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

Optimal. Leaf size=59 \[ -\frac{2 a^2}{3 b^3 \sqrt{a+b x^3}}-\frac{4 a \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{3/2}}{9 b^3} \]

[Out]

(-2*a^2)/(3*b^3*Sqrt[a + b*x^3]) - (4*a*Sqrt[a + b*x^3])/(3*b^3) + (2*(a + b*x^3
)^(3/2))/(9*b^3)

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Rubi [A]  time = 0.0886321, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^2}{3 b^3 \sqrt{a+b x^3}}-\frac{4 a \sqrt{a+b x^3}}{3 b^3}+\frac{2 \left (a+b x^3\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^3)^(3/2),x]

[Out]

(-2*a^2)/(3*b^3*Sqrt[a + b*x^3]) - (4*a*Sqrt[a + b*x^3])/(3*b^3) + (2*(a + b*x^3
)^(3/2))/(9*b^3)

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Rubi in Sympy [A]  time = 10.8284, size = 54, normalized size = 0.92 \[ - \frac{2 a^{2}}{3 b^{3} \sqrt{a + b x^{3}}} - \frac{4 a \sqrt{a + b x^{3}}}{3 b^{3}} + \frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(3*b**3*sqrt(a + b*x**3)) - 4*a*sqrt(a + b*x**3)/(3*b**3) + 2*(a + b*x**
3)**(3/2)/(9*b**3)

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Mathematica [A]  time = 0.0301514, size = 38, normalized size = 0.64 \[ \frac{2 \left (-8 a^2-4 a b x^3+b^2 x^6\right )}{9 b^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-8*a^2 - 4*a*b*x^3 + b^2*x^6))/(9*b^3*Sqrt[a + b*x^3])

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Maple [A]  time = 0.007, size = 36, normalized size = 0.6 \[ -{\frac{-2\,{b}^{2}{x}^{6}+8\,ab{x}^{3}+16\,{a}^{2}}{9\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^3+a)^(3/2),x)

[Out]

-2/9/(b*x^3+a)^(1/2)*(-b^2*x^6+4*a*b*x^3+8*a^2)/b^3

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Maxima [A]  time = 1.43941, size = 63, normalized size = 1.07 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b^{3}} - \frac{4 \, \sqrt{b x^{3} + a} a}{3 \, b^{3}} - \frac{2 \, a^{2}}{3 \, \sqrt{b x^{3} + a} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b*x^3 + a)^(3/2)/b^3 - 4/3*sqrt(b*x^3 + a)*a/b^3 - 2/3*a^2/(sqrt(b*x^3 + a)
*b^3)

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Fricas [A]  time = 0.227429, size = 46, normalized size = 0.78 \[ \frac{2 \,{\left (b^{2} x^{6} - 4 \, a b x^{3} - 8 \, a^{2}\right )}}{9 \, \sqrt{b x^{3} + a} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b^2*x^6 - 4*a*b*x^3 - 8*a^2)/(sqrt(b*x^3 + a)*b^3)

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Sympy [A]  time = 5.48458, size = 70, normalized size = 1.19 \[ \begin{cases} - \frac{16 a^{2}}{9 b^{3} \sqrt{a + b x^{3}}} - \frac{8 a x^{3}}{9 b^{2} \sqrt{a + b x^{3}}} + \frac{2 x^{6}}{9 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{9}}{9 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**3+a)**(3/2),x)

[Out]

Piecewise((-16*a**2/(9*b**3*sqrt(a + b*x**3)) - 8*a*x**3/(9*b**2*sqrt(a + b*x**3
)) + 2*x**6/(9*b*sqrt(a + b*x**3)), Ne(b, 0)), (x**9/(9*a**(3/2)), True))

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GIAC/XCAS [A]  time = 0.235441, size = 55, normalized size = 0.93 \[ \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b x^{3} + a} a - \frac{3 \, a^{2}}{\sqrt{b x^{3} + a}}\right )}}{9 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*((b*x^3 + a)^(3/2) - 6*sqrt(b*x^3 + a)*a - 3*a^2/sqrt(b*x^3 + a))/b^3