3.480 \(\int \frac{x^8}{\left (a+b x^3\right )^2 \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=123 \[ -\frac{a^2 \sqrt{c+d x^3}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} (b c-a d)^{3/2}}+\frac{2 \sqrt{c+d x^3}}{3 b^2 d} \]

[Out]

(2*Sqrt[c + d*x^3])/(3*b^2*d) - (a^2*Sqrt[c + d*x^3])/(3*b^2*(b*c - a*d)*(a + b*
x^3)) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(
3*b^(5/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.446819, antiderivative size = 123, 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 \[ -\frac{a^2 \sqrt{c+d x^3}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} (b c-a d)^{3/2}}+\frac{2 \sqrt{c+d x^3}}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]  Int[x^8/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

(2*Sqrt[c + d*x^3])/(3*b^2*d) - (a^2*Sqrt[c + d*x^3])/(3*b^2*(b*c - a*d)*(a + b*
x^3)) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(
3*b^(5/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 35.1829, size = 105, normalized size = 0.85 \[ \frac{a^{2} \sqrt{c + d x^{3}}}{3 b^{2} \left (a + b x^{3}\right ) \left (a d - b c\right )} - \frac{a \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{2 \sqrt{c + d x^{3}}}{3 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)

[Out]

a**2*sqrt(c + d*x**3)/(3*b**2*(a + b*x**3)*(a*d - b*c)) - a*(3*a*d - 4*b*c)*atan
(sqrt(b)*sqrt(c + d*x**3)/sqrt(a*d - b*c))/(3*b**(5/2)*(a*d - b*c)**(3/2)) + 2*s
qrt(c + d*x**3)/(3*b**2*d)

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Mathematica [A]  time = 0.400469, size = 107, normalized size = 0.87 \[ \frac{1}{3} \left (\frac{\sqrt{c+d x^3} \left (\frac{a^2}{\left (a+b x^3\right ) (a d-b c)}+\frac{2}{d}\right )}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

((Sqrt[c + d*x^3]*(2/d + a^2/((-(b*c) + a*d)*(a + b*x^3))))/b^2 + (a*(4*b*c - 3*
a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b^(5/2)*(b*c - a*d)^(3
/2)))/3

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Maple [C]  time = 0.061, size = 911, normalized size = 7.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^3+a)^2/(d*x^3+c)^(1/2),x)

[Out]

2/3*(d*x^3+c)^(1/2)/b^2/d+a^2/b^2*(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)*_alp
ha*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)))+2/3*I*a/b^2/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*_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^8/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228928, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \,{\left (b^{2} c - a b d\right )} x^{3} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d} +{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{3}\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right )}{6 \,{\left (a b^{3} c d - a^{2} b^{2} d^{2} +{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{3}\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \,{\left (b^{2} c - a b d\right )} x^{3} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d} +{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{3}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (a b^{3} c d - a^{2} b^{2} d^{2} +{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{3}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="fricas")

[Out]

[1/6*(2*(2*(b^2*c - a*b*d)*x^3 + 2*a*b*c - 3*a^2*d)*sqrt(d*x^3 + c)*sqrt(b^2*c -
a*b*d) + (4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^3)*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^3*c*d - a^2*b^2*d^2 + (b^4*c*d - a*b^3*d^2)*x^3)*sqrt(b^2*c - a*
b*d)), 1/3*((2*(b^2*c - a*b*d)*x^3 + 2*a*b*c - 3*a^2*d)*sqrt(d*x^3 + c)*sqrt(-b^
2*c + a*b*d) + (4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^3)*arcta
n(-(b*c - a*d)/(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d))))/((a*b^3*c*d - a^2*b^2*d^
2 + (b^4*c*d - a*b^3*d^2)*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**8/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220107, size = 181, normalized size = 1.47 \[ -\frac{\sqrt{d x^{3} + c} a^{2} d}{3 \,{\left (b^{3} c - a b^{2} d\right )}{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )}} - \frac{{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \, \sqrt{d x^{3} + c}}{3 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="giac")

[Out]

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