Integrand size = 22, antiderivative size = 139 \[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=-\frac {12 x}{5 \sqrt [4]{1+2 x^2}}+\frac {1}{5} x \left (1+2 x^2\right )^{3/4}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+2 x^2}}{1+\sqrt {1+2 x^2}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {1-\sqrt {1+2 x^2}}{\sqrt {2} x \sqrt [4]{1+2 x^2}}\right )}{\sqrt {2}}+\frac {6}{5} \sqrt {2} E\left (\left .\frac {1}{2} \arctan \left (\sqrt {2} x\right )\right |2\right ) \] Output:
-12/5*x/(2*x^2+1)^(1/4)+1/5*x*(2*x^2+1)^(3/4)+1/2*arctan(2^(1/2)*x*(2*x^2+ 1)^(1/4)/(1+(2*x^2+1)^(1/2)))*2^(1/2)-1/2*arctanh(1/2*(1-(2*x^2+1)^(1/2))* 2^(1/2)/x/(2*x^2+1)^(1/4))*2^(1/2)+6/5*2^(1/2)*EllipticE(sin(1/2*arctan(x* 2^(1/2))),2^(1/2))
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 5.42 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.12 \[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=\frac {1}{5} x \left (-2 x^2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-2 x^2,-x^2\right )+\frac {1+2 x^2+\frac {3 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-2 x^2,-x^2\right )}{\left (1+x^2\right ) \left (-3 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-2 x^2,-x^2\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-2 x^2,-x^2\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-2 x^2,-x^2\right )\right )\right )}}{\sqrt [4]{1+2 x^2}}\right ) \] Input:
Integrate[x^4/((1 + x^2)*(1 + 2*x^2)^(1/4)),x]
Output:
(x*(-2*x^2*AppellF1[3/2, 1/4, 1, 5/2, -2*x^2, -x^2] + (1 + 2*x^2 + (3*Appe llF1[1/2, 1/4, 1, 3/2, -2*x^2, -x^2])/((1 + x^2)*(-3*AppellF1[1/2, 1/4, 1, 3/2, -2*x^2, -x^2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -2*x^2, -x^2] + Ap pellF1[3/2, 5/4, 1, 5/2, -2*x^2, -x^2]))))/(1 + 2*x^2)^(1/4)))/5
Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {349, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^4}{\left (x^2+1\right ) \sqrt [4]{2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 349 |
\(\displaystyle \int \left (\frac {x^2}{\sqrt [4]{2 x^2+1}}+\frac {1}{\left (x^2+1\right ) \sqrt [4]{2 x^2+1}}-\frac {1}{\sqrt [4]{2 x^2+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2 x^2+1}+1}{\sqrt {2} x \sqrt [4]{2 x^2+1}}\right )}{\sqrt {2}}+\frac {6}{5} \sqrt {2} E\left (\left .\frac {1}{2} \arctan \left (\sqrt {2} x\right )\right |2\right )-\frac {\text {arctanh}\left (\frac {1-\sqrt {2 x^2+1}}{\sqrt {2} x \sqrt [4]{2 x^2+1}}\right )}{\sqrt {2}}+\frac {1}{5} \left (2 x^2+1\right )^{3/4} x-\frac {12 x}{5 \sqrt [4]{2 x^2+1}}\) |
Input:
Int[x^4/((1 + x^2)*(1 + 2*x^2)^(1/4)),x]
Output:
(-12*x)/(5*(1 + 2*x^2)^(1/4)) + (x*(1 + 2*x^2)^(3/4))/5 - ArcTan[(1 + Sqrt [1 + 2*x^2])/(Sqrt[2]*x*(1 + 2*x^2)^(1/4))]/Sqrt[2] - ArcTanh[(1 - Sqrt[1 + 2*x^2])/(Sqrt[2]*x*(1 + 2*x^2)^(1/4))]/Sqrt[2] + (6*Sqrt[2]*EllipticE[Ar cTan[Sqrt[2]*x]/2, 2])/5
Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol ] :> Int[ExpandIntegrand[x^m/((a + b*x^2)^(1/4)*(c + d*x^2)), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a] || In tegerQ[m/2])
\[\int \frac {x^{4}}{\left (x^{2}+1\right ) \left (2 x^{2}+1\right )^{\frac {1}{4}}}d x\]
Input:
int(x^4/(x^2+1)/(2*x^2+1)^(1/4),x)
Output:
int(x^4/(x^2+1)/(2*x^2+1)^(1/4),x)
\[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=\int { \frac {x^{4}}{{\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(x^4/(x^2+1)/(2*x^2+1)^(1/4),x, algorithm="fricas")
Output:
integral((2*x^2 + 1)^(3/4)*x^4/(2*x^4 + 3*x^2 + 1), x)
\[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=\int \frac {x^{4}}{\left (x^{2} + 1\right ) \sqrt [4]{2 x^{2} + 1}}\, dx \] Input:
integrate(x**4/(x**2+1)/(2*x**2+1)**(1/4),x)
Output:
Integral(x**4/((x**2 + 1)*(2*x**2 + 1)**(1/4)), x)
\[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=\int { \frac {x^{4}}{{\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(x^4/(x^2+1)/(2*x^2+1)^(1/4),x, algorithm="maxima")
Output:
integrate(x^4/((2*x^2 + 1)^(1/4)*(x^2 + 1)), x)
\[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=\int { \frac {x^{4}}{{\left (2 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(x^4/(x^2+1)/(2*x^2+1)^(1/4),x, algorithm="giac")
Output:
integrate(x^4/((2*x^2 + 1)^(1/4)*(x^2 + 1)), x)
Timed out. \[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=\int \frac {x^4}{\left (x^2+1\right )\,{\left (2\,x^2+1\right )}^{1/4}} \,d x \] Input:
int(x^4/((x^2 + 1)*(2*x^2 + 1)^(1/4)),x)
Output:
int(x^4/((x^2 + 1)*(2*x^2 + 1)^(1/4)), x)
\[ \int \frac {x^4}{\left (1+x^2\right ) \sqrt [4]{1+2 x^2}} \, dx=\int \frac {x^{4}}{\left (2 x^{2}+1\right )^{\frac {1}{4}} x^{2}+\left (2 x^{2}+1\right )^{\frac {1}{4}}}d x \] Input:
int(x^4/(x^2+1)/(2*x^2+1)^(1/4),x)
Output:
int(x**4/((2*x**2 + 1)**(1/4)*x**2 + (2*x**2 + 1)**(1/4)),x)