3.520 \(\int \frac{\sqrt [3]{a+b x^3}}{x^8} \, dx\)

Optimal. Leaf size=44 \[ \frac{3 b \left (a+b x^3\right )^{4/3}}{28 a^2 x^4}-\frac{\left (a+b x^3\right )^{4/3}}{7 a x^7} \]

[Out]

-(a + b*x^3)^(4/3)/(7*a*x^7) + (3*b*(a + b*x^3)^(4/3))/(28*a^2*x^4)

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Rubi [A]  time = 0.0108102, 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} \[ \frac{3 b \left (a+b x^3\right )^{4/3}}{28 a^2 x^4}-\frac{\left (a+b x^3\right )^{4/3}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/x^8,x]

[Out]

-(a + b*x^3)^(4/3)/(7*a*x^7) + (3*b*(a + b*x^3)^(4/3))/(28*a^2*x^4)

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{\sqrt [3]{a+b x^3}}{x^8} \, dx &=-\frac{\left (a+b x^3\right )^{4/3}}{7 a x^7}-\frac{(3 b) \int \frac{\sqrt [3]{a+b x^3}}{x^5} \, dx}{7 a}\\ &=-\frac{\left (a+b x^3\right )^{4/3}}{7 a x^7}+\frac{3 b \left (a+b x^3\right )^{4/3}}{28 a^2 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0085041, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^3\right )^{4/3} \left (3 b x^3-4 a\right )}{28 a^2 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/x^8,x]

[Out]

((a + b*x^3)^(4/3)*(-4*a + 3*b*x^3))/(28*a^2*x^7)

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Maple [A]  time = 0.004, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-3\,b{x}^{3}+4\,a}{28\,{a}^{2}{x}^{7}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/28*(b*x^3+a)^(4/3)*(-3*b*x^3+4*a)/a^2/x^7

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Maxima [A]  time = 1.00063, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b}{x^{4}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}}}{x^{7}}}{28 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(7*(b*x^3 + a)^(4/3)*b/x^4 - 4*(b*x^3 + a)^(7/3)/x^7)/a^2

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Fricas [A]  time = 1.99866, size = 86, normalized size = 1.95 \begin{align*} \frac{{\left (3 \, b^{2} x^{6} - a b x^{3} - 4 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{28 \, a^{2} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(3*b^2*x^6 - a*b*x^3 - 4*a^2)*(b*x^3 + a)^(1/3)/(a^2*x^7)

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Sympy [B]  time = 1.20193, size = 109, normalized size = 2.48 \begin{align*} - \frac{4 \sqrt [3]{b} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{3}\right )}{9 x^{6} \Gamma \left (- \frac{1}{3}\right )} - \frac{b^{\frac{4}{3}} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{3}\right )}{9 a x^{3} \Gamma \left (- \frac{1}{3}\right )} + \frac{b^{\frac{7}{3}} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{7}{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((b*x**3+a)**(1/3)/x**8,x)

[Out]

-4*b**(1/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(9*x**6*gamma(-1/3)) - b**(4/3)*(a/(b*x**3) + 1)**(1/3)*gamma(
-7/3)/(9*a*x**3*gamma(-1/3)) + b**(7/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-7/3)/(3*a**2*gamma(-1/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)/x^8, x)