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

Optimal. Leaf size=25 \[ \frac {\left (a x^3+b x^6\right )^{5/3}}{5 b x^5} \]

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Rubi [A]  time = 0.00, antiderivative size = 25, 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 {\left (a x^3+b x^6\right )^{5/3}}{5 b x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^3 + b*x^6)^(2/3),x]

[Out]

(a*x^3 + b*x^6)^(5/3)/(5*b*x^5)

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 \left (a x^3+b x^6\right )^{2/3} \, dx &=\frac {\left (a x^3+b x^6\right )^{5/3}}{5 b x^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (x^3 \left (a+b x^3\right )\right )^{5/3}}{5 b x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^3 + b*x^6)^(2/3),x]

[Out]

(x^3*(a + b*x^3))^(5/3)/(5*b*x^5)

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IntegrateAlgebraic [A]  time = 0.20, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (a x^3+b x^6\right )^{5/3}}{5 b x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^3 + b*x^6)^(5/3)/(5*b*x^5)

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fricas [A]  time = 1.91, size = 28, normalized size = 1.12 \begin {gather*} \frac {{\left (b x^{6} + a x^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}}{5 \, b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x^6 + a*x^3)^(2/3)*(b*x^3 + a)/(b*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^6 + a*x^3)^(2/3), x)

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maple [A]  time = 0.00, size = 29, normalized size = 1.16 \begin {gather*} \frac {\left (b \,x^{3}+a \right ) \left (b \,x^{6}+a \,x^{3}\right )^{\frac {2}{3}}}{5 b \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x^3+a)/b/x^2*(b*x^6+a*x^3)^(2/3)

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maxima [A]  time = 0.46, size = 14, normalized size = 0.56 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x^3 + a)^(5/3)/b

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mupad [B]  time = 1.15, size = 29, normalized size = 1.16 \begin {gather*} \frac {\left (\frac {a}{5\,b}+\frac {x^3}{5}\right )\,{\left (b\,x^6+a\,x^3\right )}^{2/3}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3 + b*x^6)^(2/3),x)

[Out]

((a/(5*b) + x^3/5)*(a*x^3 + b*x^6)^(2/3))/x^2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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((b*x**6+a*x**3)**(2/3),x)

[Out]

Integral((a*x**3 + b*x**6)**(2/3), x)

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