3.340 \(\int \frac{x^{11}}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=61 \[ \frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}+\frac{x^3}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.104342, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac{a^2}{b^4 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right )}{b^4}+\frac{x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + b*x^3)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3}}{6 b^{4} \left (a + b x^{3}\right )^{2}} - \frac{a^{2}}{b^{4} \left (a + b x^{3}\right )} - \frac{a \log{\left (a + b x^{3} \right )}}{b^{4}} + \frac{\int ^{x^{3}} \frac{1}{b^{3}}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**3+a)**3,x)

[Out]

a**3/(6*b**4*(a + b*x**3)**2) - a**2/(b**4*(a + b*x**3)) - a*log(a + b*x**3)/b**
4 + Integral(b**(-3), (x, x**3))/3

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Mathematica [A]  time = 0.106542, size = 48, normalized size = 0.79 \[ -\frac{\frac{a^2 \left (5 a+6 b x^3\right )}{\left (a+b x^3\right )^2}+6 a \log \left (a+b x^3\right )-2 b x^3}{6 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + b*x^3)^3,x]

[Out]

-(-2*b*x^3 + (a^2*(5*a + 6*b*x^3))/(a + b*x^3)^2 + 6*a*Log[a + b*x^3])/(6*b^4)

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Maple [A]  time = 0.008, size = 58, normalized size = 1. \[{\frac{{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{3}}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}}{{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{a\ln \left ( b{x}^{3}+a \right ) }{{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.44451, size = 89, normalized size = 1.46 \[ -\frac{6 \, a^{2} b x^{3} + 5 \, a^{3}}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac{x^{3}}{3 \, b^{3}} - \frac{a \log \left (b x^{3} + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

-1/6*(6*a^2*b*x^3 + 5*a^3)/(b^6*x^6 + 2*a*b^5*x^3 + a^2*b^4) + 1/3*x^3/b^3 - a*l
og(b*x^3 + a)/b^4

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Fricas [A]  time = 0.227307, size = 123, normalized size = 2.02 \[ \frac{2 \, b^{3} x^{9} + 4 \, a b^{2} x^{6} - 4 \, a^{2} b x^{3} - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{6} + 2 \, a^{2} b x^{3} + a^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^3*x^9 + 4*a*b^2*x^6 - 4*a^2*b*x^3 - 5*a^3 - 6*(a*b^2*x^6 + 2*a^2*b*x^3
+ a^3)*log(b*x^3 + a))/(b^6*x^6 + 2*a*b^5*x^3 + a^2*b^4)

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Sympy [A]  time = 2.75012, size = 63, normalized size = 1.03 \[ - \frac{a \log{\left (a + b x^{3} \right )}}{b^{4}} - \frac{5 a^{3} + 6 a^{2} b x^{3}}{6 a^{2} b^{4} + 12 a b^{5} x^{3} + 6 b^{6} x^{6}} + \frac{x^{3}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**3+a)**3,x)

[Out]

-a*log(a + b*x**3)/b**4 - (5*a**3 + 6*a**2*b*x**3)/(6*a**2*b**4 + 12*a*b**5*x**3
+ 6*b**6*x**6) + x**3/(3*b**3)

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GIAC/XCAS [A]  time = 0.231129, size = 84, normalized size = 1.38 \[ \frac{x^{3}}{3 \, b^{3}} - \frac{a{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{b^{4}} + \frac{9 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 4 \, a^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

1/3*x^3/b^3 - a*ln(abs(b*x^3 + a))/b^4 + 1/6*(9*a*b^2*x^6 + 12*a^2*b*x^3 + 4*a^3
)/((b*x^3 + a)^2*b^4)