3.541 \(\int \frac{(a+b x^3)^{2/3}}{x^9} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 b \left (a+b x^3\right )^{5/3}}{40 a^2 x^5}-\frac{\left (a+b x^3\right )^{5/3}}{8 a x^8} \]

[Out]

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

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Rubi [A]  time = 0.0109494, 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} \[ \frac{3 b \left (a+b x^3\right )^{5/3}}{40 a^2 x^5}-\frac{\left (a+b x^3\right )^{5/3}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(2/3)/x^9,x]

[Out]

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

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

Mathematica [A]  time = 0.009537, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^3\right )^{5/3} \left (3 b x^3-5 a\right )}{40 a^2 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(2/3)/x^9,x]

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-3\,b{x}^{3}+5\,a}{40\,{x}^{8}{a}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(2/3)/x^9,x)

[Out]

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

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Maxima [A]  time = 0.96801, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{8 \,{\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^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73844, size = 89, normalized size = 2.02 \begin{align*} \frac{{\left (3 \, b^{2} x^{6} - 2 \, a b x^{3} - 5 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{40 \, a^{2} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^9,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.96744, size = 110, normalized size = 2.5 \begin{align*} - \frac{5 b^{\frac{2}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{9 x^{6} \Gamma \left (- \frac{2}{3}\right )} - \frac{2 b^{\frac{5}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{9 a x^{3} \Gamma \left (- \frac{2}{3}\right )} + \frac{b^{\frac{8}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{8}{3}\right )}{3 a^{2} \Gamma \left (- \frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**9,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^9,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/x^9, x)