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

Optimal. Leaf size=515 \[ \frac{5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac{2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (c d^2-a e^2\right ) \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\left (-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left (-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right )-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}-\frac{2 d x^4 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*d*x^4*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (2*x^2*(a*d*e*(c*d^2 - a*e^2)*(
7*c^2*d^4 - 12*a*c*d^2*e^2 - 3*a^2*e^4) + (c*d^2 - a*e^2)*(7*c^3*d^6 - 11*a*c^2*
d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((105*c^4*d^8 - 190*a*c^3*d^6*e^2 + 36*a^2*
c^2*d^4*e^4 + 30*a^3*c*d^2*e^6 - 45*a^4*e^8 - 2*c*d*e*(35*c^3*d^6 - 61*a*c^2*d^4
*e^2 + 9*a^2*c*d^2*e^4 - 15*a^3*e^6)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2])/(12*c^3*d^3*e^4*(c*d^2 - a*e^2)^3) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*
e^4)*ArcTanh[(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*c^(7/2)*d^(7/2)*e^(9/2))

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Rubi [A]  time = 1.6607, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac{2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (c d^2-a e^2\right ) \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\left (-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left (-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right )-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}-\frac{2 d x^4 (a e+c d x)}{3 e \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*x^4*(a*e + c*d*x))/(3*e*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2)) - (2*x^2*(a*d*e*(c*d^2 - a*e^2)*(7*c^2*d^4 - 12*a*c*d^2*e^2 - 3*a^2
*e^4) + (c*d^2 - a*e^2)*(7*c^3*d^6 - 11*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^
6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
- ((105*c^4*d^8 - 190*a*c^3*d^6*e^2 + 36*a^2*c^2*d^4*e^4 + 30*a^3*c*d^2*e^6 - 4
5*a^4*e^8 - 2*c*d*e*(35*c^3*d^6 - 61*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 15*a^3*e^
6)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*c^3*d^3*e^4*(c*d^2 - a*e^
2)^3) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(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*c^(7/2)*d^(7/2)*e^(9/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.08267, size = 296, normalized size = 0.57 \[ \frac{\frac{2 (d+e x)^2 (a e+c d x)^2 \left (\frac{24 a^5 e^9}{c^3 \left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{3 \left (7 a e^2+11 c d^2\right )}{c^3}+\frac{8 d^8}{(d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{40 \left (3 a d^7 e^2-2 c d^9\right )}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{6 d e x}{c^2}\right )}{3 d^3 e^4}+\frac{5 (d+e x)^{3/2} \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) (a e+c d x)^{3/2} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{9/2}}}{8 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*(a*e + c*d*x)^2*(d + e*x)^2*((-3*(11*c*d^2 + 7*a*e^2))/c^3 + (6*d*e*x)/c^2 +
(24*a^5*e^9)/(c^3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)) + (8*d^8)/((c*d^2 - a*e^2)^2
*(d + e*x)^2) + (40*(-2*c*d^9 + 3*a*d^7*e^2))/((c*d^2 - a*e^2)^3*(d + e*x))))/(3
*d^3*e^4) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(a*e + c*d*x)^(3/2)*(d +
e*x)^(3/2)*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)])/(c^(7/2)*d^(7/2)*e^(9/2)))/(8*((a*e + c*d*x)*(d + e*x))^(3/
2))

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Maple [B]  time = 0.035, size = 1680, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/c*d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)*x*a^2+11/2/e^2*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)*x*a+15/4/e^2/c^2/d*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)*a-15/4/e^2/c^2/d*x/(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+51/8/e^4*c*d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d
^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+15/16*e^5/c^4/d^4/(-a^2*e^4+2*a*c*
d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+35/16*e^3/c^3/d^2/(
-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+21/8
/e/c*d^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)*a^2-5/4/e/c^2/d^2*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-16/3*d^7/e^4*c
^2/(a*e^2-c*d^2)^3/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-8/3*d^6/e^3*c
/(a*e^2-c*d^2)^3/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*a+51/16/e^5/c*d^2
/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-7/16/e^3/c^2/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)*a-9/4/e^3/c*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+15/8*e^4/c
^3/d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*
x*a^4+5/2*e^2/c^2/d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)*x*a^3+95/16/e^3*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)*a+51/16/e^5*c*d^6/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/8/e^4/c*d*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)+15/16*e/
c^4/d^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-35/8/e^4/c*d*x/(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)+1/2/e^2*x^3/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2)+5/16/e/c^3/d^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-15/8/c^3/d^3*x/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+2*d^4/e^5*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*
c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+2/3*d^5/e^6/(
a*e^2-c*d^2)/(x+d/e)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-8/3*d^8/e^5*c
^2/(a*e^2-c*d^2)^3/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+15/8/c^3/d^3*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)*a^2+9/8*e/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.47214, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(4*(105*a*c^4*d^10*e - 190*a^2*c^3*d^8*e^3 + 36*a^3*c^2*d^6*e^5 + 30*a^4*
c*d^4*e^7 - 45*a^5*d^2*e^9 - 6*(c^5*d^8*e^3 - 3*a*c^4*d^6*e^5 + 3*a^2*c^3*d^4*e^
7 - a^3*c^2*d^2*e^9)*x^4 + 3*(7*c^5*d^9*e^2 - 16*a*c^4*d^7*e^4 + 6*a^2*c^3*d^5*e
^6 + 8*a^3*c^2*d^3*e^8 - 5*a^4*c*d*e^10)*x^3 + (140*c^5*d^10*e - 237*a*c^4*d^8*e
^3 + 12*a^2*c^3*d^6*e^5 + 66*a^3*c^2*d^4*e^7 - 45*a^5*e^11)*x^2 + (105*c^5*d^11
- 50*a*c^4*d^9*e^2 - 222*a^2*c^3*d^7*e^4 + 84*a^3*c^2*d^5*e^6 + 45*a^4*c*d^3*e^8
- 90*a^5*d*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) - 1
5*(7*a*c^5*d^12*e - 15*a^2*c^4*d^10*e^3 + 6*a^3*c^3*d^8*e^5 + 2*a^4*c^2*d^6*e^7
+ 3*a^5*c*d^4*e^9 - 3*a^6*d^2*e^11 + (7*c^6*d^11*e^2 - 15*a*c^5*d^9*e^4 + 6*a^2*
c^4*d^7*e^6 + 2*a^3*c^3*d^5*e^8 + 3*a^4*c^2*d^3*e^10 - 3*a^5*c*d*e^12)*x^3 + (14
*c^6*d^12*e - 23*a*c^5*d^10*e^3 - 3*a^2*c^4*d^8*e^5 + 10*a^3*c^3*d^6*e^7 + 8*a^4
*c^2*d^4*e^9 - 3*a^5*c*d^2*e^11 - 3*a^6*e^13)*x^2 + (7*c^6*d^13 - a*c^5*d^11*e^2
- 24*a^2*c^4*d^9*e^4 + 14*a^3*c^3*d^7*e^6 + 7*a^4*c^2*d^5*e^8 + 3*a^5*c*d^3*e^1
0 - 6*a^6*d*e^12)*x)*log(4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2
+ a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d*e)))/((a*c^6*d^11*e^5 - 3*a^2*
c^5*d^9*e^7 + 3*a^3*c^4*d^7*e^9 - a^4*c^3*d^5*e^11 + (c^7*d^10*e^6 - 3*a*c^6*d^8
*e^8 + 3*a^2*c^5*d^6*e^10 - a^3*c^4*d^4*e^12)*x^3 + (2*c^7*d^11*e^5 - 5*a*c^6*d^
9*e^7 + 3*a^2*c^5*d^7*e^9 + a^3*c^4*d^5*e^11 - a^4*c^3*d^3*e^13)*x^2 + (c^7*d^12
*e^4 - a*c^6*d^10*e^6 - 3*a^2*c^5*d^8*e^8 + 5*a^3*c^4*d^6*e^10 - 2*a^4*c^3*d^4*e
^12)*x)*sqrt(c*d*e)), -1/24*(2*(105*a*c^4*d^10*e - 190*a^2*c^3*d^8*e^3 + 36*a^3*
c^2*d^6*e^5 + 30*a^4*c*d^4*e^7 - 45*a^5*d^2*e^9 - 6*(c^5*d^8*e^3 - 3*a*c^4*d^6*e
^5 + 3*a^2*c^3*d^4*e^7 - a^3*c^2*d^2*e^9)*x^4 + 3*(7*c^5*d^9*e^2 - 16*a*c^4*d^7*
e^4 + 6*a^2*c^3*d^5*e^6 + 8*a^3*c^2*d^3*e^8 - 5*a^4*c*d*e^10)*x^3 + (140*c^5*d^1
0*e - 237*a*c^4*d^8*e^3 + 12*a^2*c^3*d^6*e^5 + 66*a^3*c^2*d^4*e^7 - 45*a^5*e^11)
*x^2 + (105*c^5*d^11 - 50*a*c^4*d^9*e^2 - 222*a^2*c^3*d^7*e^4 + 84*a^3*c^2*d^5*e
^6 + 45*a^4*c*d^3*e^8 - 90*a^5*d*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)*sqrt(-c*d*e) - 15*(7*a*c^5*d^12*e - 15*a^2*c^4*d^10*e^3 + 6*a^3*c^3*d^8*e^
5 + 2*a^4*c^2*d^6*e^7 + 3*a^5*c*d^4*e^9 - 3*a^6*d^2*e^11 + (7*c^6*d^11*e^2 - 15*
a*c^5*d^9*e^4 + 6*a^2*c^4*d^7*e^6 + 2*a^3*c^3*d^5*e^8 + 3*a^4*c^2*d^3*e^10 - 3*a
^5*c*d*e^12)*x^3 + (14*c^6*d^12*e - 23*a*c^5*d^10*e^3 - 3*a^2*c^4*d^8*e^5 + 10*a
^3*c^3*d^6*e^7 + 8*a^4*c^2*d^4*e^9 - 3*a^5*c*d^2*e^11 - 3*a^6*e^13)*x^2 + (7*c^6
*d^13 - a*c^5*d^11*e^2 - 24*a^2*c^4*d^9*e^4 + 14*a^3*c^3*d^7*e^6 + 7*a^4*c^2*d^5
*e^8 + 3*a^5*c*d^3*e^10 - 6*a^6*d*e^12)*x)*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2
)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)))/((a*c^6*d^1
1*e^5 - 3*a^2*c^5*d^9*e^7 + 3*a^3*c^4*d^7*e^9 - a^4*c^3*d^5*e^11 + (c^7*d^10*e^6
- 3*a*c^6*d^8*e^8 + 3*a^2*c^5*d^6*e^10 - a^3*c^4*d^4*e^12)*x^3 + (2*c^7*d^11*e^
5 - 5*a*c^6*d^9*e^7 + 3*a^2*c^5*d^7*e^9 + a^3*c^4*d^5*e^11 - a^4*c^3*d^3*e^13)*x
^2 + (c^7*d^12*e^4 - a*c^6*d^10*e^6 - 3*a^2*c^5*d^8*e^8 + 5*a^3*c^4*d^6*e^10 - 2
*a^4*c^3*d^4*e^12)*x)*sqrt(-c*d*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[undef, undef, undef, undef, undef, 1]