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

Optimal. Leaf size=16 \[ \frac {\left (x^8+1\right )^{3/2}}{6 x^6} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {449} \begin {gather*} \frac {\left (x^8+1\right )^{3/2}}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^8)*Sqrt[1 + x^8])/x^7,x]

[Out]

(1 + x^8)^(3/2)/(6*x^6)

Rule 449

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*} \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}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (x^8+1\right )^{3/2}}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^8)^(3/2)/(6*x^6)

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IntegrateAlgebraic [A]  time = 2.42, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^8\right )^{3/2}}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^8)^(3/2)/(6*x^6)

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fricas [A]  time = 0.47, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.69, size = 25, normalized size = 1.56 \begin {gather*} \frac {1}{6} \, \sqrt {x^{8} + 1} x^{2} + \frac {\sqrt {\frac {1}{x^{8}} + 1}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.10, size = 13, normalized size = 0.81

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.58, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.23, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^8+1\right )}^{3/2}}{6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^8 + 1)^(3/2)/(6*x^6)

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sympy [C]  time = 3.20, size = 66, normalized size = 4.12 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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