3.1.94 \(\int \frac {(-4+x^5) (1+x^5)^{3/4}}{x^8} \, dx\)

Optimal. Leaf size=16 \[ \frac {4 \left (x^5+1\right )^{7/4}}{7 x^7} \]

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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 {4 \left (x^5+1\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4 + x^5)*(1 + x^5)^(3/4))/x^8,x]

[Out]

(4*(1 + x^5)^(7/4))/(7*x^7)

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 (-4+x^5\right ) \left (1+x^5\right )^{3/4}}{x^8} \, dx &=\frac {4 \left (1+x^5\right )^{7/4}}{7 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^5+1\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 + x^5)*(1 + x^5)^(3/4))/x^8,x]

[Out]

(4*(1 + x^5)^(7/4))/(7*x^7)

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IntegrateAlgebraic [A]  time = 0.11, size = 16, normalized size = 1.00 \begin {gather*} \frac {4 \left (1+x^5\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4 + x^5)*(1 + x^5)^(3/4))/x^8,x]

[Out]

(4*(1 + x^5)^(7/4))/(7*x^7)

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fricas [A]  time = 0.48, size = 12, normalized size = 0.75 \begin {gather*} \frac {4 \, {\left (x^{5} + 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-4)*(x^5+1)^(3/4)/x^8,x, algorithm="fricas")

[Out]

4/7*(x^5 + 1)^(7/4)/x^7

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} + 1\right )}^{\frac {3}{4}} {\left (x^{5} - 4\right )}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-4)*(x^5+1)^(3/4)/x^8,x, algorithm="giac")

[Out]

integrate((x^5 + 1)^(3/4)*(x^5 - 4)/x^8, x)

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

method	result	size
trager	\(\frac {4 \left (x^{5}+1\right )^{\frac {7}{4}}}{7 x^{7}}\)	\(13\)
risch	\(\frac {\frac {4}{7} x^{10}+\frac {8}{7} x^{5}+\frac {4}{7}}{x^{7} \left (x^{5}+1\right )^{\frac {1}{4}}}\)	\(23\)
gosper	\(\frac {4 \left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (x^{5}+1\right )^{\frac {3}{4}}}{7 x^{7}}\)	\(32\)
meijerg	\(\frac {4 \hypergeom \left (\left [-\frac {7}{5}, -\frac {3}{4}\right ], \left [-\frac {2}{5}\right ], -x^{5}\right )}{7 x^{7}}-\frac {\hypergeom \left (\left [-\frac {3}{4}, -\frac {2}{5}\right ], \left [\frac {3}{5}\right ], -x^{5}\right )}{2 x^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5-4)*(x^5+1)^(3/4)/x^8,x,method=_RETURNVERBOSE)

[Out]

4/7*(x^5+1)^(7/4)/x^7

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maxima [B]  time = 0.89, size = 33, normalized size = 2.06 \begin {gather*} \frac {4 \, {\left (x^{5} + 1\right )} {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-4)*(x^5+1)^(3/4)/x^8,x, algorithm="maxima")

[Out]

4/7*(x^5 + 1)*(x^4 - x^3 + x^2 - x + 1)^(3/4)*(x + 1)^(3/4)/x^7

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mupad [B]  time = 0.24, size = 27, normalized size = 1.69 \begin {gather*} \frac {4\,{\left (x^5+1\right )}^{3/4}+4\,x^5\,{\left (x^5+1\right )}^{3/4}}{7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5 + 1)^(3/4)*(x^5 - 4))/x^8,x)

[Out]

(4*(x^5 + 1)^(3/4) + 4*x^5*(x^5 + 1)^(3/4))/(7*x^7)

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sympy [C]  time = 2.69, size = 75, normalized size = 4.69 \begin {gather*} \frac {\Gamma \left (- \frac {2}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {2}{5} \\ \frac {3}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{2} \Gamma \left (\frac {3}{5}\right )} - \frac {4 \Gamma \left (- \frac {7}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{5}, - \frac {3}{4} \\ - \frac {2}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{7} \Gamma \left (- \frac {2}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5-4)*(x**5+1)**(3/4)/x**8,x)

[Out]

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

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