Integrand size = 22, antiderivative size = 120 \[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=-\frac {\sqrt [4]{1+2 x^2}}{x}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+2 x^2}}{1+\sqrt {1+2 x^2}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {1-\sqrt {1+2 x^2}}{\sqrt {2} x \sqrt [4]{1+2 x^2}}\right )}{\sqrt {2}}-2 \sqrt {2} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\sqrt {2} x\right ),2\right ) \] Output:
-(2*x^2+1)^(1/4)/x+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)-2*2^(1/2)*InverseJacobiAM(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 = 11.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},\frac {3}{4},1,\frac {1}{2},-2 x^2,-x^2\right )}{x} \] Input:
Integrate[1/(x^2*(1 + x^2)*(1 + 2*x^2)^(3/4)),x]
Output:
-(AppellF1[-1/2, 3/4, 1, 1/2, -2*x^2, -x^2]/x)
Time = 0.25 (sec) , antiderivative size = 121, 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 = {352, 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.
\(\displaystyle \int \frac {1}{x^2 \left (x^2+1\right ) \left (2 x^2+1\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 352 |
\(\displaystyle \int \left (\frac {1}{\left (-x^2-1\right ) \left (2 x^2+1\right )^{3/4}}+\frac {1}{x^2 \left (2 x^2+1\right )^{3/4}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2 x^2+1}+1}{\sqrt {2} x \sqrt [4]{2 x^2+1}}\right )}{\sqrt {2}}-2 \sqrt {2} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\sqrt {2} x\right ),2\right )+\frac {\text {arctanh}\left (\frac {1-\sqrt {2 x^2+1}}{\sqrt {2} x \sqrt [4]{2 x^2+1}}\right )}{\sqrt {2}}-\frac {\sqrt [4]{2 x^2+1}}{x}\) |
Input:
Int[1/(x^2*(1 + x^2)*(1 + 2*x^2)^(3/4)),x]
Output:
-((1 + 2*x^2)^(1/4)/x) - 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] - 2*Sqrt[2]*EllipticF[ArcTan[Sqrt[2]*x]/2, 2]
Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol ] :> Int[ExpandIntegrand[x^m/((a + b*x^2)^(3/4)*(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])
\[\int \frac {1}{x^{2} \left (x^{2}+1\right ) \left (2 x^{2}+1\right )^{\frac {3}{4}}}d x\]
Input:
int(1/x^2/(x^2+1)/(2*x^2+1)^(3/4),x)
Output:
int(1/x^2/(x^2+1)/(2*x^2+1)^(3/4),x)
\[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^2+1)/(2*x^2+1)^(3/4),x, algorithm="fricas")
Output:
integral((2*x^2 + 1)^(1/4)/(2*x^6 + 3*x^4 + x^2), x)
\[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int \frac {1}{x^{2} \left (x^{2} + 1\right ) \left (2 x^{2} + 1\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(1/x**2/(x**2+1)/(2*x**2+1)**(3/4),x)
Output:
Integral(1/(x**2*(x**2 + 1)*(2*x**2 + 1)**(3/4)), x)
\[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^2+1)/(2*x^2+1)^(3/4),x, algorithm="maxima")
Output:
integrate(1/((2*x^2 + 1)^(3/4)*(x^2 + 1)*x^2), x)
\[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^2+1)/(2*x^2+1)^(3/4),x, algorithm="giac")
Output:
integrate(1/((2*x^2 + 1)^(3/4)*(x^2 + 1)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int \frac {1}{x^2\,\left (x^2+1\right )\,{\left (2\,x^2+1\right )}^{3/4}} \,d x \] Input:
int(1/(x^2*(x^2 + 1)*(2*x^2 + 1)^(3/4)),x)
Output:
int(1/(x^2*(x^2 + 1)*(2*x^2 + 1)^(3/4)), x)
\[ \int \frac {1}{x^2 \left (1+x^2\right ) \left (1+2 x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (2 x^{2}+1\right )^{\frac {3}{4}} x^{4}+\left (2 x^{2}+1\right )^{\frac {3}{4}} x^{2}}d x \] Input:
int(1/x^2/(x^2+1)/(2*x^2+1)^(3/4),x)
Output:
int(1/((2*x**2 + 1)**(3/4)*x**4 + (2*x**2 + 1)**(3/4)*x**2),x)