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

Optimal. Leaf size=149 \[ -\frac{a^2 d^2+2 b^2 c^2}{3 b^2 d \sqrt{c+d x^3} (b c-a d)^2}-\frac{a^2}{3 b^2 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.559496, antiderivative size = 149, 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 \[ -\frac{a^2 d^2+2 b^2 c^2}{3 b^2 d \sqrt{c+d x^3} (b c-a d)^2}-\frac{a^2}{3 b^2 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.6482, size = 134, normalized size = 0.9 \[ \frac{a \left (a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{5}{2}}} - \frac{2 c^{2}}{3 d^{2} \left (a + b x^{3}\right ) \sqrt{c + d x^{3}} \left (a d - b c\right )} - \frac{\sqrt{c + d x^{3}} \left (a^{2} d^{2} + 2 b^{2} c^{2}\right )}{3 b d^{2} \left (a + b x^{3}\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.591203, size = 118, normalized size = 0.79 \[ \frac{1}{3} \left (\frac{-\frac{a^2 \left (c+d x^3\right )}{b \left (a+b x^3\right )}-\frac{2 c^2}{d}}{\sqrt{c+d x^3} (b c-a d)^2}+\frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b c-a d)^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.072, size = 978, normalized size = 6.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/b^2/d/(d*x^3+c)^(1/2)+a^2/b^2*(-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)))-2*a/b^2*(-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))*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*(d*x^3 + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232446, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, a^{2} b c d - a^{3} d^{2} +{\left (4 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right ) + 2 \,{\left (2 \, a b c^{2} + a^{2} c d +{\left (2 \, b^{2} c^{2} + a^{2} d^{2}\right )} x^{3}\right )} \sqrt{b^{2} c - a b d}}{6 \,{\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3} +{\left (b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}, \frac{{\left (4 \, a^{2} b c d - a^{3} d^{2} +{\left (4 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ) -{\left (2 \, a b c^{2} + a^{2} c d +{\left (2 \, b^{2} c^{2} + a^{2} d^{2}\right )} x^{3}\right )} \sqrt{-b^{2} c + a b d}}{3 \,{\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3} +{\left (b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229246, size = 263, normalized size = 1.77 \[ -\frac{{\left (4 \, a b c - a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x^{3} + c\right )} b^{2} c^{2} - 2 \, b^{2} c^{3} + 2 \, a b c^{2} d +{\left (d x^{3} + c\right )} a^{2} d^{2}}{3 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )}{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b - \sqrt{d x^{3} + c} b c + \sqrt{d x^{3} + c} a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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