3.572 \(\int \frac{1}{x^8 (a+b x^3)^{2/3}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{9 b^2 \sqrt [3]{a+b x^3}}{14 a^3 x}+\frac{3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}-\frac{\sqrt [3]{a+b x^3}}{7 a x^7} \]

[Out]

-(a + b*x^3)^(1/3)/(7*a*x^7) + (3*b*(a + b*x^3)^(1/3))/(14*a^2*x^4) - (9*b^2*(a + b*x^3)^(1/3))/(14*a^3*x)

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Rubi [A]  time = 0.0201348, antiderivative size = 68, 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, Rules used = {271, 264} \[ -\frac{9 b^2 \sqrt [3]{a+b x^3}}{14 a^3 x}+\frac{3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}-\frac{\sqrt [3]{a+b x^3}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^3)^(2/3)),x]

[Out]

-(a + b*x^3)^(1/3)/(7*a*x^7) + (3*b*(a + b*x^3)^(1/3))/(14*a^2*x^4) - (9*b^2*(a + b*x^3)^(1/3))/(14*a^3*x)

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^8 \left (a+b x^3\right )^{2/3}} \, dx &=-\frac{\sqrt [3]{a+b x^3}}{7 a x^7}-\frac{(6 b) \int \frac{1}{x^5 \left (a+b x^3\right )^{2/3}} \, dx}{7 a}\\ &=-\frac{\sqrt [3]{a+b x^3}}{7 a x^7}+\frac{3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}+\frac{\left (9 b^2\right ) \int \frac{1}{x^2 \left (a+b x^3\right )^{2/3}} \, dx}{14 a^2}\\ &=-\frac{\sqrt [3]{a+b x^3}}{7 a x^7}+\frac{3 b \sqrt [3]{a+b x^3}}{14 a^2 x^4}-\frac{9 b^2 \sqrt [3]{a+b x^3}}{14 a^3 x}\\ \end{align*}

Mathematica [A]  time = 0.0156626, size = 42, normalized size = 0.62 \[ -\frac{\sqrt [3]{a+b x^3} \left (2 a^2-3 a b x^3+9 b^2 x^6\right )}{14 a^3 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^3)^(2/3)),x]

[Out]

-((a + b*x^3)^(1/3)*(2*a^2 - 3*a*b*x^3 + 9*b^2*x^6))/(14*a^3*x^7)

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{9\,{b}^{2}{x}^{6}-3\,{x}^{3}ab+2\,{a}^{2}}{14\,{a}^{3}{x}^{7}}\sqrt [3]{b{x}^{3}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^3+a)^(2/3),x)

[Out]

-1/14*(b*x^3+a)^(1/3)*(9*b^2*x^6-3*a*b*x^3+2*a^2)/a^3/x^7

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Maxima [A]  time = 1.00377, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{14 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{2}}{x} - \frac{7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b}{x^{4}} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}}}{x^{7}}}{14 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

-1/14*(14*(b*x^3 + a)^(1/3)*b^2/x - 7*(b*x^3 + a)^(4/3)*b/x^4 + 2*(b*x^3 + a)^(7/3)/x^7)/a^3

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Fricas [A]  time = 1.51532, size = 90, normalized size = 1.32 \begin{align*} -\frac{{\left (9 \, b^{2} x^{6} - 3 \, a b x^{3} + 2 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{14 \, a^{3} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

-1/14*(9*b^2*x^6 - 3*a*b*x^3 + 2*a^2)*(b*x^3 + a)^(1/3)/(a^3*x^7)

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Sympy [B]  time = 1.91298, size = 406, normalized size = 5.97 \begin{align*} \frac{4 a^{4} b^{\frac{13}{3}} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{2}{3}\right )} + \frac{2 a^{3} b^{\frac{16}{3}} x^{3} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{2}{3}\right )} + \frac{10 a^{2} b^{\frac{19}{3}} x^{6} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{2}{3}\right )} + \frac{30 a b^{\frac{22}{3}} x^{9} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{2}{3}\right )} + \frac{18 b^{\frac{25}{3}} x^{12} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{2}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{2}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**3+a)**(2/3),x)

[Out]

4*a**4*b**(13/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2
/3) + 27*a**3*b**6*x**12*gamma(2/3)) + 2*a**3*b**(16/3)*x**3*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4
*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x**12*gamma(2/3)) + 10*a**2*b**(19/3)*x**6*(a/(
b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x*
*12*gamma(2/3)) + 30*a*b**(22/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a
**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x**12*gamma(2/3)) + 18*b**(25/3)*x**12*(a/(b*x**3) + 1)**(1/3)*gamma(-
7/3)/(27*a**5*b**4*x**6*gamma(2/3) + 54*a**4*b**5*x**9*gamma(2/3) + 27*a**3*b**6*x**12*gamma(2/3))

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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{2}{3}} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^8), x)