\(\int \frac {\sqrt [4]{2+3 x^4} (4+6 x^4+x^8)}{x^6 (1+x^4) (1+2 x^4)} \, dx\) [2125]

Optimal result

Integrand size = 41, antiderivative size = 154 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {4 \sqrt [4]{2+3 x^4} \left (-1+6 x^4\right )}{5 x^5}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )+\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{-x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )-\frac {5 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}} \]

[Out]

4/5*(3*x^4+2)^(1/4)*(6*x^4-1)/x^5+1/2*arctan(x/(3*x^4+2)^(1/4))+5/4*arctan(2^(1/2)*x*(3*x^4+2)^(1/4)/(-x^2+(3*
x^4+2)^(1/2)))*2^(1/2)-1/2*arctanh(x/(3*x^4+2)^(1/4))-5/4*arctanh(2^(1/2)*x*(3*x^4+2)^(1/4)/(x^2+(3*x^4+2)^(1/
2)))*2^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.62 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {6857, 270, 283, 338, 304, 209, 212, 524} \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {10}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )+3 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )-3 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )+\frac {6 \sqrt [4]{3 x^4+2}}{x}-\frac {2 \left (3 x^4+2\right )^{5/4}}{5 x^5} \]

[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 209

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

Rule 212

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

Rule 270

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 283

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 304

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 338

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 6857

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*} \text {integral}& = \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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )-18 \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )-\left (3 \sqrt {3}\right ) \text {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 ) \text {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 \operatorname {AppellF1}\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 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )+3 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right )-3 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {1}{20} \left (\frac {16 \sqrt [4]{2+3 x^4} \left (-1+6 x^4\right )}{x^5}+10 \arctan \left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )+25 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{-x^2+\sqrt {2+3 x^4}}\right )-10 \text {arctanh}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )-25 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{x^2+\sqrt {2+3 x^4}}\right )\right ) \]

[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]

((16*(2 + 3*x^4)^(1/4)*(-1 + 6*x^4))/x^5 + 10*ArcTan[x/(2 + 3*x^4)^(1/4)] + 25*Sqrt[2]*ArcTan[(Sqrt[2]*x*(2 +
3*x^4)^(1/4))/(-x^2 + Sqrt[2 + 3*x^4])] - 10*ArcTanh[x/(2 + 3*x^4)^(1/4)] - 25*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(2 +
3*x^4)^(1/4))/(x^2 + Sqrt[2 + 3*x^4])])/20

Maple [A] (verified)

Time = 8.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.43

method	result	size
pseudoelliptic	\(\frac {-25 \ln \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {3 x^{4}+2}}{\sqrt {3 x^{4}+2}-\left (3 x^{4}+2\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}}\right ) \sqrt {2}\, x^{5}-50 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}\, x^{5}-50 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}\, x^{5}+10 \ln \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}}-x}{x}\right ) x^{5}-10 \ln \left (\frac {x +\left (3 x^{4}+2\right )^{\frac {1}{4}}}{x}\right ) x^{5}-20 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}}}{x}\right ) x^{5}+192 \left (3 x^{4}+2\right )^{\frac {1}{4}} x^{4}-32 \left (3 x^{4}+2\right )^{\frac {1}{4}}}{40 x^{5}}\)	\(220\)
trager	\(\frac {4 \left (3 x^{4}+2\right )^{\frac {1}{4}} \left (6 x^{4}-1\right )}{5 x^{5}}+\frac {\ln \left (-\frac {\left (3 x^{4}+2\right )^{\frac {3}{4}} x -x^{2} \sqrt {3 x^{4}+2}+\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}-2 x^{4}-1}{x^{4}+1}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\left (3 x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{4}+1}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\left (3 x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{2 x^{4}+1}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {-\sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{x^{4}+1}\right )}{4}\)	\(360\)
risch	\(\text {Expression too large to display}\)	\(945\)

[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]

1/40*(-25*ln(((3*x^4+2)^(1/4)*x*2^(1/2)+x^2+(3*x^4+2)^(1/2))/((3*x^4+2)^(1/2)-(3*x^4+2)^(1/4)*x*2^(1/2)+x^2))*
2^(1/2)*x^5-50*arctan(((3*x^4+2)^(1/4)*2^(1/2)+x)/x)*2^(1/2)*x^5-50*arctan(((3*x^4+2)^(1/4)*2^(1/2)-x)/x)*2^(1
/2)*x^5+10*ln(((3*x^4+2)^(1/4)-x)/x)*x^5-10*ln((x+(3*x^4+2)^(1/4))/x)*x^5-20*arctan(1/x*(3*x^4+2)^(1/4))*x^5+1
92*(3*x^4+2)^(1/4)*x^4-32*(3*x^4+2)^(1/4))/x^5

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.01 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.73 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {-\left (25 i + 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \left (i + 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} - i + 1\right )}}{2 \, x^{4} + 1}\right ) + \left (25 i + 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} - \left (i + 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} + i - 1\right )}}{2 \, x^{4} + 1}\right ) + \left (25 i - 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {-2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} - \left (i - 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} + i + 1\right )}}{2 \, x^{4} + 1}\right ) - \left (25 i - 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {-2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \left (i - 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} - i - 1\right )}}{2 \, x^{4} + 1}\right ) + 20 \, x^{5} \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 20 \, x^{5} \log \left (-\frac {2 \, x^{4} - {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \sqrt {3 \, x^{4} + 2} x^{2} - {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{x^{4} + 1}\right ) + 64 \, {\left (6 \, x^{4} - 1\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{80 \, x^{5}} \]

[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*(-(25*I + 25)*sqrt(2)*x^5*log((2*I*(3*x^4 + 2)^(1/4)*x^3 + (I + 1)*sqrt(2)*sqrt(3*x^4 + 2)*x^2 + 2*(3*x^4
+ 2)^(3/4)*x + sqrt(2)*(-(I - 1)*x^4 - I + 1))/(2*x^4 + 1)) + (25*I + 25)*sqrt(2)*x^5*log((2*I*(3*x^4 + 2)^(1
/4)*x^3 - (I + 1)*sqrt(2)*sqrt(3*x^4 + 2)*x^2 + 2*(3*x^4 + 2)^(3/4)*x + sqrt(2)*((I - 1)*x^4 + I - 1))/(2*x^4
+ 1)) + (25*I - 25)*sqrt(2)*x^5*log((-2*I*(3*x^4 + 2)^(1/4)*x^3 - (I - 1)*sqrt(2)*sqrt(3*x^4 + 2)*x^2 + 2*(3*x
^4 + 2)^(3/4)*x + sqrt(2)*((I + 1)*x^4 + I + 1))/(2*x^4 + 1)) - (25*I - 25)*sqrt(2)*x^5*log((-2*I*(3*x^4 + 2)^
(1/4)*x^3 + (I - 1)*sqrt(2)*sqrt(3*x^4 + 2)*x^2 + 2*(3*x^4 + 2)^(3/4)*x + sqrt(2)*(-(I + 1)*x^4 - I - 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

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=-\frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) + \frac {5}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (\frac {2}{x^{4}} + 3\right )}}{5 \, x} + \frac {6 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - \frac {1}{2} \, \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - 1\right ) \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\int \frac {{\left (3\,x^4+2\right )}^{1/4}\,\left (x^8+6\,x^4+4\right )}{x^6\,\left (x^4+1\right )\,\left (2\,x^4+1\right )} \,d x \]

[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)