\(\int \frac {(1+x^5)^{2/3} (6+x^5)}{x^{11}} \, dx\) [96]

Optimal result

Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \left (1+x^5\right )^{5/3}}{5 x^{10}} \]

[Out]

-3/5*(x^5+1)^(5/3)/x^10

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 75} \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \left (x^5+1\right )^{5/3}}{5 x^{10}} \]

[In]

Int[((1 + x^5)^(2/3)*(6 + x^5))/x^11,x]

[Out]

(-3*(1 + x^5)^(5/3))/(5*x^10)

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {(1+x)^{2/3} (6+x)}{x^3} \, dx,x,x^5\right ) \\ & = -\frac {3 \left (1+x^5\right )^{5/3}}{5 x^{10}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \left (1+x^5\right )^{5/3}}{5 x^{10}} \]

[In]

Integrate[((1 + x^5)^(2/3)*(6 + x^5))/x^11,x]

[Out]

(-3*(1 + x^5)^(5/3))/(5*x^10)

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
trager	\(-\frac {3 \left (x^{5}+1\right )^{\frac {5}{3}}}{5 x^{10}}\)	\(13\)
pseudoelliptic	\(-\frac {3 \left (x^{5}+1\right )^{\frac {5}{3}}}{5 x^{10}}\)	\(13\)
risch	\(-\frac {3 \left (x^{10}+2 x^{5}+1\right )}{5 x^{10} \left (x^{5}+1\right )^{\frac {1}{3}}}\)	\(23\)
gosper	\(-\frac {3 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right ) \left (x^{5}+1\right )^{\frac {2}{3}}}{5 x^{10}}\)	\(32\)
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi }-\frac {2 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {4 \pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 4\right ], -x^{5}\right )}{81 \Gamma \left (\frac {2}{3}\right )}+\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{2 \Gamma \left (\frac {2}{3}\right ) x^{10}}+\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{5 \pi }\)	\(166\)

[In]

int((x^5+1)^(2/3)*(x^5+6)/x^11,x,method=_RETURNVERBOSE)

[Out]

-3/5*(x^5+1)^(5/3)/x^10

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \, {\left (x^{5} + 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \]

[In]

integrate((x^5+1)^(2/3)*(x^5+6)/x^11,x, algorithm="fricas")

[Out]

-3/5*(x^5 + 1)^(5/3)/x^10

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 77.78 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.38 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} - \frac {6 \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {20}{3}} \Gamma \left (\frac {7}{3}\right )} \]

[In]

integrate((x**5+1)**(2/3)*(x**5+6)/x**11,x)

[Out]

-gamma(1/3)*hyper((-2/3, 1/3), (4/3,), exp_polar(I*pi)/x**5)/(5*x**(5/3)*gamma(4/3)) - 6*gamma(4/3)*hyper((-2/
3, 4/3), (7/3,), exp_polar(I*pi)/x**5)/(5*x**(20/3)*gamma(7/3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (12) = 24\).

Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.12 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=\frac {2 \, {\left (x^{5} + 1\right )}^{\frac {5}{3}} + {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, {\left (2 \, x^{5} - {\left (x^{5} + 1\right )}^{2} + 1\right )}} - \frac {{\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]

[In]

integrate((x^5+1)^(2/3)*(x^5+6)/x^11,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3 \, {\left (x^{5} + 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \]

[In]

integrate((x^5+1)^(2/3)*(x^5+6)/x^11,x, algorithm="giac")

[Out]

-3/5*(x^5 + 1)^(5/3)/x^10

Mupad [B] (verification not implemented)

Time = 5.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (6+x^5\right )}{x^{11}} \, dx=-\frac {3\,{\left (x^5+1\right )}^{5/3}}{5\,x^{10}} \]

[In]

int(((x^5 + 1)^(2/3)*(x^5 + 6))/x^11,x)

[Out]

-(3*(x^5 + 1)^(5/3))/(5*x^10)