3.2067 \(\int \frac{1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=395 \[ \frac{105 c^3 d^3 e \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{105 c^3 d^3 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{11/2}}-\frac{35 c^2 d^2 e}{8 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{7 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{3 c d}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

1/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) + (3*c*d)/(4*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2)^(3/2)) - (7*c^2*d^2*Sqrt[d + e*x])/(4*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(3/2)) - (35*c^2*d^2*e)/(8*(c*d^2 - a*e^2)^4*Sqrt[d + e
*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3*e*Sqrt[d + e*x])
/(8*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^
3*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d
^2 - a*e^2]*Sqrt[d + e*x])])/(8*(c*d^2 - a*e^2)^(11/2))

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Rubi [A]  time = 0.906745, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{105 c^3 d^3 e \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{105 c^3 d^3 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{11/2}}-\frac{35 c^2 d^2 e}{8 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{7 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{3 c d}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) + (3*c*d)/(4*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2)^(3/2)) - (7*c^2*d^2*Sqrt[d + e*x])/(4*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(3/2)) - (35*c^2*d^2*e)/(8*(c*d^2 - a*e^2)^4*Sqrt[d + e
*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3*e*Sqrt[d + e*x])
/(8*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^
3*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d
^2 - a*e^2]*Sqrt[d + e*x])])/(8*(c*d^2 - a*e^2)^(11/2))

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Rubi in Sympy [A]  time = 168.926, size = 374, normalized size = 0.95 \[ \frac{105 c^{3} d^{3} e^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{8 \left (a e^{2} - c d^{2}\right )^{\frac{11}{2}}} - \frac{105 c^{3} d^{3} e \sqrt{d + e x}}{8 \left (a e^{2} - c d^{2}\right )^{5} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{35 c^{2} d^{2} e}{8 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{7 c^{2} d^{2} \sqrt{d + e x}}{4 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} + \frac{3 c d}{4 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{1}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

105*c**3*d**3*e**(3/2)*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**
2))/(sqrt(d + e*x)*sqrt(a*e**2 - c*d**2)))/(8*(a*e**2 - c*d**2)**(11/2)) - 105*c
**3*d**3*e*sqrt(d + e*x)/(8*(a*e**2 - c*d**2)**5*sqrt(a*d*e + c*d*e*x**2 + x*(a*
e**2 + c*d**2))) - 35*c**2*d**2*e/(8*sqrt(d + e*x)*(a*e**2 - c*d**2)**4*sqrt(a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 7*c**2*d**2*sqrt(d + e*x)/(4*(a*e**2 -
c*d**2)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)) + 3*c*d/(4*sqrt(d
+ e*x)*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2))
- 1/(3*(d + e*x)**(3/2)*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d*
*2))**(3/2))

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Mathematica [A]  time = 1.86531, size = 254, normalized size = 0.64 \[ \frac{(d+e x)^{5/2} \left (\frac{105 c^3 d^3 e^{3/2} (a e+c d x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\left (a e^2-c d^2\right )^{11/2}}+\frac{(a e+c d x)^3 \left (-\frac{192 c^3 d^3 e}{a e+c d x}+\frac{16 c^3 d^3 \left (c d^2-a e^2\right )}{(a e+c d x)^2}-\frac{8 \left (c d^2 e-a e^3\right )^2}{(d+e x)^3}+\frac{34 c d e^2 \left (a e^2-c d^2\right )}{(d+e x)^2}-\frac{123 c^2 d^2 e^2}{d+e x}\right )}{3 \left (a e^2-c d^2\right )^5}\right )}{8 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

((d + e*x)^(5/2)*(((a*e + c*d*x)^3*((16*c^3*d^3*(c*d^2 - a*e^2))/(a*e + c*d*x)^2
- (192*c^3*d^3*e)/(a*e + c*d*x) - (8*(c*d^2*e - a*e^3)^2)/(d + e*x)^3 + (34*c*d
*e^2*(-(c*d^2) + a*e^2))/(d + e*x)^2 - (123*c^2*d^2*e^2)/(d + e*x)))/(3*(-(c*d^2
) + a*e^2)^5) + (105*c^3*d^3*e^(3/2)*(a*e + c*d*x)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[a
*e + c*d*x])/Sqrt[-(c*d^2) + a*e^2]])/(-(c*d^2) + a*e^2)^(11/2)))/(8*((a*e + c*d
*x)*(d + e*x))^(5/2))

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Maple [B]  time = 0.05, size = 930, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(315*arctanh(e*(c*d*x+a*e)^(1/2)/((
a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x^4*c^4*d^4*e^5+315*arctanh(e*(c*d*x+a*
e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^3*a*c^3*d^3*e^6*(c*d*x+a*e)^(1/2)+945*arctan
h(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x^3*c^4*d^5*e^4
+945*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^2*a*c^3*d^4*e^5*(c*d
*x+a*e)^(1/2)+945*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*
e)^(1/2)*x^2*c^4*d^6*e^3-315*((a*e^2-c*d^2)*e)^(1/2)*x^4*c^4*d^4*e^4+945*arctanh
(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*a*c^3*d^5*e^4*(c*d*x+a*e)^(1/2)+
315*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x*c^4
*d^7*e^2-420*((a*e^2-c*d^2)*e)^(1/2)*x^3*a*c^3*d^3*e^5-840*((a*e^2-c*d^2)*e)^(1/
2)*x^3*c^4*d^5*e^3+315*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^
3*d^6*e^3*(c*d*x+a*e)^(1/2)-63*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^2*c^2*d^2*e^6-1134*
((a*e^2-c*d^2)*e)^(1/2)*x^2*a*c^3*d^4*e^4-693*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^4*d^
6*e^2+18*((a*e^2-c*d^2)*e)^(1/2)*x*a^3*c*d*e^7-180*((a*e^2-c*d^2)*e)^(1/2)*x*a^2
*c^2*d^3*e^5-954*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^3*d^5*e^3-144*((a*e^2-c*d^2)*e)^(
1/2)*x*c^4*d^7*e-8*((a*e^2-c*d^2)*e)^(1/2)*a^4*e^8+50*((a*e^2-c*d^2)*e)^(1/2)*a^
3*c*d^2*e^6-165*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^4*e^4-208*((a*e^2-c*d^2)*e)^(1
/2)*a*c^3*d^6*e^2+16*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^8)/(e*x+d)^(7/2)/(c*d*x+a*e)^
2/(a*e^2-c*d^2)^5/((a*e^2-c*d^2)*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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25209, 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*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/48*(315*(c^5*d^5*e^5*x^6 + a^2*c^3*d^7*e^3 + 2*(2*c^5*d^6*e^4 + a*c^4*d^4*e^6
)*x^5 + (6*c^5*d^7*e^3 + 8*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^4 + 4*(c^5*d^8*e^2
+ 3*a*c^4*d^6*e^4 + a^2*c^3*d^4*e^6)*x^3 + (c^5*d^9*e + 8*a*c^4*d^7*e^3 + 6*a^2
*c^3*d^5*e^5)*x^2 + 2*(a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^4)*x)*sqrt(-e/(c*d^2 - a*
e^2))*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d
*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/
(e^2*x^2 + 2*d*e*x + d^2)) + 2*(315*c^4*d^4*e^4*x^4 - 16*c^4*d^8 + 208*a*c^3*d^6
*e^2 + 165*a^2*c^2*d^4*e^4 - 50*a^3*c*d^2*e^6 + 8*a^4*e^8 + 420*(2*c^4*d^5*e^3 +
a*c^3*d^3*e^5)*x^3 + 63*(11*c^4*d^6*e^2 + 18*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x
^2 + 18*(8*c^4*d^7*e + 53*a*c^3*d^5*e^3 + 10*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^5*d^14*e^2 - 5*
a^3*c^4*d^12*e^4 + 10*a^4*c^3*d^10*e^6 - 10*a^5*c^2*d^8*e^8 + 5*a^6*c*d^6*e^10 -
a^7*d^4*e^12 + (c^7*d^12*e^4 - 5*a*c^6*d^10*e^6 + 10*a^2*c^5*d^8*e^8 - 10*a^3*c
^4*d^6*e^10 + 5*a^4*c^3*d^4*e^12 - a^5*c^2*d^2*e^14)*x^6 + 2*(2*c^7*d^13*e^3 - 9
*a*c^6*d^11*e^5 + 15*a^2*c^5*d^9*e^7 - 10*a^3*c^4*d^7*e^9 + 3*a^5*c^2*d^3*e^13 -
a^6*c*d*e^15)*x^5 + (6*c^7*d^14*e^2 - 22*a*c^6*d^12*e^4 + 21*a^2*c^5*d^10*e^6 +
15*a^3*c^4*d^8*e^8 - 40*a^4*c^3*d^6*e^10 + 24*a^5*c^2*d^4*e^12 - 3*a^6*c*d^2*e^
14 - a^7*e^16)*x^4 + 4*(c^7*d^15*e - 2*a*c^6*d^13*e^3 - 4*a^2*c^5*d^11*e^5 + 15*
a^3*c^4*d^9*e^7 - 15*a^4*c^3*d^7*e^9 + 4*a^5*c^2*d^5*e^11 + 2*a^6*c*d^3*e^13 - a
^7*d*e^15)*x^3 + (c^7*d^16 + 3*a*c^6*d^14*e^2 - 24*a^2*c^5*d^12*e^4 + 40*a^3*c^4
*d^10*e^6 - 15*a^4*c^3*d^8*e^8 - 21*a^5*c^2*d^6*e^10 + 22*a^6*c*d^4*e^12 - 6*a^7
*d^2*e^14)*x^2 + 2*(a*c^6*d^15*e - 3*a^2*c^5*d^13*e^3 + 10*a^4*c^3*d^9*e^7 - 15*
a^5*c^2*d^7*e^9 + 9*a^6*c*d^5*e^11 - 2*a^7*d^3*e^13)*x), -1/24*(315*(c^5*d^5*e^5
*x^6 + a^2*c^3*d^7*e^3 + 2*(2*c^5*d^6*e^4 + a*c^4*d^4*e^6)*x^5 + (6*c^5*d^7*e^3
+ 8*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^4 + 4*(c^5*d^8*e^2 + 3*a*c^4*d^6*e^4 + a^
2*c^3*d^4*e^6)*x^3 + (c^5*d^9*e + 8*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5)*x^2 + 2*(
a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^4)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(sqrt(e*x +
d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d^2 - a*e^2)))) - (315
*c^4*d^4*e^4*x^4 - 16*c^4*d^8 + 208*a*c^3*d^6*e^2 + 165*a^2*c^2*d^4*e^4 - 50*a^3
*c*d^2*e^6 + 8*a^4*e^8 + 420*(2*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 63*(11*c^4*d^
6*e^2 + 18*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 18*(8*c^4*d^7*e + 53*a*c^3*d^5
*e^3 + 10*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*
e^2)*x)*sqrt(e*x + d))/(a^2*c^5*d^14*e^2 - 5*a^3*c^4*d^12*e^4 + 10*a^4*c^3*d^10*
e^6 - 10*a^5*c^2*d^8*e^8 + 5*a^6*c*d^6*e^10 - a^7*d^4*e^12 + (c^7*d^12*e^4 - 5*a
*c^6*d^10*e^6 + 10*a^2*c^5*d^8*e^8 - 10*a^3*c^4*d^6*e^10 + 5*a^4*c^3*d^4*e^12 -
a^5*c^2*d^2*e^14)*x^6 + 2*(2*c^7*d^13*e^3 - 9*a*c^6*d^11*e^5 + 15*a^2*c^5*d^9*e^
7 - 10*a^3*c^4*d^7*e^9 + 3*a^5*c^2*d^3*e^13 - a^6*c*d*e^15)*x^5 + (6*c^7*d^14*e^
2 - 22*a*c^6*d^12*e^4 + 21*a^2*c^5*d^10*e^6 + 15*a^3*c^4*d^8*e^8 - 40*a^4*c^3*d^
6*e^10 + 24*a^5*c^2*d^4*e^12 - 3*a^6*c*d^2*e^14 - a^7*e^16)*x^4 + 4*(c^7*d^15*e
- 2*a*c^6*d^13*e^3 - 4*a^2*c^5*d^11*e^5 + 15*a^3*c^4*d^9*e^7 - 15*a^4*c^3*d^7*e^
9 + 4*a^5*c^2*d^5*e^11 + 2*a^6*c*d^3*e^13 - a^7*d*e^15)*x^3 + (c^7*d^16 + 3*a*c^
6*d^14*e^2 - 24*a^2*c^5*d^12*e^4 + 40*a^3*c^4*d^10*e^6 - 15*a^4*c^3*d^8*e^8 - 21
*a^5*c^2*d^6*e^10 + 22*a^6*c*d^4*e^12 - 6*a^7*d^2*e^14)*x^2 + 2*(a*c^6*d^15*e -
3*a^2*c^5*d^13*e^3 + 10*a^4*c^3*d^9*e^7 - 15*a^5*c^2*d^7*e^9 + 9*a^6*c*d^5*e^11
- 2*a^7*d^3*e^13)*x)]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, 2]