3.13.45 \(\int \frac {A+B x}{(d+e x)^{5/2} (b x+c x^2)^2} \, dx\) [1245]

Optimal. Leaf size=344 \[ -\frac {e \left (6 A c^2 d^2-b^2 e (2 B d-5 A e)-3 b c d (B d+2 A e)\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e \left (2 A c^3 d^3-b^2 c d e (6 B d-11 A e)+b^3 e^2 (2 B d-5 A e)-b c^2 d^2 (B d+3 A e)\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {(2 b B d-4 A c d-5 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}+\frac {c^{5/2} \left (2 b B c d-4 A c^2 d-7 b^2 B e+9 A b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}} \]

[Out]

-1/3*e*(6*A*c^2*d^2-b^2*e*(-5*A*e+2*B*d)-3*b*c*d*(2*A*e+B*d))/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)+(-A*b*(-b*e+c
*d)-c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(3/2)/(c*x^2+b*x)-(-5*A*b*e-4*A*c*d+2*B*b*d)*arctanh((
e*x+d)^(1/2)/d^(1/2))/b^3/d^(7/2)+c^(5/2)*(9*A*b*c*e-4*A*c^2*d-7*B*b^2*e+2*B*b*c*d)*arctanh(c^(1/2)*(e*x+d)^(1
/2)/(-b*e+c*d)^(1/2))/b^3/(-b*e+c*d)^(7/2)-e*(2*A*c^3*d^3-b^2*c*d*e*(-11*A*e+6*B*d)+b^3*e^2*(-5*A*e+2*B*d)-b*c
^2*d^2*(3*A*e+B*d))/b^2/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)

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Rubi [A]
time = 0.62, 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 = {836, 842, 840, 1180, 214} \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 {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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]

[Out]

-1/3*(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)))/(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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 836

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 840

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 842

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 1180

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 \text {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)) \text {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 ) \text {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 [A]
time = 1.55, size = 385, normalized size = 1.12 \begin {gather*} \frac {\frac {b \left (b B d x \left (3 c^3 d^2 (d+e x)^2-2 b^3 e^3 (4 d+3 e x)+2 b c^2 d e^2 x (10 d+9 e x)+2 b^2 c e^2 \left (10 d^2+5 d e x-3 e^2 x^2\right )\right )+A \left (-6 c^4 d^3 x (d+e x)^2-3 b c^3 d^2 (d-3 e x) (d+e x)^2+b^4 e^3 \left (3 d^2+20 d e x+15 e^2 x^2\right )+b^2 c^2 d e \left (9 d^3+9 d^2 e x-35 d e^2 x^2-33 e^3 x^3\right )+b^3 c e^2 \left (-9 d^3-41 d^2 e x-13 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{d^3 (c d-b e)^3 x (b+c x) (d+e x)^{3/2}}-\frac {3 c^{5/2} \left (4 A c^2 d+7 b^2 B e-b c (2 B d+9 A e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}+\frac {3 (-2 b B d+4 A c d+5 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{7/2}}}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]

[Out]

((b*(b*B*d*x*(3*c^3*d^2*(d + e*x)^2 - 2*b^3*e^3*(4*d + 3*e*x) + 2*b*c^2*d*e^2*x*(10*d + 9*e*x) + 2*b^2*c*e^2*(
10*d^2 + 5*d*e*x - 3*e^2*x^2)) + A*(-6*c^4*d^3*x*(d + e*x)^2 - 3*b*c^3*d^2*(d - 3*e*x)*(d + e*x)^2 + b^4*e^3*(
3*d^2 + 20*d*e*x + 15*e^2*x^2) + b^2*c^2*d*e*(9*d^3 + 9*d^2*e*x - 35*d*e^2*x^2 - 33*e^3*x^3) + b^3*c*e^2*(-9*d
^3 - 41*d^2*e*x - 13*d*e^2*x^2 + 15*e^3*x^3))))/(d^3*(c*d - b*e)^3*x*(b + c*x)*(d + e*x)^(3/2)) - (3*c^(5/2)*(
4*A*c^2*d + 7*b^2*B*e - b*c*(2*B*d + 9*A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e
)^(7/2) + (3*(-2*b*B*d + 4*A*c*d + 5*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(7/2))/(3*b^3)

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Maple [A]
time = 0.78, size = 267, normalized size = 0.78

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {c^{3} \left (\frac {\left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (9 A b c e -4 A \,c^{2} d -7 b^{2} B e +2 B d c b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} e^{2} b^{3}}+\frac {-\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (5 A b e +4 A c d -2 B b d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{3} e^{2} b^{3}}-\frac {A e -B d}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}\right )\)	\(267\)
default	\(2 e^{2} \left (\frac {c^{3} \left (\frac {\left (\frac {1}{2} A b c e -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (9 A b c e -4 A \,c^{2} d -7 b^{2} B e +2 B d c b \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} e^{2} b^{3}}+\frac {-\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (5 A b e +4 A c d -2 B b d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{d^{3} e^{2} b^{3}}-\frac {A e -B d}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}\right )\)	\(267\)
risch	\(-\frac {A \sqrt {e x +d}}{d^{3} b^{2} x}+\frac {e \,c^{4} \sqrt {e x +d}\, A}{b^{2} \left (b e -c d \right )^{3} \left (c e x +b e \right )}-\frac {e \,c^{3} \sqrt {e x +d}\, B}{b \left (b e -c d \right )^{3} \left (c e x +b e \right )}+\frac {9 e \,c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A}{b^{2} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {4 d \,c^{5} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) A}{b^{3} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {7 e \,c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B}{b \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {2 d \,c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) B}{b^{2} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {4 b \,e^{4} A}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {8 e^{3} A c}{d^{2} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {2 b \,e^{3} B}{d^{2} \left (b e -c d \right )^{3} \sqrt {e x +d}}-\frac {6 e^{2} B c}{d \left (b e -c d \right )^{3} \sqrt {e x +d}}-\frac {2 e^{3} A}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 e^{2} B}{3 d \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {5 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) A}{b^{2} d^{\frac {7}{2}}}+\frac {4 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) A c}{b^{3} d^{\frac {5}{2}}}-\frac {2 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) B}{b^{2} d^{\frac {5}{2}}}\)	\(535\)

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,method=_RETURNVERBOSE)

[Out]

2*e^2*(c^3/(b*e-c*d)^3/e^2/b^3*((1/2*A*b*c*e-1/2*b^2*B*e)*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(9*A*b*c*e-4*A
*c^2*d-7*B*b^2*e+2*B*b*c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))+1/d^3/e^2/b^3*(-1
/2*A*b*(e*x+d)^(1/2)/x+1/2*(5*A*b*e+4*A*c*d-2*B*b*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))-1/3*(A*e-B*d)/d^2
/(b*e-c*d)^2/(e*x+d)^(3/2)-(2*A*b*e^2-4*A*c*d*e-B*b*d*e+3*B*c*d^2)/d^3/(b*e-c*d)^3/(e*x+d)^(1/2))

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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)^(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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1426 vs. \(2 (343) = 686\).
time = 106.65, size = 5737, normalized size = 16.68 \begin {gather*} \text {Too large to display} \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="fricas")

[Out]

[-1/6*(3*(2*(B*b*c^4 - 2*A*c^5)*d^7*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^7*x - ((7*B*b^2*c^3 - 9*A*b*c^4)*d^4*x^4
+ (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^4*x^3)*e^3 + 2*((B*b*c^4 - 2*A*c^5)*d^5*x^4 - (6*B*b^2*c^3 - 7*A*b*c^4)*d^5*x
^3 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^5*x^2)*e^2 + (4*(B*b*c^4 - 2*A*c^5)*d^6*x^3 - (3*B*b^2*c^3 - A*b*c^4)*d^6*x
^2 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^6*x)*e)*sqrt(c/(c*d - b*e))*log((2*c*d - 2*(c*d - b*e)*sqrt(x*e + d)*sqrt(c
/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + 3*(2*(B*b*c^4 - 2*A*c^5)*d^6*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^6*x +
5*(A*b^4*c*x^4 + A*b^5*x^3)*e^6 + (10*A*b^5*d*x^2 - (2*B*b^4*c + 11*A*b^3*c^2)*d*x^4 - (2*B*b^5 + A*b^4*c)*d*
x^3)*e^5 + (5*A*b^5*d^2*x + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^2*x^4 + (2*B*b^4*c - 19*A*b^3*c^2)*d^2*x^3 - (4*B*b^
5 + 17*A*b^4*c)*d^2*x^2)*e^4 - ((6*B*b^2*c^3 - 7*A*b*c^4)*d^3*x^4 - (6*B*b^3*c^2 + 13*A*b^2*c^3)*d^3*x^3 - 5*(
2*B*b^4*c - A*b^3*c^2)*d^3*x^2 + (2*B*b^5 + 11*A*b^4*c)*d^3*x)*e^3 + (2*(B*b*c^4 - 2*A*c^5)*d^4*x^4 - 10*(B*b^
2*c^3 - A*b*c^4)*d^4*x^3 - (6*B*b^3*c^2 - 17*A*b^2*c^3)*d^4*x^2 + 3*(2*B*b^4*c + A*b^3*c^2)*d^4*x)*e^2 + (4*(B
*b*c^4 - 2*A*c^5)*d^5*x^3 - (2*B*b^2*c^3 + A*b*c^4)*d^5*x^2 - (6*B*b^3*c^2 - 7*A*b^2*c^3)*d^5*x)*e)*sqrt(d)*lo
g((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(3*A*b^2*c^3*d^6 - 3*(B*b^2*c^3 - 2*A*b*c^4)*d^6*x - 15*(A*b^4*
c*d*x^3 + A*b^5*d*x^2)*e^5 - (20*A*b^5*d^2*x - 3*(2*B*b^4*c + 11*A*b^3*c^2)*d^2*x^3 - (6*B*b^5 + 13*A*b^4*c)*d
^2*x^2)*e^4 - (3*A*b^5*d^3 + 9*(2*B*b^3*c^2 + A*b^2*c^3)*d^3*x^3 + 5*(2*B*b^4*c - 7*A*b^3*c^2)*d^3*x^2 - (8*B*
b^5 + 41*A*b^4*c)*d^3*x)*e^3 + (9*A*b^4*c*d^4 - 3*(B*b^2*c^3 - 2*A*b*c^4)*d^4*x^3 - 5*(4*B*b^3*c^2 + 3*A*b^2*c
^3)*d^4*x^2 - (20*B*b^4*c + 9*A*b^3*c^2)*d^4*x)*e^2 - 3*(A*b^2*c^3*d^5*x + 3*A*b^3*c^2*d^5 + 2*(B*b^2*c^3 - 2*
A*b*c^4)*d^5*x^2)*e)*sqrt(x*e + d))/(b^3*c^4*d^9*x^2 + b^4*c^3*d^9*x - (b^6*c*d^4*x^4 + b^7*d^4*x^3)*e^5 + (3*
b^5*c^2*d^5*x^4 + b^6*c*d^5*x^3 - 2*b^7*d^5*x^2)*e^4 - (3*b^4*c^3*d^6*x^4 - 3*b^5*c^2*d^6*x^3 - 5*b^6*c*d^6*x^
2 + b^7*d^6*x)*e^3 + (b^3*c^4*d^7*x^4 - 5*b^4*c^3*d^7*x^3 - 3*b^5*c^2*d^7*x^2 + 3*b^6*c*d^7*x)*e^2 + (2*b^3*c^
4*d^8*x^3 - b^4*c^3*d^8*x^2 - 3*b^5*c^2*d^8*x)*e), 1/6*(6*(2*(B*b*c^4 - 2*A*c^5)*d^7*x^2 + 2*(B*b^2*c^3 - 2*A*
b*c^4)*d^7*x - ((7*B*b^2*c^3 - 9*A*b*c^4)*d^4*x^4 + (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^4*x^3)*e^3 + 2*((B*b*c^4 - 2
*A*c^5)*d^5*x^4 - (6*B*b^2*c^3 - 7*A*b*c^4)*d^5*x^3 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^5*x^2)*e^2 + (4*(B*b*c^4 -
2*A*c^5)*d^6*x^3 - (3*B*b^2*c^3 - A*b*c^4)*d^6*x^2 - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^6*x)*e)*sqrt(-c/(c*d - b*e
))*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) - 3*(2*(B*b*c^4 - 2*A*c^5)*d^6*x^2 +
2*(B*b^2*c^3 - 2*A*b*c^4)*d^6*x + 5*(A*b^4*c*x^4 + A*b^5*x^3)*e^6 + (10*A*b^5*d*x^2 - (2*B*b^4*c + 11*A*b^3*c^
2)*d*x^4 - (2*B*b^5 + A*b^4*c)*d*x^3)*e^5 + (5*A*b^5*d^2*x + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^2*x^4 + (2*B*b^4*c
- 19*A*b^3*c^2)*d^2*x^3 - (4*B*b^5 + 17*A*b^4*c)*d^2*x^2)*e^4 - ((6*B*b^2*c^3 - 7*A*b*c^4)*d^3*x^4 - (6*B*b^3*
c^2 + 13*A*b^2*c^3)*d^3*x^3 - 5*(2*B*b^4*c - A*b^3*c^2)*d^3*x^2 + (2*B*b^5 + 11*A*b^4*c)*d^3*x)*e^3 + (2*(B*b*
c^4 - 2*A*c^5)*d^4*x^4 - 10*(B*b^2*c^3 - A*b*c^4)*d^4*x^3 - (6*B*b^3*c^2 - 17*A*b^2*c^3)*d^4*x^2 + 3*(2*B*b^4*
c + A*b^3*c^2)*d^4*x)*e^2 + (4*(B*b*c^4 - 2*A*c^5)*d^5*x^3 - (2*B*b^2*c^3 + A*b*c^4)*d^5*x^2 - (6*B*b^3*c^2 -
7*A*b^2*c^3)*d^5*x)*e)*sqrt(d)*log((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) - 2*(3*A*b^2*c^3*d^6 - 3*(B*b^2*c^
3 - 2*A*b*c^4)*d^6*x - 15*(A*b^4*c*d*x^3 + A*b^5*d*x^2)*e^5 - (20*A*b^5*d^2*x - 3*(2*B*b^4*c + 11*A*b^3*c^2)*d
^2*x^3 - (6*B*b^5 + 13*A*b^4*c)*d^2*x^2)*e^4 - (3*A*b^5*d^3 + 9*(2*B*b^3*c^2 + A*b^2*c^3)*d^3*x^3 + 5*(2*B*b^4
*c - 7*A*b^3*c^2)*d^3*x^2 - (8*B*b^5 + 41*A*b^4*c)*d^3*x)*e^3 + (9*A*b^4*c*d^4 - 3*(B*b^2*c^3 - 2*A*b*c^4)*d^4
*x^3 - 5*(4*B*b^3*c^2 + 3*A*b^2*c^3)*d^4*x^2 - (20*B*b^4*c + 9*A*b^3*c^2)*d^4*x)*e^2 - 3*(A*b^2*c^3*d^5*x + 3*
A*b^3*c^2*d^5 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^5*x^2)*e)*sqrt(x*e + d))/(b^3*c^4*d^9*x^2 + b^4*c^3*d^9*x - (b^6*c
*d^4*x^4 + b^7*d^4*x^3)*e^5 + (3*b^5*c^2*d^5*x^4 + b^6*c*d^5*x^3 - 2*b^7*d^5*x^2)*e^4 - (3*b^4*c^3*d^6*x^4 - 3
*b^5*c^2*d^6*x^3 - 5*b^6*c*d^6*x^2 + b^7*d^6*x)*e^3 + (b^3*c^4*d^7*x^4 - 5*b^4*c^3*d^7*x^3 - 3*b^5*c^2*d^7*x^2
+ 3*b^6*c*d^7*x)*e^2 + (2*b^3*c^4*d^8*x^3 - b^4*c^3*d^8*x^2 - 3*b^5*c^2*d^8*x)*e), 1/6*(6*(2*(B*b*c^4 - 2*A*c
^5)*d^6*x^2 + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^6*x + 5*(A*b^4*c*x^4 + A*b^5*x^3)*e^6 + (10*A*b^5*d*x^2 - (2*B*b^4*c
+ 11*A*b^3*c^2)*d*x^4 - (2*B*b^5 + A*b^4*c)*d*x^3)*e^5 + (5*A*b^5*d^2*x + 3*(2*B*b^3*c^2 + A*b^2*c^3)*d^2*x^4
+ (2*B*b^4*c - 19*A*b^3*c^2)*d^2*x^3 - (4*B*b^5 + 17*A*b^4*c)*d^2*x^2)*e^4 - ((6*B*b^2*c^3 - 7*A*b*c^4)*d^3*x
^4 - (6*B*b^3*c^2 + 13*A*b^2*c^3)*d^3*x^3 - 5*(2*B*b^4*c - A*b^3*c^2)*d^3*x^2 + (2*B*b^5 + 11*A*b^4*c)*d^3*x)*
e^3 + (2*(B*b*c^4 - 2*A*c^5)*d^4*x^4 - 10*(B*b^2*c^3 - A*b*c^4)*d^4*x^3 - (6*B*b^3*c^2 - 17*A*b^2*c^3)*d^4*x^2
+ 3*(2*B*b^4*c + A*b^3*c^2)*d^4*x)*e^2 + (4*(B*b*c^4 - 2*A*c^5)*d^5*x^3 - (2*B*b^2*c^3 + A*b*c^4)*d^5*x^2 - (
6*B*b^3*c^2 - 7*A*b^2*c^3)*d^5*x)*e)*sqrt(-d)*a...

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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.27, 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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Mupad [B]
time = 6.10, size = 2500, normalized size = 7.27 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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