3.571 \(\int \frac {1}{x^5 (a+b x^3)^{2/3}} \, dx\)

Optimal. Leaf size=44 \[ \frac {3 b \sqrt [3]{a+b x^3}}{4 a^2 x}-\frac {\sqrt [3]{a+b x^3}}{4 a x^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, 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} \[ \frac {3 b \sqrt [3]{a+b x^3}}{4 a^2 x}-\frac {\sqrt [3]{a+b x^3}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^3)^(2/3)),x]

[Out]

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

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]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 29, normalized size = 0.66 \[ -\frac {\left (a-3 b x^3\right ) \sqrt [3]{a+b x^3}}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^3)^(2/3)),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 27, normalized size = 0.61 \[ \frac {{\left (3 \, b x^{3} - a\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{4 \, a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^5), x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 26, normalized size = 0.59 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (-3 b \,x^{3}+a \right )}{4 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^3+a)^(2/3),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 35, normalized size = 0.80 \[ \frac {\frac {4 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x} - \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{x^{4}}}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.08, size = 25, normalized size = 0.57 \[ -\frac {{\left (b\,x^3+a\right )}^{1/3}\,\left (a-3\,b\,x^3\right )}{4\,a^2\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x^3)^(2/3)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.54, size = 68, normalized size = 1.55 \[ - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {4}{3}\right )}{9 a x^{3} \Gamma \left (\frac {2}{3}\right )} + \frac {b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {4}{3}\right )}{3 a^{2} \Gamma \left (\frac {2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x**3+a)**(2/3),x)

[Out]

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

________________________________________________________________________________________