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

Optimal. Leaf size=295 \[ -\frac{\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{5/2} d^{7/2} e^{5/2}}+\frac{\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d x^4}-\frac{\left (\frac{3 c}{a e}-\frac{5 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 x^3} \]

[Out]

((c*d^2 - a*e^2)*(3*c*d^2 + 5*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*a^2*d^3*e^2*x^2) - (a*d*e + (c*d^2 + a*e^2)*
x + c*d*e*x^2)^(3/2)/(4*d*x^4) - (((3*c)/(a*e) - (5*e)/d^2)*(a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2)^(3/2))/(24*x^3) - ((c*d^2 - a*e^2)^3*(3*c*d^2 + 5*a*e^2)*Arc
Tanh[(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])])/(128*a^(5/2)*d^(7/2)*e^(5/2))

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Rubi [A]  time = 1.07445, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{5/2} d^{7/2} e^{5/2}}+\frac{\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d x^4}-\frac{\left (\frac{3 c}{a e}-\frac{5 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 x^3} \]

Antiderivative was successfully verified.

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

[Out]

((c*d^2 - a*e^2)*(3*c*d^2 + 5*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*a^2*d^3*e^2*x^2) - (a*d*e + (c*d^2 + a*e^2)*
x + c*d*e*x^2)^(3/2)/(4*d*x^4) - (((3*c)/(a*e) - (5*e)/d^2)*(a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2)^(3/2))/(24*x^3) - ((c*d^2 - a*e^2)^3*(3*c*d^2 + 5*a*e^2)*Arc
Tanh[(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])])/(128*a^(5/2)*d^(7/2)*e^(5/2))

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Rubi in Sympy [A]  time = 106.171, size = 274, normalized size = 0.93 \[ - \frac{\left (2 a d e + x \left (a e^{2} + c d^{2}\right )\right ) \left (\frac{5 e^{2}}{64 d^{3}} - \frac{c}{32 a d} - \frac{3 c^{2} d}{64 a^{2} e^{2}}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{x^{2}} + \frac{\left (\frac{5 e}{24 d^{2}} - \frac{c}{8 a e}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{x^{3}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 d x^{4}} + \frac{\left (a e^{2} - c d^{2}\right )^{3} \left (5 a e^{2} + 3 c d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{128 a^{\frac{5}{2}} d^{\frac{7}{2}} e^{\frac{5}{2}}} \]

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)**(3/2)/x**5/(e*x+d),x)

[Out]

-(2*a*d*e + x*(a*e**2 + c*d**2))*(5*e**2/(64*d**3) - c/(32*a*d) - 3*c**2*d/(64*a
**2*e**2))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/x**2 + (5*e/(24*d**2)
- c/(8*a*e))*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/x**3 - (a*d*e + c
*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(4*d*x**4) + (a*e**2 - c*d**2)**3*(5*a*e
**2 + 3*c*d**2)*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))))/(128*a**(5/2)*d**(7/2)*e**(5/2
))

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Mathematica [A]  time = 0.468948, size = 323, normalized size = 1.09 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (-2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^3 e^3 \left (48 d^3+8 d^2 e x-10 d e^2 x^2+15 e^3 x^3\right )+a^2 c d^2 e^2 x \left (72 d^2+20 d e x-31 e^2 x^2\right )+3 a c^2 d^4 e x^2 (2 d+3 e x)-9 c^3 d^6 x^3\right )+3 x^4 \log (x) \left (c d^2-a e^2\right )^3 \left (5 a e^2+3 c d^2\right )-3 x^4 \left (c d^2-a e^2\right )^3 \left (5 a e^2+3 c d^2\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )\right )}{384 a^{5/2} d^{7/2} e^{5/2} x^4 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*S
qrt[d + e*x]*(-9*c^3*d^6*x^3 + 3*a*c^2*d^4*e*x^2*(2*d + 3*e*x) + a^2*c*d^2*e^2*x
*(72*d^2 + 20*d*e*x - 31*e^2*x^2) + a^3*e^3*(48*d^3 + 8*d^2*e*x - 10*d*e^2*x^2 +
15*e^3*x^3)) + 3*(c*d^2 - a*e^2)^3*(3*c*d^2 + 5*a*e^2)*x^4*Log[x] - 3*(c*d^2 -
a*e^2)^3*(3*c*d^2 + 5*a*e^2)*x^4*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)]))/(384*a^(5/2)*d^(7/2)*e^(5/2)*x^4*S
qrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.036, size = 2427, normalized size = 8.2 \[ \text{result too large to display} \]

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)^(3/2)/x^5/(e*x+d),x)

[Out]

1/16/d^6*e^9*a^3/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)+3/16/d^2*e^5*a*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)-1/16/d^6*e^9*a^3/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)-3/16/d^2*e^5*a*
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)+1/64*d^2/a^4/e^3*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)*x-1/64*d/a^4/e^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3+1/8/d/a^2/e^
2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-3/64/d/a*e^2*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*x*c^2-13/48/d^2/a^2/e/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(5/2)*c+3/64*d^3/a^3/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^4+19/192/d
/a^3/e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-3/128*d^5/a^2/e^2/(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^4-3/32/d*a*e^4/(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-133/192/d^4/a*e^3*c*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-91/192/d^2/a^2*e*c^2*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(3/2)*x-1/16*e^3*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)-5/64/d^4*a*e
^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/32/d^2*e^3*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)*c+11/24/d^3/a/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-29/96/
d/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2-1/32/a*e*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*c^2+1/16*e^3*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)+1/8/d^2*e^3*c*(c*d*e*(
x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/3/d^5*e^4*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)
*(x+d/e))^(3/2)-23/64/d^5*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/4/d^5*e^
6*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3/16/d^4*e^7*a^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)+1/4/d^3*e^4*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/
2)*x+1/4/d^5*e^6*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+1/8/d^6*e^7*a^2/c*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/16/d^4*e^7*a^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)-1/
32/a^3/e^3/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-19/192/a^3/e*c^3*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+91/192/d^3/a^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(5/2)*c-21/64/d^3*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c+1/32*
d^3/a/(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^3+3/64*d*e^2/(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+5/64*d/a^2*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3-53/96/d^3/a*e^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*c+1/32*d^2/a^2/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c
^3+133/192/d^5/a*e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+5/128/d^3*a^2*e^6
/(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)-5/96*d/a^3/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^
3+1/64*d^3/a^4/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^4+3/64*d^4/a^3/e^3*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^4-59/96/d^4/a*e/x^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(5/2)-1/4/d^2/a/e/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-1/
8/d^6*e^7*a^2/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.93651, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} x^{4} \log \left (\frac{4 \,{\left (2 \, a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{a d e}}{x^{2}}\right ) + 4 \,{\left (48 \, a^{3} d^{3} e^{3} -{\left (9 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 31 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )} x^{3} + 2 \,{\left (3 \, a c^{2} d^{5} e + 10 \, a^{2} c d^{3} e^{3} - 5 \, a^{3} d e^{5}\right )} x^{2} + 8 \,{\left (9 \, a^{2} c d^{4} e^{2} + a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{a d e}}{768 \, \sqrt{a d e} a^{2} d^{3} e^{2} x^{4}}, -\frac{3 \,{\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} a d e}\right ) + 2 \,{\left (48 \, a^{3} d^{3} e^{3} -{\left (9 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 31 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )} x^{3} + 2 \,{\left (3 \, a c^{2} d^{5} e + 10 \, a^{2} c d^{3} e^{3} - 5 \, a^{3} d e^{5}\right )} x^{2} + 8 \,{\left (9 \, a^{2} c d^{4} e^{2} + a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-a d e}}{384 \, \sqrt{-a d e} a^{2} d^{3} e^{2} x^{4}}\right ] \]

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

[Out]

[-1/768*(3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 -
5*a^4*e^8)*x^4*log((4*(2*a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)*sqrt(c*d*e*x^
2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2
*e^4)*x^2 + 8*(a*c*d^3*e + a^2*d*e^3)*x)*sqrt(a*d*e))/x^2) + 4*(48*a^3*d^3*e^3 -
(9*c^3*d^6 - 9*a*c^2*d^4*e^2 + 31*a^2*c*d^2*e^4 - 15*a^3*e^6)*x^3 + 2*(3*a*c^2*
d^5*e + 10*a^2*c*d^3*e^3 - 5*a^3*d*e^5)*x^2 + 8*(9*a^2*c*d^4*e^2 + a^3*d^2*e^4)*
x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e))/(sqrt(a*d*e)*a^2*d^3
*e^2*x^4), -1/384*(3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 + 12*a^3*c
*d^2*e^6 - 5*a^4*e^8)*x^4*arctan(1/2*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/
(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*a*d*e)) + 2*(48*a^3*d^3*e^3 - (9*c^
3*d^6 - 9*a*c^2*d^4*e^2 + 31*a^2*c*d^2*e^4 - 15*a^3*e^6)*x^3 + 2*(3*a*c^2*d^5*e
+ 10*a^2*c*d^3*e^3 - 5*a^3*d*e^5)*x^2 + 8*(9*a^2*c*d^4*e^2 + a^3*d^2*e^4)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e))/(sqrt(-a*d*e)*a^2*d^3*e^2
*x^4)]

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

[Out]

Exception raised: TypeError