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

Optimal. Leaf size=134 \[ \frac{a}{3 b \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{a d+2 b c}{3 b \sqrt{c+d x^3} (b c-a d)^2}-\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} (b c-a d)^{5/2}} \]

[Out]

(2*b*c + a*d)/(3*b*(b*c - a*d)^2*Sqrt[c + d*x^3]) + a/(3*b*(b*c - a*d)*(a + b*x^
3)*Sqrt[c + d*x^3]) - ((2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c
- a*d]])/(3*Sqrt[b]*(b*c - a*d)^(5/2))

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Rubi [A]  time = 0.332811, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{a}{3 b \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{a d+2 b c}{3 b \sqrt{c+d x^3} (b c-a d)^2}-\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*b*c + a*d)/(3*b*(b*c - a*d)^2*Sqrt[c + d*x^3]) + a/(3*b*(b*c - a*d)*(a + b*x^
3)*Sqrt[c + d*x^3]) - ((2*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c
- a*d]])/(3*Sqrt[b]*(b*c - a*d)^(5/2))

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Rubi in Sympy [A]  time = 32.9731, size = 112, normalized size = 0.84 \[ - \frac{a}{3 b \left (a + b x^{3}\right ) \sqrt{c + d x^{3}} \left (a d - b c\right )} + \frac{2 \left (\frac{a d}{2} + b c\right )}{3 b \sqrt{c + d x^{3}} \left (a d - b c\right )^{2}} + \frac{2 \left (\frac{a d}{2} + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 \sqrt{b} \left (a d - b c\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(3*b*(a + b*x**3)*sqrt(c + d*x**3)*(a*d - b*c)) + 2*(a*d/2 + b*c)/(3*b*sqrt(c
+ d*x**3)*(a*d - b*c)**2) + 2*(a*d/2 + b*c)*atan(sqrt(b)*sqrt(c + d*x**3)/sqrt(
a*d - b*c))/(3*sqrt(b)*(a*d - b*c)**(5/2))

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Mathematica [A]  time = 0.292997, size = 111, normalized size = 0.83 \[ \frac{1}{3} \left (\frac{3 a c+a d x^3+2 b c x^3}{\left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)^2}-\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{\sqrt{b} (b c-a d)^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((3*a*c + 2*b*c*x^3 + a*d*x^3)/((b*c - a*d)^2*(a + b*x^3)*Sqrt[c + d*x^3]) - ((2
*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(Sqrt[b]*(b*c -
a*d)^(5/2)))/3

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Maple [C]  time = 0.016, size = 958, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b*(-2/3/(a*d-b*c)/((x^3+c/d)*d)^(1/2)-1/3*I/d^2*b*2^(1/2)*sum(1/(-a*d+b*c)/(a*
d-b*c)*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3
)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)
*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1
/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2
*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*E
llipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))
*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-
I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/(a*
d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^
2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b+a)))-a/b*(-1/3*b/(a*d-b*c)^2*(d*x^3+c)^(1
/2)/(b*x^3+a)-2/3*d/(a*d-b*c)^2/((x^3+c/d)*d)^(1/2)+1/2*I*b/d*2^(1/2)*sum(1/(a*d
-b*c)^3*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/
3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2
)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(
1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+
2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*
EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)
)*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d
-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/(a
*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d
^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b+a)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23321, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right ) + 2 \,{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c\right )} \sqrt{b^{2} c - a b d}}{6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}, -\frac{{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt{d x^{3} + c} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ) -{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c\right )} \sqrt{-b^{2} c + a b d}}{3 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(((2*b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*sqrt(d*x^3 + c)*log(((b*d*x^3 +
2*b*c - a*d)*sqrt(b^2*c - a*b*d) - 2*sqrt(d*x^3 + c)*(b^2*c - a*b*d))/(b*x^3 + a
)) + 2*((2*b*c + a*d)*x^3 + 3*a*c)*sqrt(b^2*c - a*b*d))/((a*b^2*c^2 - 2*a^2*b*c*
d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(b^2*
c - a*b*d)), -1/3*(((2*b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*sqrt(d*x^3 + c)*arc
tan(-(b*c - a*d)/(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d))) - ((2*b*c + a*d)*x^3 +
3*a*c)*sqrt(-b^2*c + a*b*d))/((a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*
a*b^2*c*d + a^2*b*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224442, size = 244, normalized size = 1.82 \[ \frac{\frac{{\left (2 \, b c d + a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (d x^{3} + c\right )} b c d - 2 \, b c^{2} d +{\left (d x^{3} + c\right )} a d^{2} + 2 \, a c d^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b - \sqrt{d x^{3} + c} b c + \sqrt{d x^{3} + c} a d\right )}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((2*b*c*d + a*d^2)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/((b^2*c^2
- 2*a*b*c*d + a^2*d^2)*sqrt(-b^2*c + a*b*d)) + (2*(d*x^3 + c)*b*c*d - 2*b*c^2*d
+ (d*x^3 + c)*a*d^2 + 2*a*c*d^2)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*((d*x^3 + c)^(
3/2)*b - sqrt(d*x^3 + c)*b*c + sqrt(d*x^3 + c)*a*d)))/d