\(\int \frac {\sqrt [3]{a+b x^3}}{x^6} \, dx\) [526]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6} \, dx=-\frac {\left (a+b x^3\right )^{4/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,-\frac {2}{3},-\frac {b x^3}{a}\right )}{5 a x^5} \]

[Out]

-1/5*(b*x^3+a)^(4/3)*hypergeom([-1/3, 1],[-2/3],-b*x^3/a)/a/x^5

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} \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6} \, dx=-\frac {\sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {1}{3},-\frac {2}{3},-\frac {b x^3}{a}\right )}{5 x^5 \sqrt [3]{\frac {b x^3}{a}+1}} \]

[In]

Int[(a + b*x^3)^(1/3)/x^6,x]

[Out]

-1/5*((a + b*x^3)^(1/3)*Hypergeometric2F1[-5/3, -1/3, -2/3, -((b*x^3)/a)])/(x^5*(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 \frac {\sqrt [3]{1+\frac {b x^3}{a}}}{x^6} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}} \\ & = -\frac {\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {5}{3},-\frac {1}{3};-\frac {2}{3};-\frac {b x^3}{a}\right )}{5 x^5 \sqrt [3]{1+\frac {b x^3}{a}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6} \, dx=-\frac {\sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {1}{3},-\frac {2}{3},-\frac {b x^3}{a}\right )}{5 x^5 \sqrt [3]{1+\frac {b x^3}{a}}} \]

[In]

Integrate[(a + b*x^3)^(1/3)/x^6,x]

[Out]

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

Maple [F]

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

[In]

int((b*x^3+a)^(1/3)/x^6,x)

[Out]

int((b*x^3+a)^(1/3)/x^6,x)

Fricas [F]

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

[In]

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

[Out]

integral((b*x^3 + a)^(1/3)/x^6, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6} \, dx=\frac {\sqrt [3]{b} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)/x^6, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x^3 + a)^(1/3)/x^6, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^6} \,d x \]

[In]

int((a + b*x^3)^(1/3)/x^6,x)

[Out]

int((a + b*x^3)^(1/3)/x^6, x)