\(\int \frac {(-2+x^3) (1+x^3)^{3/2}}{x^6} \, dx\) [74]

Optimal result

Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \left (1+x^3\right )^{5/2}}{5 x^5} \]

[Out]

2/5*(x^3+1)^(5/2)/x^5

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \left (x^3+1\right )^{5/2}}{5 x^5} \]

[In]

Int[((-2 + x^3)*(1 + x^3)^(3/2))/x^6,x]

[Out]

(2*(1 + x^3)^(5/2))/(5*x^5)

Rule 460

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (1+x^3\right )^{5/2}}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \left (1+x^3\right )^{5/2}}{5 x^5} \]

[In]

Integrate[((-2 + x^3)*(1 + x^3)^(3/2))/x^6,x]

[Out]

(2*(1 + x^3)^(5/2))/(5*x^5)

Maple [A] (verified)

Time = 2.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {2 \left (x^{3}+1\right )^{\frac {5}{2}}}{5 x^{5}}\)	\(13\)
trager	\(\frac {2 \left (x^{6}+2 x^{3}+1\right ) \sqrt {x^{3}+1}}{5 x^{5}}\)	\(23\)
gosper	\(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+1\right )^{\frac {3}{2}}}{5 x^{5}}\)	\(24\)
risch	\(\frac {\frac {2}{5} x^{9}+\frac {6}{5} x^{6}+\frac {6}{5} x^{3}+\frac {2}{5}}{x^{5} \sqrt {x^{3}+1}}\)	\(28\)
meijerg	\(\frac {2 \operatorname {hypergeom}\left (\left [-\frac {5}{3}, -\frac {3}{2}\right ], \left [-\frac {2}{3}\right ], -x^{3}\right )}{5 x^{5}}-\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{2}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], -x^{3}\right )}{2 x^{2}}\)	\(34\)
default	\(\frac {4 \sqrt {x^{3}+1}}{5 x^{2}}+\frac {2 x \sqrt {x^{3}+1}}{5}+\frac {2 \sqrt {x^{3}+1}}{5 x^{5}}\)	\(36\)
elliptic	\(\frac {4 \sqrt {x^{3}+1}}{5 x^{2}}+\frac {2 x \sqrt {x^{3}+1}}{5}+\frac {2 \sqrt {x^{3}+1}}{5 x^{5}}\)	\(36\)

[In]

int((x^3-2)*(x^3+1)^(3/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

2/5*(x^3+1)^(5/2)/x^5

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} \sqrt {x^{3} + 1}}{5 \, x^{5}} \]

[In]

integrate((x^3-2)*(x^3+1)^(3/2)/x^6,x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 6.56 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {2 \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \]

[In]

integrate((x**3-2)*(x**3+1)**(3/2)/x**6,x)

[Out]

x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), x**3*exp_polar(I*pi))/(3*gamma(4/3)) - gamma(-2/3)*hyper((-2/3, -1/2),
(1/3,), x**3*exp_polar(I*pi))/(3*x**2*gamma(1/3)) - 2*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3,), x**3*exp_polar
(I*pi))/(3*x**5*gamma(-2/3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{5 \, x^{5}} \]

[In]

integrate((x^3-2)*(x^3+1)^(3/2)/x^6,x, algorithm="maxima")

[Out]

2/5*(x^6 + 2*x^3 + 1)*sqrt(x^2 - x + 1)*sqrt(x + 1)/x^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2}{5} \, \sqrt {x^{3} + 1} x + \frac {2}{5} \, \sqrt {\frac {1}{x} + \frac {1}{x^{4}}} {\left (\frac {1}{x^{3}} + 2\right )} \]

[In]

integrate((x^3-2)*(x^3+1)^(3/2)/x^6,x, algorithm="giac")

[Out]

2/5*sqrt(x^3 + 1)*x + 2/5*sqrt(1/x + 1/x^4)*(1/x^3 + 2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2\,{\left (x^3+1\right )}^{5/2}}{5\,x^5} \]

[In]

int(((x^3 + 1)^(3/2)*(x^3 - 2))/x^6,x)

[Out]

(2*(x^3 + 1)^(5/2))/(5*x^5)