3.559 \(\int \frac{1}{x^9 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}+\frac{3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac{\left (a+b x^3\right )^{2/3}}{8 a x^8} \]

[Out]

-(a + b*x^3)^(2/3)/(8*a*x^8) + (3*b*(a + b*x^3)^(2/3))/(20*a^2*x^5) - (9*b^2*(a + b*x^3)^(2/3))/(40*a^3*x^2)

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Rubi [A]  time = 0.0176911, 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} \[ -\frac{9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}+\frac{3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac{\left (a+b x^3\right )^{2/3}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(8*a*x^8) + (3*b*(a + b*x^3)^(2/3))/(20*a^2*x^5) - (9*b^2*(a + b*x^3)^(2/3))/(40*a^3*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^9 \sqrt [3]{a+b x^3}} \, dx &=-\frac{\left (a+b x^3\right )^{2/3}}{8 a x^8}-\frac{(3 b) \int \frac{1}{x^6 \sqrt [3]{a+b x^3}} \, dx}{4 a}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{8 a x^8}+\frac{3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}+\frac{\left (9 b^2\right ) \int \frac{1}{x^3 \sqrt [3]{a+b x^3}} \, dx}{20 a^2}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{8 a x^8}+\frac{3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac{9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0167355, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^3\right )^{2/3} \left (5 a^2-6 a b x^3+9 b^2 x^6\right )}{40 a^3 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^3)^(1/3)),x]

[Out]

-((a + b*x^3)^(2/3)*(5*a^2 - 6*a*b*x^3 + 9*b^2*x^6))/(40*a^3*x^8)

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Maple [A]  time = 0.002, size = 39, normalized size = 0.6 \begin{align*} -{\frac{9\,{b}^{2}{x}^{6}-6\,{x}^{3}ab+5\,{a}^{2}}{40\,{x}^{8}{a}^{3}} \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^9/(b*x^3+a)^(1/3),x)

[Out]

-1/40*(b*x^3+a)^(2/3)*(9*b^2*x^6-6*a*b*x^3+5*a^2)/x^8/a^3

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Maxima [A]  time = 1.55631, size = 70, normalized size = 1.03 \begin{align*} -\frac{\frac{20 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{2}}{x^{2}} - \frac{16 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b}{x^{5}} + \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}}}{x^{8}}}{40 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

-1/40*(20*(b*x^3 + a)^(2/3)*b^2/x^2 - 16*(b*x^3 + a)^(5/3)*b/x^5 + 5*(b*x^3 + a)^(8/3)/x^8)/a^3

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Fricas [A]  time = 1.53844, size = 90, normalized size = 1.32 \begin{align*} -\frac{{\left (9 \, b^{2} x^{6} - 6 \, a b x^{3} + 5 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{40 \, a^{3} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

-1/40*(9*b^2*x^6 - 6*a*b*x^3 + 5*a^2)*(b*x^3 + a)^(2/3)/(a^3*x^8)

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Sympy [B]  time = 2.12868, size = 406, normalized size = 5.97 \begin{align*} \frac{10 a^{4} b^{\frac{14}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{8 a^{3} b^{\frac{17}{3}} x^{3} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{4 a^{2} b^{\frac{20}{3}} x^{6} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{24 a b^{\frac{23}{3}} x^{9} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} + \frac{18 b^{\frac{26}{3}} x^{12} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac{1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac{1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**3+a)**(1/3),x)

[Out]

10*a**4*b**(14/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(
1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 8*a**3*b**(17/3)*x**3*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**
4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 4*a**2*b**(20/3)*x**6*(a/(
b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x*
*12*gamma(1/3)) + 24*a*b**(23/3)*x**9*(a/(b*x**3) + 1)**(2/3)*gamma(-8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a
**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*gamma(1/3)) + 18*b**(26/3)*x**12*(a/(b*x**3) + 1)**(2/3)*gamma(-
8/3)/(27*a**5*b**4*x**6*gamma(1/3) + 54*a**4*b**5*x**9*gamma(1/3) + 27*a**3*b**6*x**12*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^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^9), x)