3.22.25 \(\int \frac {\sqrt [4]{2+3 x^4} (4+6 x^4+x^8)}{x^6 (1+x^4) (1+2 x^4)} \, dx\)

Optimal. Leaf size=154 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{3 x^4+2}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{3 x^4+2}}\right )+\frac {4 \sqrt [4]{3 x^4+2} \left (6 x^4-1\right )}{5 x^5}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{3 x^4+2}}{\sqrt {3 x^4+2}-x^2}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{3 x^4+2}}{\sqrt {3 x^4+2}+x^2}\right )}{2 \sqrt {2}} \]

________________________________________________________________________________________

Rubi [C]  time = 0.97, antiderivative size = 157, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {6725, 264, 277, 331, 298, 203, 206, 510} \begin {gather*} \frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )+\frac {6 \sqrt [4]{3 x^4+2}}{x}+3 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )-3 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )-\frac {2 \left (3 x^4+2\right )^{5/4}}{5 x^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((2 + 3*x^4)^(1/4)*(4 + 6*x^4 + x^8))/(x^6*(1 + x^4)*(1 + 2*x^4)),x]

[Out]

(6*(2 + 3*x^4)^(1/4))/x - (2*(2 + 3*x^4)^(5/4))/(5*x^5) + (10*2^(1/4)*x^3*AppellF1[3/4, 1, -1/4, 7/4, -2*x^4,
(-3*x^4)/2])/3 + (2^(1/4)*x^3*AppellF1[3/4, 1, -1/4, 7/4, -x^4, (-3*x^4)/2])/3 + 3*3^(1/4)*ArcTan[(3^(1/4)*x)/
(2 + 3*x^4)^(1/4)] - 3*3^(1/4)*ArcTanh[(3^(1/4)*x)/(2 + 3*x^4)^(1/4)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx &=\int \left (\frac {4 \sqrt [4]{2+3 x^4}}{x^6}-\frac {6 \sqrt [4]{2+3 x^4}}{x^2}+\frac {x^2 \sqrt [4]{2+3 x^4}}{1+x^4}+\frac {10 x^2 \sqrt [4]{2+3 x^4}}{1+2 x^4}\right ) \, dx\\ &=4 \int \frac {\sqrt [4]{2+3 x^4}}{x^6} \, dx-6 \int \frac {\sqrt [4]{2+3 x^4}}{x^2} \, dx+10 \int \frac {x^2 \sqrt [4]{2+3 x^4}}{1+2 x^4} \, dx+\int \frac {x^2 \sqrt [4]{2+3 x^4}}{1+x^4} \, dx\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )-18 \int \frac {x^2}{\left (2+3 x^4\right )^{3/4}} \, dx\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )-18 \operatorname {Subst}\left (\int \frac {x^2}{1-3 x^4} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right )\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )-\left (3 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right )+\left (3 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right )\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )+3 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right )-3 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.21, size = 94, normalized size = 0.61 \begin {gather*} \frac {x^3 \left (5 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {x^4}{3 x^4+2}\right )-\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {x^4}{3 x^4+2}\right )\right )}{3 \left (3 x^4+2\right )^{3/4}}+\left (\frac {24}{5 x}-\frac {4}{5 x^5}\right ) \sqrt [4]{3 x^4+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x^4)^(1/4)*(4 + 6*x^4 + x^8))/(x^6*(1 + x^4)*(1 + 2*x^4)),x]

[Out]

(-4/(5*x^5) + 24/(5*x))*(2 + 3*x^4)^(1/4) + (x^3*(5*Hypergeometric2F1[3/4, 1, 7/4, -(x^4/(2 + 3*x^4))] - Hyper
geometric2F1[3/4, 1, 7/4, x^4/(2 + 3*x^4)]))/(3*(2 + 3*x^4)^(3/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.43, size = 154, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{2+3 x^4} \left (-1+6 x^4\right )}{5 x^5}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{-x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + 3*x^4)^(1/4)*(4 + 6*x^4 + x^8))/(x^6*(1 + x^4)*(1 + 2*x^4)),x]

[Out]

(4*(2 + 3*x^4)^(1/4)*(-1 + 6*x^4))/(5*x^5) + ArcTan[x/(2 + 3*x^4)^(1/4)]/2 + (5*ArcTan[(Sqrt[2]*x*(2 + 3*x^4)^
(1/4))/(-x^2 + Sqrt[2 + 3*x^4])])/(2*Sqrt[2]) - ArcTanh[x/(2 + 3*x^4)^(1/4)]/2 - (5*ArcTanh[(Sqrt[2]*x*(2 + 3*
x^4)^(1/4))/(x^2 + Sqrt[2 + 3*x^4])])/(2*Sqrt[2])

________________________________________________________________________________________

fricas [B]  time = 6.13, size = 724, normalized size = 4.70 \begin {gather*} \frac {100 \, \sqrt {2} x^{5} \arctan \left (-\frac {4 \, x^{8} + 4 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (4 \, x^{7} + 3 \, x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (2 \, x^{6} + x^{2}\right )} \sqrt {3 \, x^{4} + 2} - {\left (4 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x^{5} + \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} - \sqrt {2} {\left (4 \, x^{8} + x^{4} - 1\right )} + 2 \, {\left (2 \, x^{7} + x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {2 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}} + 1}{8 \, x^{8} + 4 \, x^{4} - 1}\right ) - 100 \, \sqrt {2} x^{5} \arctan \left (-\frac {4 \, x^{8} + 4 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (4 \, x^{7} + 3 \, x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (2 \, x^{6} + x^{2}\right )} \sqrt {3 \, x^{4} + 2} - {\left (4 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x^{5} - \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + \sqrt {2} {\left (4 \, x^{8} + x^{4} - 1\right )} + 2 \, {\left (2 \, x^{7} + x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {2 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}} + 1}{8 \, x^{8} + 4 \, x^{4} - 1}\right ) - 25 \, \sqrt {2} x^{5} \log \left (\frac {4 \, {\left (2 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1\right )}}{2 \, x^{4} + 1}\right ) + 25 \, \sqrt {2} x^{5} \log \left (\frac {4 \, {\left (2 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1\right )}}{2 \, x^{4} + 1}\right ) + 20 \, x^{5} \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 20 \, x^{5} \log \left (-\frac {2 \, x^{4} - {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \sqrt {3 \, x^{4} + 2} x^{2} - {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{x^{4} + 1}\right ) + 64 \, {\left (6 \, x^{4} - 1\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{80 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x, algorithm="fricas")

[Out]

1/80*(100*sqrt(2)*x^5*arctan(-(4*x^8 + 4*x^4 + sqrt(2)*(3*x^4 + 2)^(3/4)*x + sqrt(2)*(4*x^7 + 3*x^3)*(3*x^4 +
2)^(1/4) + 2*(2*x^6 + x^2)*sqrt(3*x^4 + 2) - (4*(3*x^4 + 2)^(3/4)*x^5 + sqrt(2)*sqrt(3*x^4 + 2)*x^2 - sqrt(2)*
(4*x^8 + x^4 - 1) + 2*(2*x^7 + x^3)*(3*x^4 + 2)^(1/4))*sqrt((2*x^4 + sqrt(2)*(3*x^4 + 2)^(1/4)*x^3 + 2*sqrt(3*
x^4 + 2)*x^2 + sqrt(2)*(3*x^4 + 2)^(3/4)*x + 1)/(2*x^4 + 1)) + 1)/(8*x^8 + 4*x^4 - 1)) - 100*sqrt(2)*x^5*arcta
n(-(4*x^8 + 4*x^4 - sqrt(2)*(3*x^4 + 2)^(3/4)*x - sqrt(2)*(4*x^7 + 3*x^3)*(3*x^4 + 2)^(1/4) + 2*(2*x^6 + x^2)*
sqrt(3*x^4 + 2) - (4*(3*x^4 + 2)^(3/4)*x^5 - sqrt(2)*sqrt(3*x^4 + 2)*x^2 + sqrt(2)*(4*x^8 + x^4 - 1) + 2*(2*x^
7 + x^3)*(3*x^4 + 2)^(1/4))*sqrt((2*x^4 - sqrt(2)*(3*x^4 + 2)^(1/4)*x^3 + 2*sqrt(3*x^4 + 2)*x^2 - sqrt(2)*(3*x
^4 + 2)^(3/4)*x + 1)/(2*x^4 + 1)) + 1)/(8*x^8 + 4*x^4 - 1)) - 25*sqrt(2)*x^5*log(4*(2*x^4 + sqrt(2)*(3*x^4 + 2
)^(1/4)*x^3 + 2*sqrt(3*x^4 + 2)*x^2 + sqrt(2)*(3*x^4 + 2)^(3/4)*x + 1)/(2*x^4 + 1)) + 25*sqrt(2)*x^5*log(4*(2*
x^4 - sqrt(2)*(3*x^4 + 2)^(1/4)*x^3 + 2*sqrt(3*x^4 + 2)*x^2 - sqrt(2)*(3*x^4 + 2)^(3/4)*x + 1)/(2*x^4 + 1)) +
20*x^5*arctan(((3*x^4 + 2)^(1/4)*x^3 + (3*x^4 + 2)^(3/4)*x)/(x^4 + 1)) + 20*x^5*log(-(2*x^4 - (3*x^4 + 2)^(1/4
)*x^3 + sqrt(3*x^4 + 2)*x^2 - (3*x^4 + 2)^(3/4)*x + 1)/(x^4 + 1)) + 64*(6*x^4 - 1)*(3*x^4 + 2)^(1/4))/x^5

________________________________________________________________________________________

giac [A]  time = 0.17, size = 221, normalized size = 1.44 \begin {gather*} -\frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) + \frac {5}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (\frac {2}{x^{4}} + 3\right )}}{5 \, x} + \frac {6 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - \frac {1}{2} \, \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x, algorithm="giac")

[Out]

-5/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(3*x^4 + 2)^(1/4)/x)) - 5/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)
- 2*(3*x^4 + 2)^(1/4)/x)) - 5/8*sqrt(2)*log(sqrt(2)*(3*x^4 + 2)^(1/4)/x + sqrt(3*x^4 + 2)/x^2 + 1) + 5/8*sqrt(
2)*log(-sqrt(2)*(3*x^4 + 2)^(1/4)/x + sqrt(3*x^4 + 2)/x^2 + 1) - 2/5*(3*x^4 + 2)^(1/4)*(2/x^4 + 3)/x + 6*(3*x^
4 + 2)^(1/4)/x - 1/2*arctan((3*x^4 + 2)^(1/4)/x) - 1/4*log((3*x^4 + 2)^(1/4)/x + 1) + 1/4*log((3*x^4 + 2)^(1/4
)/x - 1)

________________________________________________________________________________________

maple [C]  time = 12.62, size = 360, normalized size = 2.34

method	result	size
trager	\(\frac {4 \left (3 x^{4}+2\right )^{\frac {1}{4}} \left (6 x^{4}-1\right )}{5 x^{5}}-\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\sqrt {3 x^{4}+2}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\left (3 x^{4}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{2 x^{4}+1}\right )}{4}-\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\left (3 x^{4}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\sqrt {3 x^{4}+2}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{4}+1}\right )}{4}-\frac {\ln \left (-\frac {\left (3 x^{4}+2\right )^{\frac {3}{4}} x +x^{2} \sqrt {3 x^{4}+2}+\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}+2 x^{4}+1}{x^{4}+1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {\sqrt {3 x^{4}+2}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{x^{4}+1}\right )}{4}\)	\(360\)
risch	\(\frac {\frac {36}{5} x^{4}-\frac {8}{5}+\frac {72}{5} x^{8}}{x^{5} \left (3 x^{4}+2\right )^{\frac {3}{4}}}+\frac {\left (\frac {\ln \left (\frac {-18 x^{12}+9 \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{9}-3 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{6}-33 x^{8}+\left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {3}{4}} x^{3}+12 \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{5}-2 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{2}-20 x^{4}+4 \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x -4}{\left (x^{4}+1\right ) \left (3 x^{4}+2\right )^{2}}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {18 x^{12}-9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{9}-3 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{6}+33 x^{8}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {3}{4}} x^{3}-12 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{5}-2 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{2}+20 x^{4}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x +4}{\left (x^{4}+1\right ) \left (3 x^{4}+2\right )^{2}}\right )}{4}-\frac {5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{12}+9 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{9}+21 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+12 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{5}+3 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{6}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {3}{4}} x^{3}+16 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x +2 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (2 x^{4}+1\right ) \left (3 x^{4}+2\right )^{2}}\right )}{4}-\frac {5 \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-9 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{12}+9 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{9}-21 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {3}{4}} x^{3}+3 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{6}+12 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x^{5}-16 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \sqrt {27 x^{12}+54 x^{8}+36 x^{4}+8}\, x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \left (27 x^{12}+54 x^{8}+36 x^{4}+8\right )^{\frac {1}{4}} x -4 \RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (2 x^{4}+1\right ) \left (3 x^{4}+2\right )^{2}}\right )}{4}\right ) \left (\left (3 x^{4}+2\right )^{3}\right )^{\frac {1}{4}}}{\left (3 x^{4}+2\right )^{\frac {3}{4}}}\)	\(986\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x,method=_RETURNVERBOSE)

[Out]

4/5*(3*x^4+2)^(1/4)*(6*x^4-1)/x^5-5/4*RootOf(_Z^4+1)^3*ln(-((3*x^4+2)^(1/2)*RootOf(_Z^4+1)^3*x^2-(3*x^4+2)^(1/
4)*RootOf(_Z^4+1)^2*x^3-RootOf(_Z^4+1)*x^4+(3*x^4+2)^(3/4)*x-RootOf(_Z^4+1))/(2*x^4+1))-5/4*RootOf(_Z^4+1)*ln(
-(-RootOf(_Z^4+1)^3*x^4+(3*x^4+2)^(1/4)*RootOf(_Z^4+1)^2*x^3+(3*x^4+2)^(1/2)*RootOf(_Z^4+1)*x^2+(3*x^4+2)^(3/4
)*x-RootOf(_Z^4+1)^3)/(2*x^4+1))-1/4*ln(-((3*x^4+2)^(3/4)*x+x^2*(3*x^4+2)^(1/2)+(3*x^4+2)^(1/4)*x^3+2*x^4+1)/(
x^4+1))-1/4*RootOf(_Z^4+1)^2*ln(-((3*x^4+2)^(1/2)*RootOf(_Z^4+1)^2*x^2-2*RootOf(_Z^4+1)^2*x^4+(3*x^4+2)^(3/4)*
x-(3*x^4+2)^(1/4)*x^3-RootOf(_Z^4+1)^2)/(x^4+1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + 6 \, x^{4} + 4\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4+2)^(1/4)*(x^8+6*x^4+4)/x^6/(x^4+1)/(2*x^4+1),x, algorithm="maxima")

[Out]

integrate((x^8 + 6*x^4 + 4)*(3*x^4 + 2)^(1/4)/((2*x^4 + 1)*(x^4 + 1)*x^6), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x^4+2\right )}^{1/4}\,\left (x^8+6\,x^4+4\right )}{x^6\,\left (x^4+1\right )\,\left (2\,x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^4 + 2)^(1/4)*(6*x^4 + x^8 + 4))/(x^6*(x^4 + 1)*(2*x^4 + 1)),x)

[Out]

int(((3*x^4 + 2)^(1/4)*(6*x^4 + x^8 + 4))/(x^6*(x^4 + 1)*(2*x^4 + 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{3 x^{4} + 2} \left (x^{8} + 6 x^{4} + 4\right )}{x^{6} \left (x^{4} + 1\right ) \left (2 x^{4} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**4+2)**(1/4)*(x**8+6*x**4+4)/x**6/(x**4+1)/(2*x**4+1),x)

[Out]

Integral((3*x**4 + 2)**(1/4)*(x**8 + 6*x**4 + 4)/(x**6*(x**4 + 1)*(2*x**4 + 1)), x)

________________________________________________________________________________________