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

Optimal. Leaf size=38 \[ \frac{x^2 \left (a+b x^3\right )^{5/3} \, _2F_1\left (1,\frac{7}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a} \]

[Out]

(x^2*(a + b*x^3)^(5/3)*Hypergeometric2F1[1, 7/3, 5/3, -((b*x^3)/a)])/(2*a)

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Rubi [A]  time = 0.0128214, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {365, 364} \[ \frac{x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^(2/3),x]

[Out]

(x^2*(a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(1 + (b*x^3)/a)^(2/3))

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x \left (a+b x^3\right )^{2/3} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int x \left (1+\frac{b x^3}{a}\right )^{2/3} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=\frac{x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0094732, size = 51, normalized size = 1.34 \[ \frac{x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3)^(2/3),x]

[Out]

(x^2*(a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(1 + (b*x^3)/a)^(2/3))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int x \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(2/3)*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3 + a)^(2/3)*x, x)

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Sympy [C]  time = 1.02409, size = 39, normalized size = 1.03 \begin{align*} \frac{a^{\frac{2}{3}} x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(2/3)*x**2*gamma(2/3)*hyper((-2/3, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(2/3)*x, x)