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

Optimal result

Integrand size = 15, antiderivative size = 23 \[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=-\frac {\sqrt [3]{a x^3+b x^6}}{a x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2025} \[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=-\frac {\sqrt [3]{a x^3+b x^6}}{a x^2} \]

[In]

Int[(a*x^3 + b*x^6)^(-2/3),x]

[Out]

-((a*x^3 + b*x^6)^(1/3)/(a*x^2))

Rule 2025

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*} \text {integral}& = -\frac {\sqrt [3]{a x^3+b x^6}}{a x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=-\frac {\sqrt [3]{x^3 \left (a+b x^3\right )}}{a x^2} \]

[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] (verified)

Time = 0.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
trager	\(-\frac {\left (b \,x^{6}+a \,x^{3}\right )^{\frac {1}{3}}}{a \,x^{2}}\)	\(22\)
pseudoelliptic	\(-\frac {\left (x^{3} \left (b \,x^{3}+a \right )\right )^{\frac {1}{3}}}{a \,x^{2}}\)	\(22\)
gosper	\(-\frac {x \left (b \,x^{3}+a \right )}{a \left (b \,x^{6}+a \,x^{3}\right )^{\frac {2}{3}}}\)	\(27\)
risch	\(-\frac {x \left (b \,x^{3}+a \right )}{\left (x^{3} \left (b \,x^{3}+a \right )\right )^{\frac {2}{3}} a}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=-\frac {{\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}}{a x^{2}} \]

[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]

\[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=\int \frac {1}{\left (a x^{3} + b x^{6}\right )^{\frac {2}{3}}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=-\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{a x} \]

[In]

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

[Out]

-(b*x^3 + a)^(1/3)/(a*x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=-\frac {{\left (b + \frac {a}{x^{3}}\right )}^{\frac {1}{3}}}{a} \]

[In]

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

[Out]

-(b + a/x^3)^(1/3)/a

Mupad [B] (verification not implemented)

Time = 8.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx=-\frac {{\left (b\,x^6+a\,x^3\right )}^{1/3}}{a\,x^2} \]

[In]

int(1/(a*x^3 + b*x^6)^(2/3),x)

[Out]

-(a*x^3 + b*x^6)^(1/3)/(a*x^2)