3.1.93 \(\int \frac {(-6+x^5) (-1+x^5)^{2/3}}{x^{11}} \, dx\)

Optimal. Leaf size=16 \[ -\frac {3 \left (x^5-1\right )^{5/3}}{5 x^{10}} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {446, 74} \begin {gather*} -\frac {3 \left (x^5-1\right )^{5/3}}{5 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-1 + x^5)^(5/3))/(5*x^10)

Rule 74

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 446

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*} \int \frac {\left (-6+x^5\right ) \left (-1+x^5\right )^{2/3}}{x^{11}} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {(-6+x) (-1+x)^{2/3}}{x^3} \, dx,x,x^5\right )\\ &=-\frac {3 \left (-1+x^5\right )^{5/3}}{5 x^{10}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^5-1\right )^{5/3}}{5 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-1 + x^5)^(5/3))/(5*x^10)

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IntegrateAlgebraic [A]  time = 0.09, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^5-1\right )^{5/3}}{5 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-1 + x^5)^(5/3))/(5*x^10)

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fricas [A]  time = 0.40, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3 \, {\left (x^{5} - 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*(x^5 - 1)^(5/3)/x^10

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giac [A]  time = 0.43, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3 \, {\left (x^{5} - 1\right )}^{\frac {5}{3}}}{5 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*(x^5 - 1)^(5/3)/x^10

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maple [B]  time = 0.01, size = 28, normalized size = 1.75 \begin {gather*} -\frac {3 \left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5/x^10*(-1+x)*(x^4+x^3+x^2+x+1)*(x^5-1)^(2/3)

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maxima [B]  time = 0.66, size = 50, normalized size = 3.12 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 0.27, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3\,{\left (x^5-1\right )}^{5/3}}{5\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(x^5 - 1)^(5/3))/(5*x^10)

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sympy [C]  time = 4.16, size = 71, normalized size = 4.44 \begin {gather*} - \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^{2 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^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {20}{3}} \Gamma \left (\frac {7}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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