3.523 \(\int x^3 \sqrt [3]{a+b x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^4 \left (a+b x^3\right )^{4/3} \, _2F_1\left (1,\frac{8}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 a} \]

[Out]

(x^4*(a + b*x^3)^(4/3)*Hypergeometric2F1[1, 8/3, 7/3, -((b*x^3)/a)])/(4*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0158705, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac{x^4 \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^3)^(1/3),x]

[Out]

(x^4*(a + b*x^3)^(1/3)*Hypergeometric2F1[-1/3, 4/3, 7/3, -((b*x^3)/a)])/(4*(1 + (b*x^3)/a)^(1/3))

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^3 \sqrt [3]{a+b x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int x^3 \sqrt [3]{1+\frac{b x^3}{a}} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{x^4 \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}

Mathematica [A]  time = 0.009431, size = 51, normalized size = 1.34 \[ \frac{x^4 \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^3)^(1/3),x]

[Out]

(x^4*(a + b*x^3)^(1/3)*Hypergeometric2F1[-1/3, 4/3, 7/3, -((b*x^3)/a)])/(4*(1 + (b*x^3)/a)^(1/3))

________________________________________________________________________________________

Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\sqrt [3]{b{x}^{3}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)*x^3, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3 + a)^(1/3)*x^3, x)

________________________________________________________________________________________

Sympy [C]  time = 1.06233, size = 39, normalized size = 1.03 \begin{align*} \frac{\sqrt [3]{a} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(1/3)*x**4*gamma(4/3)*hyper((-1/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)*x^3, x)