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

Optimal. Leaf size=21 \[ -\frac {\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {264} \[ -\frac {\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ -\frac {\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a + b*x^3)^(2/3)/(a*x^2)

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fricas [A]  time = 0.66, size = 17, normalized size = 0.81 \[ -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*x^3 + a)^(2/3)/(a*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{2 a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 17, normalized size = 0.81 \[ -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*x^3 + a)^(2/3)/(a*x^2)

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mupad [B]  time = 1.03, size = 17, normalized size = 0.81 \[ -\frac {{\left (b\,x^3+a\right )}^{2/3}}{2\,a\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^3)^(1/3)),x)

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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sympy [A]  time = 1.29, size = 31, normalized size = 1.48 \[ \frac {b^{\frac {2}{3}} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {2}{3}\right )}{3 a \Gamma \left (\frac {1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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