3.376 \(\int \frac{1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=277 \[ -\frac{c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}+\frac{(5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{7/2}}-\frac{e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{c (2 c d-b e)}{b^2 d (b+c x) (d+e x)^{3/2} (c d-b e)}-\frac{1}{b d x (b+c x) (d+e x)^{3/2}} \]

[Out]

-(e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2))/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2
)) - (c*(2*c*d - b*e))/(b^2*d*(c*d - b*e)*(b + c*x)*(d + e*x)^(3/2)) - 1/(b*d*x*
(b + c*x)*(d + e*x)^(3/2)) - (e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2))/(
b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) + ((4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sq
rt[d]])/(b^3*d^(7/2)) - (c^(7/2)*(4*c*d - 9*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(7/2))

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Rubi [A]  time = 1.33751, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}+\frac{(5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{7/2}}-\frac{e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{c (2 c d-b e)}{b^2 d (b+c x) (d+e x)^{3/2} (c d-b e)}-\frac{1}{b d x (b+c x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]

[Out]

-(e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2))/(3*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2
)) - (c*(2*c*d - b*e))/(b^2*d*(c*d - b*e)*(b + c*x)*(d + e*x)^(3/2)) - 1/(b*d*x*
(b + c*x)*(d + e*x)^(3/2)) - (e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2))/(
b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) + ((4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sq
rt[d]])/(b^3*d^(7/2)) - (c^(7/2)*(4*c*d - 9*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(7/2))

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Rubi in Sympy [A]  time = 157.637, size = 252, normalized size = 0.91 \[ - \frac{c}{b x \left (b + c x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} - \frac{b e - 2 c d}{b^{2} d x \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} - \frac{e \left (5 b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{3 b^{2} d^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )^{2}} - \frac{e \left (b e - 2 c d\right ) \left (5 b^{2} e^{2} - b c d e + c^{2} d^{2}\right )}{b^{2} d^{3} \sqrt{d + e x} \left (b e - c d\right )^{3}} + \frac{c^{\frac{7}{2}} \left (9 b e - 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \left (b e - c d\right )^{\frac{7}{2}}} + \frac{\left (5 b e + 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**2,x)

[Out]

-c/(b*x*(b + c*x)*(d + e*x)**(3/2)*(b*e - c*d)) - (b*e - 2*c*d)/(b**2*d*x*(d + e
*x)**(3/2)*(b*e - c*d)) - e*(5*b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(3*b**2*d**2
*(d + e*x)**(3/2)*(b*e - c*d)**2) - e*(b*e - 2*c*d)*(5*b**2*e**2 - b*c*d*e + c**
2*d**2)/(b**2*d**3*sqrt(d + e*x)*(b*e - c*d)**3) + c**(7/2)*(9*b*e - 4*c*d)*atan
(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b**3*(b*e - c*d)**(7/2)) + (5*b*e + 4*c
*d)*atanh(sqrt(d + e*x)/sqrt(d))/(b**3*d**(7/2))

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Mathematica [A]  time = 1.39131, size = 200, normalized size = 0.72 \[ -\frac{c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}+\frac{(5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{7/2}}+\sqrt{d+e x} \left (\frac{c^4}{b^2 (b+c x) (b e-c d)^3}-\frac{1}{b^2 d^3 x}+\frac{4 e^3 (b e-2 c d)}{d^3 (d+e x) (c d-b e)^3}-\frac{2 e^3}{3 d^2 (d+e x)^2 (c d-b e)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]

[Out]

Sqrt[d + e*x]*(-(1/(b^2*d^3*x)) + c^4/(b^2*(-(c*d) + b*e)^3*(b + c*x)) - (2*e^3)
/(3*d^2*(c*d - b*e)^2*(d + e*x)^2) + (4*e^3*(-2*c*d + b*e))/(d^3*(c*d - b*e)^3*(
d + e*x))) + ((4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(7/2)) - (c
^(7/2)*(4*c*d - 9*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c
*d - b*e)^(7/2))

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Maple [A]  time = 0.039, size = 280, normalized size = 1. \[ -{\frac{2\,{e}^{3}}{3\,{d}^{2} \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-4\,{\frac{{e}^{4}b}{{d}^{3} \left ( be-cd \right ) ^{3}\sqrt{ex+d}}}+8\,{\frac{{e}^{3}c}{{d}^{2} \left ( be-cd \right ) ^{3}\sqrt{ex+d}}}+{\frac{e{c}^{4}}{{b}^{2} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) }\sqrt{ex+d}}+9\,{\frac{e{c}^{4}}{{b}^{2} \left ( be-cd \right ) ^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-4\,{\frac{{c}^{5}d}{{b}^{3} \left ( be-cd \right ) ^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{b}^{2}{d}^{3}x}\sqrt{ex+d}}+5\,{\frac{e}{{b}^{2}{d}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{c}{{b}^{3}{d}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x)

[Out]

-2/3*e^3/d^2/(b*e-c*d)^2/(e*x+d)^(3/2)-4*e^4/d^3/(b*e-c*d)^3/(e*x+d)^(1/2)*b+8*e
^3/d^2/(b*e-c*d)^3/(e*x+d)^(1/2)*c+e*c^4/b^2/(b*e-c*d)^3*(e*x+d)^(1/2)/(c*e*x+b*
e)+9*e*c^4/b^2/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)
*c)^(1/2))-4*c^5/b^3/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*
e-c*d)*c)^(1/2))*d-1/b^2/d^3*(e*x+d)^(1/2)/x+5*e/b^2/d^(7/2)*arctanh((e*x+d)^(1/
2)/d^(1/2))+4/b^3/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.57696, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/6*(3*((4*c^5*d^4*e - 9*b*c^4*d^3*e^2)*x^3 + (4*c^5*d^5 - 5*b*c^4*d^4*e - 9*b^
2*c^3*d^3*e^2)*x^2 + (4*b*c^4*d^5 - 9*b^2*c^3*d^4*e)*x)*sqrt(e*x + d)*sqrt(d)*sq
rt(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*((4*c^5*d^4*e - 7*b*c^4*d^3*e^2 - 3*b^2*c^3*d^2*e^3
+ 11*b^3*c^2*d*e^4 - 5*b^4*c*e^5)*x^3 + (4*c^5*d^5 - 3*b*c^4*d^4*e - 10*b^2*c^3
*d^3*e^2 + 8*b^3*c^2*d^2*e^3 + 6*b^4*c*d*e^4 - 5*b^5*e^5)*x^2 + (4*b*c^4*d^5 - 7
*b^2*c^3*d^4*e - 3*b^3*c^2*d^3*e^2 + 11*b^4*c*d^2*e^3 - 5*b^5*d*e^4)*x)*sqrt(e*x
+ d)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x) - 2*(3*b^2*c^3*d^5 - 9*b^
3*c^2*d^4*e + 9*b^4*c*d^3*e^2 - 3*b^5*d^2*e^3 + 3*(2*b*c^4*d^3*e^2 - 3*b^2*c^3*d
^2*e^3 + 11*b^3*c^2*d*e^4 - 5*b^4*c*e^5)*x^3 + (12*b*c^4*d^4*e - 15*b^2*c^3*d^3*
e^2 + 35*b^3*c^2*d^2*e^3 + 13*b^4*c*d*e^4 - 15*b^5*e^5)*x^2 + (6*b*c^4*d^5 - 3*b
^2*c^3*d^4*e - 9*b^3*c^2*d^3*e^2 + 41*b^4*c*d^2*e^3 - 20*b^5*d*e^4)*x)*sqrt(d))/
(((b^3*c^4*d^6*e - 3*b^4*c^3*d^5*e^2 + 3*b^5*c^2*d^4*e^3 - b^6*c*d^3*e^4)*x^3 +
(b^3*c^4*d^7 - 2*b^4*c^3*d^6*e + 2*b^6*c*d^4*e^3 - b^7*d^3*e^4)*x^2 + (b^4*c^3*d
^7 - 3*b^5*c^2*d^6*e + 3*b^6*c*d^5*e^2 - b^7*d^4*e^3)*x)*sqrt(e*x + d)*sqrt(d)),
-1/6*(6*((4*c^5*d^4*e - 9*b*c^4*d^3*e^2)*x^3 + (4*c^5*d^5 - 5*b*c^4*d^4*e - 9*b
^2*c^3*d^3*e^2)*x^2 + (4*b*c^4*d^5 - 9*b^2*c^3*d^4*e)*x)*sqrt(e*x + d)*sqrt(d)*s
qrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c))
- 3*((4*c^5*d^4*e - 7*b*c^4*d^3*e^2 - 3*b^2*c^3*d^2*e^3 + 11*b^3*c^2*d*e^4 - 5*b
^4*c*e^5)*x^3 + (4*c^5*d^5 - 3*b*c^4*d^4*e - 10*b^2*c^3*d^3*e^2 + 8*b^3*c^2*d^2*
e^3 + 6*b^4*c*d*e^4 - 5*b^5*e^5)*x^2 + (4*b*c^4*d^5 - 7*b^2*c^3*d^4*e - 3*b^3*c^
2*d^3*e^2 + 11*b^4*c*d^2*e^3 - 5*b^5*d*e^4)*x)*sqrt(e*x + d)*log(((e*x + 2*d)*sq
rt(d) + 2*sqrt(e*x + d)*d)/x) + 2*(3*b^2*c^3*d^5 - 9*b^3*c^2*d^4*e + 9*b^4*c*d^3
*e^2 - 3*b^5*d^2*e^3 + 3*(2*b*c^4*d^3*e^2 - 3*b^2*c^3*d^2*e^3 + 11*b^3*c^2*d*e^4
- 5*b^4*c*e^5)*x^3 + (12*b*c^4*d^4*e - 15*b^2*c^3*d^3*e^2 + 35*b^3*c^2*d^2*e^3
+ 13*b^4*c*d*e^4 - 15*b^5*e^5)*x^2 + (6*b*c^4*d^5 - 3*b^2*c^3*d^4*e - 9*b^3*c^2*
d^3*e^2 + 41*b^4*c*d^2*e^3 - 20*b^5*d*e^4)*x)*sqrt(d))/(((b^3*c^4*d^6*e - 3*b^4*
c^3*d^5*e^2 + 3*b^5*c^2*d^4*e^3 - b^6*c*d^3*e^4)*x^3 + (b^3*c^4*d^7 - 2*b^4*c^3*
d^6*e + 2*b^6*c*d^4*e^3 - b^7*d^3*e^4)*x^2 + (b^4*c^3*d^7 - 3*b^5*c^2*d^6*e + 3*
b^6*c*d^5*e^2 - b^7*d^4*e^3)*x)*sqrt(e*x + d)*sqrt(d)), 1/6*(3*((4*c^5*d^4*e - 9
*b*c^4*d^3*e^2)*x^3 + (4*c^5*d^5 - 5*b*c^4*d^4*e - 9*b^2*c^3*d^3*e^2)*x^2 + (4*b
*c^4*d^5 - 9*b^2*c^3*d^4*e)*x)*sqrt(e*x + d)*sqrt(-d)*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))
- 6*((4*c^5*d^4*e - 7*b*c^4*d^3*e^2 - 3*b^2*c^3*d^2*e^3 + 11*b^3*c^2*d*e^4 - 5*
b^4*c*e^5)*x^3 + (4*c^5*d^5 - 3*b*c^4*d^4*e - 10*b^2*c^3*d^3*e^2 + 8*b^3*c^2*d^2
*e^3 + 6*b^4*c*d*e^4 - 5*b^5*e^5)*x^2 + (4*b*c^4*d^5 - 7*b^2*c^3*d^4*e - 3*b^3*c
^2*d^3*e^2 + 11*b^4*c*d^2*e^3 - 5*b^5*d*e^4)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x
+ d)*sqrt(-d))) - 2*(3*b^2*c^3*d^5 - 9*b^3*c^2*d^4*e + 9*b^4*c*d^3*e^2 - 3*b^5*
d^2*e^3 + 3*(2*b*c^4*d^3*e^2 - 3*b^2*c^3*d^2*e^3 + 11*b^3*c^2*d*e^4 - 5*b^4*c*e^
5)*x^3 + (12*b*c^4*d^4*e - 15*b^2*c^3*d^3*e^2 + 35*b^3*c^2*d^2*e^3 + 13*b^4*c*d*
e^4 - 15*b^5*e^5)*x^2 + (6*b*c^4*d^5 - 3*b^2*c^3*d^4*e - 9*b^3*c^2*d^3*e^2 + 41*
b^4*c*d^2*e^3 - 20*b^5*d*e^4)*x)*sqrt(-d))/(((b^3*c^4*d^6*e - 3*b^4*c^3*d^5*e^2
+ 3*b^5*c^2*d^4*e^3 - b^6*c*d^3*e^4)*x^3 + (b^3*c^4*d^7 - 2*b^4*c^3*d^6*e + 2*b^
6*c*d^4*e^3 - b^7*d^3*e^4)*x^2 + (b^4*c^3*d^7 - 3*b^5*c^2*d^6*e + 3*b^6*c*d^5*e^
2 - b^7*d^4*e^3)*x)*sqrt(e*x + d)*sqrt(-d)), -1/3*(3*((4*c^5*d^4*e - 9*b*c^4*d^3
*e^2)*x^3 + (4*c^5*d^5 - 5*b*c^4*d^4*e - 9*b^2*c^3*d^3*e^2)*x^2 + (4*b*c^4*d^5 -
9*b^2*c^3*d^4*e)*x)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d -
b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + 3*((4*c^5*d^4*e - 7*b*c^4*d^3*e^2
- 3*b^2*c^3*d^2*e^3 + 11*b^3*c^2*d*e^4 - 5*b^4*c*e^5)*x^3 + (4*c^5*d^5 - 3*b*c^
4*d^4*e - 10*b^2*c^3*d^3*e^2 + 8*b^3*c^2*d^2*e^3 + 6*b^4*c*d*e^4 - 5*b^5*e^5)*x^
2 + (4*b*c^4*d^5 - 7*b^2*c^3*d^4*e - 3*b^3*c^2*d^3*e^2 + 11*b^4*c*d^2*e^3 - 5*b^
5*d*e^4)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt(-d))) + (3*b^2*c^3*d^5 -
9*b^3*c^2*d^4*e + 9*b^4*c*d^3*e^2 - 3*b^5*d^2*e^3 + 3*(2*b*c^4*d^3*e^2 - 3*b^2*c
^3*d^2*e^3 + 11*b^3*c^2*d*e^4 - 5*b^4*c*e^5)*x^3 + (12*b*c^4*d^4*e - 15*b^2*c^3*
d^3*e^2 + 35*b^3*c^2*d^2*e^3 + 13*b^4*c*d*e^4 - 15*b^5*e^5)*x^2 + (6*b*c^4*d^5 -
3*b^2*c^3*d^4*e - 9*b^3*c^2*d^3*e^2 + 41*b^4*c*d^2*e^3 - 20*b^5*d*e^4)*x)*sqrt(
-d))/(((b^3*c^4*d^6*e - 3*b^4*c^3*d^5*e^2 + 3*b^5*c^2*d^4*e^3 - b^6*c*d^3*e^4)*x
^3 + (b^3*c^4*d^7 - 2*b^4*c^3*d^6*e + 2*b^6*c*d^4*e^3 - b^7*d^3*e^4)*x^2 + (b^4*
c^3*d^7 - 3*b^5*c^2*d^6*e + 3*b^6*c*d^5*e^2 - b^7*d^4*e^3)*x)*sqrt(e*x + d)*sqrt
(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (b + c x\right )^{2} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**2,x)

[Out]

Integral(1/(x**2*(b + c*x)**2*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.234888, size = 649, normalized size = 2.34 \[ \frac{{\left (4 \, c^{5} d - 9 \, b c^{4} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{4} d^{3} e - 2 \, \sqrt{x e + d} c^{4} d^{4} e - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{3} d^{2} e^{2} + 4 \, \sqrt{x e + d} b c^{3} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{2} d e^{3} - 6 \, \sqrt{x e + d} b^{2} c^{2} d^{2} e^{3} -{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c e^{4} + 4 \, \sqrt{x e + d} b^{3} c d e^{4} - \sqrt{x e + d} b^{4} e^{5}}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )}{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}} - \frac{2 \,{\left (12 \,{\left (x e + d\right )} c d e^{3} + c d^{2} e^{3} - 6 \,{\left (x e + d\right )} b e^{4} - b d e^{4}\right )}}{3 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} - \frac{{\left (4 \, c d + 5 \, b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

(4*c^5*d - 9*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)) - (2*(x*e +
d)^(3/2)*c^4*d^3*e - 2*sqrt(x*e + d)*c^4*d^4*e - 3*(x*e + d)^(3/2)*b*c^3*d^2*e^2
+ 4*sqrt(x*e + d)*b*c^3*d^3*e^2 + 3*(x*e + d)^(3/2)*b^2*c^2*d*e^3 - 6*sqrt(x*e
+ d)*b^2*c^2*d^2*e^3 - (x*e + d)^(3/2)*b^3*c*e^4 + 4*sqrt(x*e + d)*b^3*c*d*e^4 -
sqrt(x*e + d)*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*(12*(x*e + d)*c*d*e^3 + c*d^2*e^3 - 6*(x*e + d)*b*e^4 - 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)) - (4*c*d + 5*b*e)
*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^3)