3.4.36 \(\int \frac {(-2+x^6) (1-x^4+x^6)}{x^8 \sqrt [4]{1+x^6}} \, dx\)

Optimal. Leaf size=28 \[ \frac {2 \left (x^6+1\right )^{3/4} \left (3 x^6-7 x^4+3\right )}{21 x^7} \]

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Rubi [A]  time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.18, 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 )^{7/4}}{7 x^7}-\frac {2 \left (x^6+1\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-2 + x^6)*(1 - x^4 + x^6))/(x^8*(1 + x^6)^(1/4)),x]

[Out]

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

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

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Mathematica [C]  time = 0.05, size = 108, normalized size = 3.86 \begin {gather*} \frac {\, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};-x^6\right )}{x}+\frac {2 \, _2F_1\left (-\frac {7}{6},\frac {1}{4};-\frac {1}{6};-x^6\right )}{7 x^7}+\frac {1}{5} x^5 \, _2F_1\left (\frac {1}{4},\frac {5}{6};\frac {11}{6};-x^6\right )-\frac {1}{3} x^3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^6\right )-\frac {2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};-x^6\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2 + x^6)*(1 - x^4 + x^6))/(x^8*(1 + x^6)^(1/4)),x]

[Out]

(2*Hypergeometric2F1[-7/6, 1/4, -1/6, -x^6])/(7*x^7) - (2*Hypergeometric2F1[-1/2, 1/4, 1/2, -x^6])/(3*x^3) + H
ypergeometric2F1[-1/6, 1/4, 5/6, -x^6]/x - (x^3*Hypergeometric2F1[1/4, 1/2, 3/2, -x^6])/3 + (x^5*Hypergeometri
c2F1[1/4, 5/6, 11/6, -x^6])/5

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IntegrateAlgebraic [A]  time = 2.09, size = 28, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+x^6\right )^{3/4} \left (3-7 x^4+3 x^6\right )}{21 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2 + x^6)*(1 - x^4 + x^6))/(x^8*(1 + x^6)^(1/4)),x]

[Out]

(2*(1 + x^6)^(3/4)*(3 - 7*x^4 + 3*x^6))/(21*x^7)

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fricas [A]  time = 0.48, size = 24, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (3 \, x^{6} - 7 \, x^{4} + 3\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6-x^4+1)/x^8/(x^6+1)^(1/4),x, algorithm="fricas")

[Out]

2/21*(3*x^6 - 7*x^4 + 3)*(x^6 + 1)^(3/4)/x^7

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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 {1}{4}} x^{8}}\,{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^8/(x^6+1)^(1/4),x, algorithm="giac")

[Out]

integrate((x^6 - x^4 + 1)*(x^6 - 2)/((x^6 + 1)^(1/4)*x^8), x)

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maple [A]  time = 0.10, size = 25, normalized size = 0.89

method	result	size
trager	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}+3\right )}{21 x^{7}}\)	\(25\)
risch	\(\frac {\frac {2}{7} x^{12}+\frac {4}{7} x^{6}-\frac {2}{3} x^{10}-\frac {2}{3} x^{4}+\frac {2}{7}}{x^{7} \left (x^{6}+1\right )^{\frac {1}{4}}}\)	\(35\)
gosper	\(\frac {2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right ) \left (3 x^{6}-7 x^{4}+3\right )}{21 x^{7} \left (x^{6}+1\right )^{\frac {1}{4}}}\)	\(40\)
meijerg	\(\frac {2 \hypergeom \left (\left [-\frac {7}{6}, \frac {1}{4}\right ], \left [-\frac {1}{6}\right ], -x^{6}\right )}{7 x^{7}}+\frac {\hypergeom \left (\left [-\frac {1}{6}, \frac {1}{4}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}-\frac {2 \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], -x^{6}\right )}{3 x^{3}}+\frac {\hypergeom \left (\left [\frac {1}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], -x^{6}\right ) x^{5}}{5}-\frac {\hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{6}\right ) x^{3}}{3}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-2)*(x^6-x^4+1)/x^8/(x^6+1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

2/21*(x^6+1)^(3/4)*(3*x^6-7*x^4+3)/x^7

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maxima [A]  time = 0.67, size = 46, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (3 \, x^{12} - 7 \, x^{10} + 6 \, x^{6} - 7 \, x^{4} + 3\right )}}{21 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6-x^4+1)/x^8/(x^6+1)^(1/4),x, algorithm="maxima")

[Out]

2/21*(3*x^12 - 7*x^10 + 6*x^6 - 7*x^4 + 3)/((x^4 - x^2 + 1)^(1/4)*(x^2 + 1)^(1/4)*x^7)

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mupad [B]  time = 0.18, size = 39, normalized size = 1.39 \begin {gather*} \frac {6\,{\left (x^6+1\right )}^{3/4}-14\,x^4\,{\left (x^6+1\right )}^{3/4}+6\,x^6\,{\left (x^6+1\right )}^{3/4}}{21\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 2)*(x^6 - x^4 + 1))/(x^8*(x^6 + 1)^(1/4)),x)

[Out]

(6*(x^6 + 1)^(3/4) - 14*x^4*(x^6 + 1)^(3/4) + 6*x^6*(x^6 + 1)^(3/4))/(21*x^7)

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sympy [C]  time = 4.61, size = 143, normalized size = 5.11 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3} - \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {1}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {2 {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{3}} - \frac {\Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, \frac {1}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{7} \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**8/(x**6+1)**(1/4),x)

[Out]

x**5*gamma(5/6)*hyper((1/4, 5/6), (11/6,), x**6*exp_polar(I*pi))/(6*gamma(11/6)) - x**3*hyper((1/4, 1/2), (3/2
,), x**6*exp_polar(I*pi))/3 - gamma(-1/6)*hyper((-1/6, 1/4), (5/6,), x**6*exp_polar(I*pi))/(6*x*gamma(5/6)) -
2*hyper((-1/2, 1/4), (1/2,), x**6*exp_polar(I*pi))/(3*x**3) - gamma(-7/6)*hyper((-7/6, 1/4), (-1/6,), x**6*exp
_polar(I*pi))/(3*x**7*gamma(-1/6))

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