 432*A*b^3*c^16*d^26*e^4 - 2720*A*b^4*c^15*d^25*e^5 + 10600
*A*b^5*c^14*d^24*e^6 - 28608*A*b^6*c^13*d^23*e^7 + 56672*A*b^7*c^12*d^22*e^8 - 85184*A*b^8*c^11*d^21*e^9 + 990
00*A*b^9*c^10*d^20*e^10 - 89760*A*b^10*c^9*d^19*e^11 + 63536*A*b^11*c^8*d^18*e^12 - 34848*A*b^12*c^7*d^17*e^13
+ 14552*A*b^13*c^6*d^16*e^14 - 4480*A*b^14*c^5*d^15*e^15 + 960*A*b^15*c^4*d^14*e^16 - 128*A*b^16*c^3*d^13*e^1
7 + 8*A*b^17*c^2*d^12*e^18 + 8*B*b^2*c^17*d^28*e^2 - 96*B*b^3*c^16*d^27*e^3 + 528*B*b^4*c^15*d^26*e^4 - 1760*B
*b^5*c^14*d^25*e^5 + 3960*B*b^6*c^13*d^24*e^6 - 6336*B*b^7*c^12*d^23*e^7 + 7392*B*b^8*c^11*d^22*e^8 - 6336*B*b
^9*c^10*d^21*e^9 + 3960*B*b^10*c^9*d^20*e^10 - 1760*B*b^11*c^8*d^19*e^11 + 528*B*b^12*c^7*d^18*e^12 - 96*B*b^1
3*c^6*d^17*e^13 + 8*B*b^14*c^5*d^16*e^14))/(b*(d^7)^(1/2))))/(b*(d^7)^(1/2)) - 48*A^3*c^17*d^20*e^3 - 2176*A^3
*b^2*c^15*d^18*e^5 + 5904*A^3*b^3*c^14*d^17*e^6 - 10656*A^3*b^4*c^13*d^16*e^7 + 13440*A^3*b^5*c^12*d^15*e^8 -
12096*A^3*b^6*c^11*d^14*e^9 + 7776*A^3*b^7*c^10*d^13*e^10 - 3504*A^3*b^8*c^9*d^12*e^11 + 1056*A^3*b^9*c^8*d^11
*e^12 - 192*A^3*b^10*c^7*d^10*e^13 + 16*A^3*b^11*c^6*d^9*e^14 + 16*A^2*B*c^17*d^21*e^2 + 480*A^3*b*c^16*d^19*e
^4 + 144*A*B^2*b^2*c^15*d^20*e^3 - 576*A*B^2*b^3*c^14*d^19*e^4 + 1344*A*B^2*b^4*c^13*d^18*e^5 - 2016*A*B^2*b^5
*c^12*d^17*e^6 + 2016*A*B^2*b^6*c^11*d^16*e^7 - 1344*A*B^2*b^7*c^10*d^15*e^8 + 576*A*B^2*b^8*c^9*d^14*e^9 - 14
4*A*B^2*b^9*c^8*d^13*e^10 + 16*A*B^2*b^10*c^7*d^12*e^11 + 96*A^2*B*b^2*c^15*d^19*e^4 + 832*A^2*B*b^3*c^14*d^18
*e^5 - 3888*A^2*B*b^4*c^13*d^17*e^6 + 8640*A^2*B*b^5*c^12*d^16*e^7 - 12096*A^2*B*b^6*c^11*d^15*e^8 + 11520*A^2
*B*b^7*c^10*d^14*e^9 - 7632*A^2*B*b^8*c^9*d^13*e^10 + 3488*A^2*B*b^9*c^8*d^12*e^11 - 1056*A^2*B*b^10*c^7*d^11*
e^12 + 192*A^2*B*b^11*c^6*d^10*e^13 - 16*A^2*B*b^12*c^5*d^9*e^14 - 16*A*B^2*b*c^16*d^21*e^2 - 96*A^2*B*b*c^16*
d^20*e^3))*2i)/(b*(d^7)^(1/2))
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sympy [A] time = 78.32, size = 228, normalized size = 1.01 \begin {gather*} \frac {2 A \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d^{3} \sqrt {- d}} - \frac {2 \left (- A e + B d\right )}{5 d \left (d + e x\right )^{\frac {5}{2}} \left (b e - c d\right )} + \frac {2 \left (A b e^{2} - 2 A c d e + B c d^{2}\right )}{3 d^{2} \left (d + e x\right )^{\frac {3}{2}} \left (b e - c d\right )^{2}} + \frac {2 \left (A b^{2} e^{3} - 3 A b c d e^{2} + 3 A c^{2} d^{2} e - B c^{2} d^{3}\right )}{d^{3} \sqrt {d + e x} \left (b e - c d\right )^{3}} - \frac {2 c^{2} \left (- A c + B b\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b \sqrt {\frac {b e - c d}{c}} \left (b e - c d\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**(7/2)/(c*x**2+b*x),x)
[Out]
2*A*atan(sqrt(d + e*x)/sqrt(-d))/(b*d**3*sqrt(-d)) - 2*(-A*e + B*d)/(5*d*(d + e*x)**(5/2)*(b*e - c*d)) + 2*(A*
b*e**2 - 2*A*c*d*e + B*c*d**2)/(3*d**2*(d + e*x)**(3/2)*(b*e - c*d)**2) + 2*(A*b**2*e**3 - 3*A*b*c*d*e**2 + 3*
A*c**2*d**2*e - B*c**2*d**3)/(d**3*sqrt(d + e*x)*(b*e - c*d)**3) - 2*c**2*(-A*c + B*b)*atan(sqrt(d + e*x)/sqrt
((b*e - c*d)/c))/(b*sqrt((b*e - c*d)/c)*(b*e - c*d)**3)
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