3.1.3 \(\int \frac {1}{(a x^3+b x^6)^{2/3}} \, dx\)

Optimal. Leaf size=23 \[ -\frac {\sqrt [3]{a x^3+b x^6}}{a x^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 23, 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 = {2000} \begin {gather*} -\frac {\sqrt [3]{a x^3+b x^6}}{a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^3 + b*x^6)^(-2/3),x]

[Out]

-((a*x^3 + b*x^6)^(1/3)/(a*x^2))

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx &=-\frac {\sqrt [3]{a x^3+b x^6}}{a x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [3]{x^3 \left (a+b x^3\right )}}{a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^3 + b*x^6)^(-2/3),x]

[Out]

-((x^3*(a + b*x^3))^(1/3)/(a*x^2))

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IntegrateAlgebraic [A]  time = 0.41, size = 23, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [3]{a x^3+b x^6}}{a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*x^3 + b*x^6)^(-2/3),x]

[Out]

-((a*x^3 + b*x^6)^(1/3)/(a*x^2))

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fricas [A]  time = 1.17, size = 21, normalized size = 0.91 \begin {gather*} -\frac {{\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}}{a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^6+a*x^3)^(2/3),x, algorithm="fricas")

[Out]

-(b*x^6 + a*x^3)^(1/3)/(a*x^2)

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giac [A]  time = 25.00, size = 14, normalized size = 0.61 \begin {gather*} -\frac {{\left (b + \frac {a}{x^{3}}\right )}^{\frac {1}{3}}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^6+a*x^3)^(2/3),x, algorithm="giac")

[Out]

-(b + a/x^3)^(1/3)/a

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maple [A]  time = 0.00, size = 27, normalized size = 1.17 \begin {gather*} -\frac {\left (b \,x^{3}+a \right ) x}{\left (b \,x^{6}+a \,x^{3}\right )^{\frac {2}{3}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^6+a*x^3)^(2/3),x)

[Out]

-x*(b*x^3+a)/a/(b*x^6+a*x^3)^(2/3)

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maxima [A]  time = 0.45, size = 17, normalized size = 0.74 \begin {gather*} -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^6+a*x^3)^(2/3),x, algorithm="maxima")

[Out]

-(b*x^3 + a)^(1/3)/(a*x)

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mupad [B]  time = 1.15, size = 21, normalized size = 0.91 \begin {gather*} -\frac {{\left (b\,x^6+a\,x^3\right )}^{1/3}}{a\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^3 + b*x^6)^(2/3),x)

[Out]

-(a*x^3 + b*x^6)^(1/3)/(a*x^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a x^{3} + b x^{6}\right )^{\frac {2}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**6+a*x**3)**(2/3),x)

[Out]

Integral((a*x**3 + b*x**6)**(-2/3), x)

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