3.558 \(\int \frac{1}{x^6 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2}-\frac{\left (a+b x^3\right )^{2/3}}{5 a x^5} \]

[Out]

-(a + b*x^3)^(2/3)/(5*a*x^5) + (3*b*(a + b*x^3)^(2/3))/(10*a^2*x^2)

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Rubi [A]  time = 0.010515, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2}-\frac{\left (a+b x^3\right )^{2/3}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(5*a*x^5) + (3*b*(a + b*x^3)^(2/3))/(10*a^2*x^2)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^6 \sqrt [3]{a+b x^3}} \, dx &=-\frac{\left (a+b x^3\right )^{2/3}}{5 a x^5}-\frac{(3 b) \int \frac{1}{x^3 \sqrt [3]{a+b x^3}} \, dx}{5 a}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{5 a x^5}+\frac{3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.011073, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^3\right )^{2/3} \left (3 b x^3-2 a\right )}{10 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^3)^(1/3)),x]

[Out]

((a + b*x^3)^(2/3)*(-2*a + 3*b*x^3))/(10*a^2*x^5)

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Maple [A]  time = 0.005, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-3\,b{x}^{3}+2\,a}{10\,{x}^{5}{a}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^3+a)^(1/3),x)

[Out]

-1/10*(b*x^3+a)^(2/3)*(-3*b*x^3+2*a)/x^5/a^2

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Maxima [A]  time = 1.03354, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{5 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}}}{x^{5}}}{10 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

1/10*(5*(b*x^3 + a)^(2/3)*b/x^2 - 2*(b*x^3 + a)^(5/3)/x^5)/a^2

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Fricas [A]  time = 1.44987, size = 68, normalized size = 1.55 \begin{align*} \frac{{\left (3 \, b x^{3} - 2 \, a\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{10 \, a^{2} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

1/10*(3*b*x^3 - 2*a)*(b*x^3 + a)^(2/3)/(a^2*x^5)

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Sympy [A]  time = 1.13034, size = 70, normalized size = 1.59 \begin{align*} - \frac{2 b^{\frac{2}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{5}{3}\right )}{9 a x^{3} \Gamma \left (\frac{1}{3}\right )} + \frac{b^{\frac{5}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{5}{3}\right )}{3 a^{2} \Gamma \left (\frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**3+a)**(1/3),x)

[Out]

-2*b**(2/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-5/3)/(9*a*x**3*gamma(1/3)) + b**(5/3)*(a/(b*x**3) + 1)**(2/3)*gamma
(-5/3)/(3*a**2*gamma(1/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^6), x)