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

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

[Out]

-((a*x^3 + b*x^6)^(1/3)/(a*x^2))

________________________________________________________________________________________

Rubi [A]  time = 0.0049352, antiderivative size = 23, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt [3]{a x^3+b x^6}}{a x^2} \]

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*}

Mathematica [A]  time = 0.0081178, size = 23, normalized size = 1. \[ -\frac{\sqrt [3]{x^3 \left (a+b x^3\right )}}{a x^2} \]

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))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 27, normalized size = 1.2 \begin{align*} -{\frac{x \left ( b{x}^{3}+a \right ) }{a} \left ( b{x}^{6}+a{x}^{3} \right ) ^{-{\frac{2}{3}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.09956, size = 23, normalized size = 1. \begin{align*} -\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{a x} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.99811, size = 43, normalized size = 1.87 \begin{align*} -\frac{{\left (b x^{6} + a x^{3}\right )}^{\frac{1}{3}}}{a x^{2}} \end{align*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{6} + a x^{3}\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

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]

integrate((b*x^6 + a*x^3)^(-2/3), x)