\(\int \frac {(-1+x^8) \sqrt {1+x^8}}{x^7} \, dx\) [115]

Optimal result

Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {\left (1+x^8\right )^{3/2}}{6 x^6} \]

[Out]

1/6*(x^8+1)^(3/2)/x^6

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} \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {\left (x^8+1\right )^{3/2}}{6 x^6} \]

[In]

Int[((-1 + x^8)*Sqrt[1 + x^8])/x^7,x]

[Out]

(1 + x^8)^(3/2)/(6*x^6)

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 {\left (1+x^8\right )^{3/2}}{6 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {\left (1+x^8\right )^{3/2}}{6 x^6} \]

[In]

Integrate[((-1 + x^8)*Sqrt[1 + x^8])/x^7,x]

[Out]

(1 + x^8)^(3/2)/(6*x^6)

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {\left (x^{8}+1\right )^{\frac {3}{2}}}{6 x^{6}}\)	\(13\)
trager	\(\frac {\left (x^{8}+1\right )^{\frac {3}{2}}}{6 x^{6}}\)	\(13\)
pseudoelliptic	\(\frac {\left (x^{8}+1\right ) \sqrt {\frac {x^{8}+1}{x^{2}}}}{6 x^{5}}\)	\(22\)
risch	\(\frac {x^{16}+2 x^{8}+1}{6 x^{6} \sqrt {x^{8}+1}}\)	\(23\)
meijerg	\(\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{8}\right )}{6 x^{6}}+\frac {x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{8}\right )}{2}\)	\(34\)

[In]

int((x^8-1)*(x^8+1)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

1/6*(x^8+1)^(3/2)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]

[In]

integrate((x^8-1)*(x^8+1)^(1/2)/x^7,x, algorithm="fricas")

[Out]

1/6*(x^8 + 1)^(3/2)/x^6

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.12 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {x^{2} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} - \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 x^{6} \Gamma \left (\frac {1}{4}\right )} \]

[In]

integrate((x**8-1)*(x**8+1)**(1/2)/x**7,x)

[Out]

x**2*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), x**8*exp_polar(I*pi))/(8*gamma(5/4)) - gamma(-3/4)*hyper((-3/4, -1/
2), (1/4,), x**8*exp_polar(I*pi))/(8*x**6*gamma(1/4))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]

[In]

integrate((x^8-1)*(x^8+1)^(1/2)/x^7,x, algorithm="maxima")

[Out]

1/6*(x^8 + 1)^(3/2)/x^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {1}{6} \, \sqrt {x^{8} + 1} x^{2} + \frac {\sqrt {\frac {1}{x^{8}} + 1}}{6 \, x^{2}} \]

[In]

integrate((x^8-1)*(x^8+1)^(1/2)/x^7,x, algorithm="giac")

[Out]

1/6*sqrt(x^8 + 1)*x^2 + 1/6*sqrt(1/x^8 + 1)/x^2

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {{\left (x^8+1\right )}^{3/2}}{6\,x^6} \]

[In]

int(((x^8 - 1)*(x^8 + 1)^(1/2))/x^7,x)

[Out]

(x^8 + 1)^(3/2)/(6*x^6)