3.573 \(\int \frac{1}{x^{11} (a+b x^3)^{2/3}} \, dx\)

Optimal. Leaf size=92 \[ \frac{81 b^3 \sqrt [3]{a+b x^3}}{140 a^4 x}-\frac{27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}+\frac{9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac{\sqrt [3]{a+b x^3}}{10 a x^{10}} \]

[Out]

-(a + b*x^3)^(1/3)/(10*a*x^10) + (9*b*(a + b*x^3)^(1/3))/(70*a^2*x^7) - (27*b^2*(a + b*x^3)^(1/3))/(140*a^3*x^
4) + (81*b^3*(a + b*x^3)^(1/3))/(140*a^4*x)

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Rubi [A]  time = 0.0305809, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{81 b^3 \sqrt [3]{a+b x^3}}{140 a^4 x}-\frac{27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}+\frac{9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac{\sqrt [3]{a+b x^3}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^11*(a + b*x^3)^(2/3)),x]

[Out]

-(a + b*x^3)^(1/3)/(10*a*x^10) + (9*b*(a + b*x^3)^(1/3))/(70*a^2*x^7) - (27*b^2*(a + b*x^3)^(1/3))/(140*a^3*x^
4) + (81*b^3*(a + b*x^3)^(1/3))/(140*a^4*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^{11} \left (a+b x^3\right )^{2/3}} \, dx &=-\frac{\sqrt [3]{a+b x^3}}{10 a x^{10}}-\frac{(9 b) \int \frac{1}{x^8 \left (a+b x^3\right )^{2/3}} \, dx}{10 a}\\ &=-\frac{\sqrt [3]{a+b x^3}}{10 a x^{10}}+\frac{9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}+\frac{\left (27 b^2\right ) \int \frac{1}{x^5 \left (a+b x^3\right )^{2/3}} \, dx}{35 a^2}\\ &=-\frac{\sqrt [3]{a+b x^3}}{10 a x^{10}}+\frac{9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac{27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}-\frac{\left (81 b^3\right ) \int \frac{1}{x^2 \left (a+b x^3\right )^{2/3}} \, dx}{140 a^3}\\ &=-\frac{\sqrt [3]{a+b x^3}}{10 a x^{10}}+\frac{9 b \sqrt [3]{a+b x^3}}{70 a^2 x^7}-\frac{27 b^2 \sqrt [3]{a+b x^3}}{140 a^3 x^4}+\frac{81 b^3 \sqrt [3]{a+b x^3}}{140 a^4 x}\\ \end{align*}

Mathematica [A]  time = 0.0185051, size = 53, normalized size = 0.58 \[ \frac{\sqrt [3]{a+b x^3} \left (18 a^2 b x^3-14 a^3-27 a b^2 x^6+81 b^3 x^9\right )}{140 a^4 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^11*(a + b*x^3)^(2/3)),x]

[Out]

((a + b*x^3)^(1/3)*(-14*a^3 + 18*a^2*b*x^3 - 27*a*b^2*x^6 + 81*b^3*x^9))/(140*a^4*x^10)

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Maple [A]  time = 0.004, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-81\,{b}^{3}{x}^{9}+27\,a{b}^{2}{x}^{6}-18\,{a}^{2}b{x}^{3}+14\,{a}^{3}}{140\,{x}^{10}{a}^{4}}\sqrt [3]{b{x}^{3}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^11/(b*x^3+a)^(2/3),x)

[Out]

-1/140*(b*x^3+a)^(1/3)*(-81*b^3*x^9+27*a*b^2*x^6-18*a^2*b*x^3+14*a^3)/x^10/a^4

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Maxima [A]  time = 0.974876, size = 93, normalized size = 1.01 \begin{align*} \frac{\frac{140 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{3}}{x} - \frac{105 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b^{2}}{x^{4}} + \frac{60 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} b}{x^{7}} - \frac{14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}}}{x^{10}}}{140 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

1/140*(140*(b*x^3 + a)^(1/3)*b^3/x - 105*(b*x^3 + a)^(4/3)*b^2/x^4 + 60*(b*x^3 + a)^(7/3)*b/x^7 - 14*(b*x^3 +
a)^(10/3)/x^10)/a^4

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Fricas [A]  time = 1.75413, size = 119, normalized size = 1.29 \begin{align*} \frac{{\left (81 \, b^{3} x^{9} - 27 \, a b^{2} x^{6} + 18 \, a^{2} b x^{3} - 14 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{140 \, a^{4} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

1/140*(81*b^3*x^9 - 27*a*b^2*x^6 + 18*a^2*b*x^3 - 14*a^3)*(b*x^3 + a)^(1/3)/(a^4*x^10)

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Sympy [B]  time = 3.44999, size = 692, normalized size = 7.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**11/(b*x**3+a)**(2/3),x)

[Out]

-28*a**6*b**(28/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*g
amma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 48*a**5*b**(31/3)*x**3*(a/(b*x
**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11
*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 30*a**4*b**(34/3)*x**6*(a/(b*x**3) + 1)**(1/3)*gamma(-10
/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**
4*b**12*x**18*gamma(2/3)) + 80*a**3*b**(37/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gam
ma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3))
+ 360*a**2*b**(40/3)*x**12*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10
*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 432*a*b**(43/3)*x**15*
(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**
5*b**11*x**15*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 162*b**(46/3)*x**18*(a/(b*x**3) + 1)**(1/3)*gamma
(-10/3)/(81*a**7*b**9*x**9*gamma(2/3) + 243*a**6*b**10*x**12*gamma(2/3) + 243*a**5*b**11*x**15*gamma(2/3) + 81
*a**4*b**12*x**18*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^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^11), x)