3.5.96 \(\int \frac {(-2+x^6) (1+x^6)^{3/4} (1-x^4+x^6)}{x^{12}} \, dx\)

Optimal. Leaf size=38 \[ \frac {2 \left (x^6+1\right )^{3/4} \left (7 x^{12}-11 x^{10}+14 x^6-11 x^4+7\right )}{77 x^{11}} \]

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Rubi [A]  time = 0.09, antiderivative size = 33, normalized size of antiderivative = 0.87, 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 )^{11/4}}{11 x^{11}}-\frac {2 \left (x^6+1\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-2 + x^6)*(1 + x^6)^(3/4)*(1 - x^4 + x^6))/x^12,x]

[Out]

(-2*(1 + x^6)^(7/4))/(7*x^7) + (2*(1 + x^6)^(11/4))/(11*x^11)

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^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx &=\int \left (\frac {\left (1+x^6\right )^{3/4} \left (2 x^3-x^9\right )}{x^{11}}+\frac {\left (1+x^6\right )^{3/4} \left (-2-x^6+x^{12}\right )}{x^{12}}\right ) \, dx\\ &=\int \frac {\left (1+x^6\right )^{3/4} \left (2 x^3-x^9\right )}{x^{11}} \, dx+\int \frac {\left (1+x^6\right )^{3/4} \left (-2-x^6+x^{12}\right )}{x^{12}} \, dx\\ &=\int \frac {\left (2-x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx+\int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{7/4}}{x^{12}} \, dx\\ &=-\frac {2 \left (1+x^6\right )^{7/4}}{7 x^7}+\frac {2 \left (1+x^6\right )^{11/4}}{11 x^{11}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 103, normalized size = 2.71 \begin {gather*} \frac {\, _2F_1\left (-\frac {3}{4},-\frac {1}{6};\frac {5}{6};-x^6\right )}{x}+x \, _2F_1\left (-\frac {3}{4},\frac {1}{6};\frac {7}{6};-x^6\right )+\frac {2 \, _2F_1\left (-\frac {11}{6},-\frac {3}{4};-\frac {5}{6};-x^6\right )}{11 x^{11}}-\frac {2 \, _2F_1\left (-\frac {7}{6},-\frac {3}{4};-\frac {1}{6};-x^6\right )}{7 x^7}+\frac {\, _2F_1\left (-\frac {5}{6},-\frac {3}{4};\frac {1}{6};-x^6\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2 + x^6)*(1 + x^6)^(3/4)*(1 - x^4 + x^6))/x^12,x]

[Out]

(2*Hypergeometric2F1[-11/6, -3/4, -5/6, -x^6])/(11*x^11) - (2*Hypergeometric2F1[-7/6, -3/4, -1/6, -x^6])/(7*x^
7) + Hypergeometric2F1[-5/6, -3/4, 1/6, -x^6]/(5*x^5) + Hypergeometric2F1[-3/4, -1/6, 5/6, -x^6]/x + x*Hyperge
ometric2F1[-3/4, 1/6, 7/6, -x^6]

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IntegrateAlgebraic [A]  time = 2.33, size = 28, normalized size = 0.74 \begin {gather*} \frac {2 \left (1+x^6\right )^{7/4} \left (7-11 x^4+7 x^6\right )}{77 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2 + x^6)*(1 + x^6)^(3/4)*(1 - x^4 + x^6))/x^12,x]

[Out]

(2*(1 + x^6)^(7/4)*(7 - 11*x^4 + 7*x^6))/(77*x^11)

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fricas [A]  time = 0.47, size = 34, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (7 \, x^{12} - 11 \, x^{10} + 14 \, x^{6} - 11 \, x^{4} + 7\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x, algorithm="fricas")

[Out]

2/77*(7*x^12 - 11*x^10 + 14*x^6 - 11*x^4 + 7)*(x^6 + 1)^(3/4)/x^11

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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} + 1\right )}^{\frac {3}{4}} {\left (x^{6} - 2\right )}}{x^{12}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x, algorithm="giac")

[Out]

integrate((x^6 - x^4 + 1)*(x^6 + 1)^(3/4)*(x^6 - 2)/x^12, x)

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maple [A]  time = 0.14, size = 35, normalized size = 0.92

method	result	size
trager	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}} \left (7 x^{12}-11 x^{10}+14 x^{6}-11 x^{4}+7\right )}{77 x^{11}}\)	\(35\)
gosper	\(\frac {2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right ) \left (7 x^{6}-11 x^{4}+7\right ) \left (x^{6}+1\right )^{\frac {3}{4}}}{77 x^{11}}\)	\(40\)
risch	\(\frac {\frac {2}{11} x^{18}+\frac {6}{11} x^{12}+\frac {6}{11} x^{6}+\frac {2}{11}-\frac {2}{7} x^{16}-\frac {4}{7} x^{10}-\frac {2}{7} x^{4}}{x^{11} \left (x^{6}+1\right )^{\frac {1}{4}}}\)	\(45\)
meijerg	\(\frac {\hypergeom \left (\left [-\frac {5}{6}, -\frac {3}{4}\right ], \left [\frac {1}{6}\right ], -x^{6}\right )}{5 x^{5}}-\frac {2 \hypergeom \left (\left [-\frac {7}{6}, -\frac {3}{4}\right ], \left [-\frac {1}{6}\right ], -x^{6}\right )}{7 x^{7}}+\frac {2 \hypergeom \left (\left [-\frac {11}{6}, -\frac {3}{4}\right ], \left [-\frac {5}{6}\right ], -x^{6}\right )}{11 x^{11}}+\hypergeom \left (\left [-\frac {3}{4}, \frac {1}{6}\right ], \left [\frac {7}{6}\right ], -x^{6}\right ) x +\frac {\hypergeom \left (\left [-\frac {3}{4}, -\frac {1}{6}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x,method=_RETURNVERBOSE)

[Out]

2/77*(x^6+1)^(3/4)*(7*x^12-11*x^10+14*x^6-11*x^4+7)/x^11

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maxima [A]  time = 0.77, size = 46, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (7 \, x^{12} - 11 \, x^{10} + 14 \, x^{6} - 11 \, x^{4} + 7\right )} {\left (x^{4} - x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x, algorithm="maxima")

[Out]

2/77*(7*x^12 - 11*x^10 + 14*x^6 - 11*x^4 + 7)*(x^4 - x^2 + 1)^(3/4)*(x^2 + 1)^(3/4)/x^11

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mupad [B]  time = 0.72, size = 59, normalized size = 1.55 \begin {gather*} \frac {2\,x\,{\left (x^6+1\right )}^{3/4}}{11}-\frac {2\,{\left (x^6+1\right )}^{3/4}}{7\,x}+\frac {4\,{\left (x^6+1\right )}^{3/4}}{11\,x^5}-\frac {2\,{\left (x^6+1\right )}^{3/4}}{7\,x^7}+\frac {2\,{\left (x^6+1\right )}^{3/4}}{11\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 + 1)^(3/4)*(x^6 - 2)*(x^6 - x^4 + 1))/x^12,x)

[Out]

(2*x*(x^6 + 1)^(3/4))/11 - (2*(x^6 + 1)^(3/4))/(7*x) + (4*(x^6 + 1)^(3/4))/(11*x^5) - (2*(x^6 + 1)^(3/4))/(7*x
^7) + (2*(x^6 + 1)^(3/4))/(11*x^11)

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sympy [C]  time = 6.68, size = 177, normalized size = 4.66 \begin {gather*} \frac {x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} - \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {\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} e^{i \pi }} \right )}}{6 x^{5} \Gamma \left (\frac {1}{6}\right )} + \frac {\Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {3}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{7} \Gamma \left (- \frac {1}{6}\right )} - \frac {\Gamma \left (- \frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{6}, - \frac {3}{4} \\ - \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{11} \Gamma \left (- \frac {5}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-2)*(x**6+1)**(3/4)*(x**6-x**4+1)/x**12,x)

[Out]

x*gamma(1/6)*hyper((-3/4, 1/6), (7/6,), x**6*exp_polar(I*pi))/(6*gamma(7/6)) - gamma(-1/6)*hyper((-3/4, -1/6),
(5/6,), x**6*exp_polar(I*pi))/(6*x*gamma(5/6)) - gamma(-5/6)*hyper((-5/6, -3/4), (1/6,), x**6*exp_polar(I*pi)
)/(6*x**5*gamma(1/6)) + gamma(-7/6)*hyper((-7/6, -3/4), (-1/6,), x**6*exp_polar(I*pi))/(3*x**7*gamma(-1/6)) -
gamma(-11/6)*hyper((-11/6, -3/4), (-5/6,), x**6*exp_polar(I*pi))/(3*x**11*gamma(-5/6))

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