3.3.91 \(\int \frac {(2+x^6) (-1-x^4+x^6)}{x^6 (-1+x^6)^{3/4}} \, dx\)

Optimal. Leaf size=26 \[ \frac {2 \sqrt [4]{x^6-1} \left (x^6-5 x^4-1\right )}{5 x^5} \]

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Rubi [A]  time = 0.10, antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1833, 1584, 449, 1478} \begin {gather*} \frac {2 \left (x^6-1\right )^{5/4}}{5 x^5}-\frac {2 \sqrt [4]{x^6-1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + x^6)*(-1 - x^4 + x^6))/(x^6*(-1 + x^6)^(3/4)),x]

[Out]

(-2*(-1 + x^6)^(1/4))/x + (2*(-1 + x^6)^(5/4))/(5*x^5)

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]

Rule 1478

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Sym
bol] :> Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && Eq
Q[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1833

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)*Sum[Coeff[Pq, x, j + (k*n)/2]*x^((k*n)/2), {k, 0, (2*(q - j))/n + 1}]*(a + b*x^n)^p)/c^j, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rubi steps

\begin {align*} \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx &=\int \left (\frac {-2 x^3-x^9}{x^5 \left (-1+x^6\right )^{3/4}}+\frac {-2+x^6+x^{12}}{x^6 \left (-1+x^6\right )^{3/4}}\right ) \, dx\\ &=\int \frac {-2 x^3-x^9}{x^5 \left (-1+x^6\right )^{3/4}} \, dx+\int \frac {-2+x^6+x^{12}}{x^6 \left (-1+x^6\right )^{3/4}} \, dx\\ &=\int \frac {-2-x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx+\int \frac {\sqrt [4]{-1+x^6} \left (2+x^6\right )}{x^6} \, dx\\ &=-\frac {2 \sqrt [4]{-1+x^6}}{x}+\frac {2 \left (-1+x^6\right )^{5/4}}{5 x^5}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 117, normalized size = 4.50 \begin {gather*} \frac {\left (1-x^6\right )^{3/4} \left (14 \, _2F_1\left (-\frac {5}{6},\frac {3}{4};\frac {1}{6};x^6\right )+x^4 \left (-7 x^6 \, _2F_1\left (\frac {3}{4},\frac {5}{6};\frac {11}{6};x^6\right )+70 \, _2F_1\left (-\frac {1}{6},\frac {3}{4};\frac {5}{6};x^6\right )+5 x^8 \, _2F_1\left (\frac {3}{4},\frac {7}{6};\frac {13}{6};x^6\right )+35 x^2 \, _2F_1\left (\frac {1}{6},\frac {3}{4};\frac {7}{6};x^6\right )\right )\right )}{35 x^5 \left (x^6-1\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + x^6)*(-1 - x^4 + x^6))/(x^6*(-1 + x^6)^(3/4)),x]

[Out]

((1 - x^6)^(3/4)*(14*Hypergeometric2F1[-5/6, 3/4, 1/6, x^6] + x^4*(70*Hypergeometric2F1[-1/6, 3/4, 5/6, x^6] +
35*x^2*Hypergeometric2F1[1/6, 3/4, 7/6, x^6] - 7*x^6*Hypergeometric2F1[3/4, 5/6, 11/6, x^6] + 5*x^8*Hypergeom
etric2F1[3/4, 7/6, 13/6, x^6])))/(35*x^5*(-1 + x^6)^(3/4))

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IntegrateAlgebraic [A]  time = 4.18, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{-1+x^6} \left (-1-5 x^4+x^6\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + x^6)*(-1 - x^4 + x^6))/(x^6*(-1 + x^6)^(3/4)),x]

[Out]

(2*(-1 + x^6)^(1/4)*(-1 - 5*x^4 + x^6))/(5*x^5)

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fricas [A]  time = 0.48, size = 22, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (x^{6} - 5 \, x^{4} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+2)*(x^6-x^4-1)/x^6/(x^6-1)^(3/4),x, algorithm="fricas")

[Out]

2/5*(x^6 - 5*x^4 - 1)*(x^6 - 1)^(1/4)/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} - 1\right )}^{\frac {3}{4}} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+2)*(x^6-x^4-1)/x^6/(x^6-1)^(3/4),x, algorithm="giac")

[Out]

integrate((x^6 - x^4 - 1)*(x^6 + 2)/((x^6 - 1)^(3/4)*x^6), x)

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maple [A]  time = 0.12, size = 23, normalized size = 0.88

method	result	size
trager	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}} \left (x^{6}-5 x^{4}-1\right )}{5 x^{5}}\)	\(23\)
risch	\(\frac {\frac {2}{5} x^{12}-\frac {4}{5} x^{6}+\frac {2}{5}-2 x^{10}+2 x^{4}}{x^{5} \left (x^{6}-1\right )^{\frac {3}{4}}}\)	\(33\)
gosper	\(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{6}-5 x^{4}-1\right )}{5 x^{5} \left (x^{6}-1\right )^{\frac {3}{4}}}\)	\(43\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {3}{4}, \frac {7}{6}\right ], \left [\frac {13}{6}\right ], x^{6}\right ) x^{7}}{7 \mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}-\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right ) x^{5}}{5 \mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}+\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {7}{6}\right ], x^{6}\right ) x}{\mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}+\frac {2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {5}{6}, \frac {3}{4}\right ], \left [\frac {1}{6}\right ], x^{6}\right )}{5 \mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x^{5}}+\frac {2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{\mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x}\)	\(159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+2)*(x^6-x^4-1)/x^6/(x^6-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

2/5*(x^6-1)^(1/4)*(x^6-5*x^4-1)/x^5

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maxima [B]  time = 0.71, size = 53, normalized size = 2.04 \begin {gather*} \frac {2 \, {\left (x^{12} - 5 \, x^{10} - 2 \, x^{6} + 5 \, x^{4} + 1\right )}}{5 \, {\left (x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+2)*(x^6-x^4-1)/x^6/(x^6-1)^(3/4),x, algorithm="maxima")

[Out]

2/5*(x^12 - 5*x^10 - 2*x^6 + 5*x^4 + 1)/((x^2 + x + 1)^(3/4)*(x^2 - x + 1)^(3/4)*(x + 1)^(3/4)*(x - 1)^(3/4)*x
^5)

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mupad [B]  time = 0.30, size = 27, normalized size = 1.04 \begin {gather*} \frac {2\,{\left (x^6-1\right )}^{5/4}-10\,x^4\,{\left (x^6-1\right )}^{1/4}}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^6 + 2)*(x^4 - x^6 + 1))/(x^6*(x^6 - 1)^(3/4)),x)

[Out]

(2*(x^6 - 1)^(5/4) - 10*x^4*(x^6 - 1)^(1/4))/(5*x^5)

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sympy [C]  time = 4.50, size = 168, normalized size = 6.46 \begin {gather*} \frac {x^{7} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {13}{6}\right )} - \frac {x^{5} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} + \frac {x e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{4} \\ \frac {7}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} + \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{4} \\ \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x^{5} \Gamma \left (\frac {1}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+2)*(x**6-x**4-1)/x**6/(x**6-1)**(3/4),x)

[Out]

x**7*exp(-3*I*pi/4)*gamma(7/6)*hyper((3/4, 7/6), (13/6,), x**6)/(6*gamma(13/6)) - x**5*exp(-3*I*pi/4)*gamma(5/
6)*hyper((3/4, 5/6), (11/6,), x**6)/(6*gamma(11/6)) + x*exp(-3*I*pi/4)*gamma(1/6)*hyper((1/6, 3/4), (7/6,), x*
*6)/(6*gamma(7/6)) + exp(I*pi/4)*gamma(-1/6)*hyper((-1/6, 3/4), (5/6,), x**6)/(3*x*gamma(5/6)) + exp(I*pi/4)*g
amma(-5/6)*hyper((-5/6, 3/4), (1/6,), x**6)/(3*x**5*gamma(1/6))

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