\(\int x^3 \sqrt [3]{a+b x^3} \, dx\) [523]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {x^4 \left (a+b x^3\right )^{4/3} \operatorname {Hypergeometric2F1}\left (1,\frac {8}{3},\frac {7}{3},-\frac {b x^3}{a}\right )}{4 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {372, 371} \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {x^4 \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {7}{3},-\frac {b x^3}{a}\right )}{4 \sqrt [3]{\frac {b x^3}{a}+1}} \]

[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 4.37 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {x^4 \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {7}{3},-\frac {b x^3}{a}\right )}{4 \sqrt [3]{1+\frac {b x^3}{a}}} \]

[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]

\[\int x^{3} \left (b \,x^{3}+a \right )^{\frac {1}{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[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] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {\sqrt [3]{a} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \]

[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))

Maxima [F]

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

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)*x^3, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\int x^3\,{\left (b\,x^3+a\right )}^{1/3} \,d x \]

[In]

int(x^3*(a + b*x^3)^(1/3),x)

[Out]

int(x^3*(a + b*x^3)^(1/3), x)