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

Optimal. Leaf size=99 \[ \frac{a \sqrt{c+d x^3}}{3 b \left (a+b x^3\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.258734, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{a \sqrt{c+d x^3}}{3 b \left (a+b x^3\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.9063, size = 82, normalized size = 0.83 \[ - \frac{a \sqrt{c + d x^{3}}}{3 b \left (a + b x^{3}\right ) \left (a d - b c\right )} + \frac{2 \left (\frac{a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.125087, size = 99, normalized size = 1. \[ \frac{a \sqrt{c+d x^3}}{3 b \left (a+b x^3\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.016, size = 892, normalized size = 9. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*I/b/d^2*2^(1/2)*sum(1/(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))*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*_alp
ha^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*_a
lpha*(-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/(a*d-b*c)*(d*x^3+c)^(1/2)/(b*x^3+a)-1/6*I/d*2^(1/2)*sum(1/(a*d-b*c)^2*(-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*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225233, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d} a +{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right )}{6 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{3}\right )} \sqrt{b^{2} c - a b d}}, \frac{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d} a -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{3}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(2*sqrt(d*x^3 + c)*sqrt(b^2*c - a*b*d)*a + ((2*b^2*c - a*b*d)*x^3 + 2*a*b*c
- a^2*d)*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)))/((a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^3)*sqrt(
b^2*c - a*b*d)), 1/3*(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d)*a - ((2*b^2*c - a*b*d
)*x^3 + 2*a*b*c - a^2*d)*arctan(-(b*c - a*d)/(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*
d))))/((a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^3)*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21929, size = 157, normalized size = 1.59 \[ \frac{\frac{\sqrt{d x^{3} + c} a d^{2}}{{\left (b^{2} c - a b d\right )}{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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