3.463 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx\)

Optimal. Leaf size=339 \[ \frac{3 \sqrt{c} \sqrt{d} \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{e}}-\frac{3 \sqrt{a} \sqrt{e} \left (a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{d}}-\frac{(a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 x^2}-\frac{3 \left (a e \left (a e^2+3 c d^2\right )-c d x \left (3 a e^2+c d^2\right )\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 x} \]

[Out]

(-3*(a*e*(3*c*d^2 + a*e^2) - c*d*(c*d^2 + 3*a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])/(4*x) - ((a*e - c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(3/2))/(2*x^2) + (3*Sqrt[c]*Sqrt[d]*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*Arc
Tanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])])/(8*Sqrt[e]) - (3*Sqrt[a]*Sqrt[e]*(5*c^2*d^4 + 10*a*c*
d^2*e^2 + a^2*e^4)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt
[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*Sqrt[d])

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Rubi [A]  time = 1.14973, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{3 \sqrt{c} \sqrt{d} \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{e}}-\frac{3 \sqrt{a} \sqrt{e} \left (a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{d}}-\frac{(a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 x^2}-\frac{3 \left (a e \left (a e^2+3 c d^2\right )-c d x \left (3 a e^2+c d^2\right )\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)),x]

[Out]

(-3*(a*e*(3*c*d^2 + a*e^2) - c*d*(c*d^2 + 3*a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])/(4*x) - ((a*e - c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(3/2))/(2*x^2) + (3*Sqrt[c]*Sqrt[d]*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*Arc
Tanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])])/(8*Sqrt[e]) - (3*Sqrt[a]*Sqrt[e]*(5*c^2*d^4 + 10*a*c*
d^2*e^2 + a^2*e^4)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt
[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*Sqrt[d])

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Rubi in Sympy [A]  time = 127.867, size = 338, normalized size = 1. \[ - \frac{3 \sqrt{a} \sqrt{e} \left (a^{2} e^{4} + 10 a c d^{2} e^{2} + 5 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 \sqrt{d}} + \frac{3 \sqrt{c} \sqrt{d} \left (5 a^{2} e^{4} + 10 a c d^{2} e^{2} + c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 \sqrt{e}} - \frac{3 \left (2 a e \left (a e^{2} + 3 c d^{2}\right ) - 2 c d x \left (3 a e^{2} + c d^{2}\right )\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 x} - \frac{\left (2 a e - 2 c d x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**3/(e*x+d),x)

[Out]

-3*sqrt(a)*sqrt(e)*(a**2*e**4 + 10*a*c*d**2*e**2 + 5*c**2*d**4)*atanh((2*a*d*e +
x*(a*e**2 + c*d**2))/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*
e**2 + c*d**2))))/(8*sqrt(d)) + 3*sqrt(c)*sqrt(d)*(5*a**2*e**4 + 10*a*c*d**2*e**
2 + c**2*d**4)*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sq
rt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(8*sqrt(e)) - 3*(2*a*e*(a*e**2 +
3*c*d**2) - 2*c*d*x*(3*a*e**2 + c*d**2))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c
*d**2))/(8*x) - (2*a*e - 2*c*d*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3
/2)/(4*x**2)

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Mathematica [A]  time = 0.969937, size = 361, normalized size = 1.06 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (-2 \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^2 e^2 (2 d+5 e x)+9 a c d e x (d-e x)-c^2 d^2 x^2 (5 d+2 e x)\right )+3 \sqrt{a} e x^2 \log (x) \left (a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right )-3 \sqrt{a} e x^2 \left (a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )+3 \sqrt{c} d x^2 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )\right )}{8 \sqrt{d} \sqrt{e} x^2 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)),x]

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-2*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d +
e*x]*(9*a*c*d*e*x*(d - e*x) - c^2*d^2*x^2*(5*d + 2*e*x) + a^2*e^2*(2*d + 5*e*x))
+ 3*Sqrt[a]*e*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*x^2*Log[x] - 3*Sqrt[a]*e*(
5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*x^2*Log[c*d^2*x + 2*Sqrt[a]*Sqrt[d]*Sqrt[e
]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + a*e*(2*d + e*x)] + 3*Sqrt[c]*d*(c^2*d^4 + 10
*a*c*d^2*e^2 + 5*a^2*e^4)*x^2*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c
*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)]))/(8*Sqrt[d]*Sqrt[e]*x^2*Sqrt[(a*e + c*d*
x)*(d + e*x)])

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Maple [B]  time = 0.029, size = 2688, normalized size = 7.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d),x)

[Out]

3/64/d^4*e^7*a^3/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3/256/d^5*e^1
0*a^5/c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+
(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+15/128*d*e^4*a^2*c*ln((1/2*a*e^2-1/2
*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2
))/(c*d*e)^(1/2)-15/256*d^3*e^2*a*c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*
d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+15/256/d
^3*e^8*a^4/c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)
^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-3/64/d^4*e^7*a^3/c*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)*x+3/256/d^5*e^10*a^5/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e
*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-15/256/
d^3*e^8*a^4/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-15/4*d^2*a^2*e^3/(a*d*e)^(1/2)*ln((2*a*d*e+(
a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c-15/
8*d^4*a*e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^2+225/128*d*a^2*e^4*c*ln((1/2*a*e^2+1/2*c*d^2+
c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)+97
5/256*d^3*a*e^2*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-3/4/d/a^2/e^2/x*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(7/2)*c-1/4/d^2/a*e*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+1/4/
d^3/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+3/256*d^5*c^3*ln((1/2*a*e^2-1/2*
c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)
)/(c*d*e)^(1/2)+1/d/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-3/8*a^3*e^5/(a*d
*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2))/x)+93/256*d^5*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)*c^3+3/4/d*a^2*e^4*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/4/d^2*a*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)-1/20/d^3*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+387/128*d^3*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2-1/5/d^3*e^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/
e))^(5/2)+1/16*e*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/128*d^3*c^2*(
c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+31/16*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)*c-3/64/d^3*e^6*a^3/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+
1/8/d*e^2*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-1/16/d^4*e^5*a^2/c*(
c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-9/64/d^2*e^5*a^2*(c*d*e*(x+d/e)^2+(
a*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/128/d^5*e^8*a^4/c^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^
2)*(x+d/e))^(1/2)+3/64*d*e^2*a*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1
5/128/d*e^6*a^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d
/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-3/64*d^2*e*c^2*(c*d*e*(x+d/e)^
2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/64/d^3*e^6*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)+1/8/d^3*e^4*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+1/16/d^4*e^5
*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+9/64/d^2*e^5*a^2*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*x-3/128/d^5*e^8*a^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)+39/64*a*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c+3/4/a^2/e*c^2*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+333/64*d*a*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*c+5/4*d^2/a/e*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/8/d*e
^2*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+147/64*d^2*e*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)*x*c^2+5/4*d/a*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+
15/128/d*a^3*e^6*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)+3/4*d/a^2/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(5/2)*c^2-1/2/d^2/a/e/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-1/8/d^3
*e^4*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x+9/64*e^3*a*c*(c*d*e*(x+d/
e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x

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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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.65457, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^3),x, algorithm="fricas")

[Out]

[1/16*(3*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(c*d/e)*x^2*log(8*c^2*d^2*e^
2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sq
rt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3
)*x) + 3*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*sqrt(a*e/d)*x^2*log((8*a^2*d^2*e
^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(2*a*d^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a*e/d) + 8*(a*c*d^3*e + a^2*
d*e^3)*x)/x^2) + 4*(2*c^2*d^2*e*x^3 - 2*a^2*d*e^2 + (5*c^2*d^3 + 9*a*c*d*e^2)*x^
2 - (9*a*c*d^2*e + 5*a^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/x^
2, 1/16*(6*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(-c*d/e)*x^2*arctan(1/2*(2
*c*d*e*x + c*d^2 + a*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d
/e)*e)) + 3*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*sqrt(a*e/d)*x^2*log((8*a^2*d^
2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*
d^2 + a*e^2)*x)*(2*a*d^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a*e/d) + 8*(a*c*d^3*e + a
^2*d*e^3)*x)/x^2) + 4*(2*c^2*d^2*e*x^3 - 2*a^2*d*e^2 + (5*c^2*d^3 + 9*a*c*d*e^2)
*x^2 - (9*a*c*d^2*e + 5*a^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))
/x^2, -1/16*(6*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*sqrt(-a*e/d)*x^2*arctan(1/
2*(2*a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*d*s
qrt(-a*e/d))) - 3*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(c*d/e)*x^2*log(8*c
^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*(2*c*d*e^2*x + c*d^2*e +
a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e +
a*c*d*e^3)*x) - 4*(2*c^2*d^2*e*x^3 - 2*a^2*d*e^2 + (5*c^2*d^3 + 9*a*c*d*e^2)*x^2
- (9*a*c*d^2*e + 5*a^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/x^2
, -1/8*(3*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*sqrt(-a*e/d)*x^2*arctan(1/2*(2*
a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*d*sqrt(-
a*e/d))) - 3*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(-c*d/e)*x^2*arctan(1/2*
(2*c*d*e*x + c*d^2 + a*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c
*d/e)*e)) - 2*(2*c^2*d^2*e*x^3 - 2*a^2*d*e^2 + (5*c^2*d^3 + 9*a*c*d*e^2)*x^2 - (
9*a*c*d^2*e + 5*a^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/x^2]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**3/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError