Integrand size = 24, antiderivative size = 102 \[ \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=-\frac {17 x \sqrt {2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac {563 x \sqrt {2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac {191 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{9800}-\frac {1251 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{24500}+\frac {9879 \operatorname {EllipticPi}\left (-\frac {10}{7},\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{343000} \]
191/9800*EllipticE(1/2*x*2^(1/2),I*2^(1/2))-1251/24500*EllipticF(1/2*x*2^( 1/2),I*2^(1/2))+9879/343000*EllipticPi(1/2*x*2^(1/2),-10/7,I*2^(1/2))-17/3 50*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)^2+563/9800*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7 )
Result contains complex when optimal does not.
Time = 10.39 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.39 \[ \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\frac {485100 x+636650 x^3-45500 x^5-197050 x^7+13370 i \sqrt {2} \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )-2541 i \sqrt {2} \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )-484071 i \sqrt {2} \sqrt {2+x^2-x^4} \operatorname {EllipticPi}\left (\frac {5}{7},i \text {arcsinh}(x),-\frac {1}{2}\right )-691530 i \sqrt {2} x^2 \sqrt {2+x^2-x^4} \operatorname {EllipticPi}\left (\frac {5}{7},i \text {arcsinh}(x),-\frac {1}{2}\right )-246975 i \sqrt {2} x^4 \sqrt {2+x^2-x^4} \operatorname {EllipticPi}\left (\frac {5}{7},i \text {arcsinh}(x),-\frac {1}{2}\right )}{686000 \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \]
(485100*x + 636650*x^3 - 45500*x^5 - 197050*x^7 + (13370*I)*Sqrt[2]*(7 + 5 *x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] - (2541*I)*Sqrt[ 2]*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] - (4840 71*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (6 91530*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/ 2] - (246975*I)*Sqrt[2]*x^4*Sqrt[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[ x], -1/2])/(686000*(7 + 5*x^2)^2*Sqrt[2 + x^2 - x^4])
Time = 0.59 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1556, 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.
| \(\displaystyle \int \frac {\left (-x^4+x^2+2\right )^{3/2}}{\left (5 x^2+7\right )^3} \, dx\) |
| \(\Big \downarrow \) 1556 |
| \(\displaystyle \int \left (\frac {x^2}{125 \sqrt {-x^4+x^2+2}}+\frac {429}{625 \left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}}-\frac {1292}{625 \left (5 x^2+7\right )^2 \sqrt {-x^4+x^2+2}}+\frac {1156}{625 \left (5 x^2+7\right )^3 \sqrt {-x^4+x^2+2}}-\frac {31}{625 \sqrt {-x^4+x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1251 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{24500}+\frac {191 E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{9800}+\frac {9879 \operatorname {EllipticPi}\left (-\frac {10}{7},\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{343000}+\frac {563 \sqrt {-x^4+x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac {17 \sqrt {-x^4+x^2+2} x}{350 \left (5 x^2+7\right )^2}\) |
(-17*x*Sqrt[2 + x^2 - x^4])/(350*(7 + 5*x^2)^2) + (563*x*Sqrt[2 + x^2 - x^ 4])/(9800*(7 + 5*x^2)) + (191*EllipticE[ArcSin[x/Sqrt[2]], -2])/9800 - (12 51*EllipticF[ArcSin[x/Sqrt[2]], -2])/24500 + (9879*EllipticPi[-10/7, ArcSi n[x/Sqrt[2]], -2])/343000
3.4.31.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Module[{aa, bb, cc}, Int[ExpandIntegrand[1/Sqrt[aa + bb*x^2 + c c*x^4], (d + e*x^2)^q*(aa + bb*x^2 + cc*x^4)^(p + 1/2), x] /. {aa -> a, bb -> b, cc -> c}, x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]
Time = 3.79 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.85
| method | result | size |
| default | \(-\frac {17 x \sqrt {-x^{4}+x^{2}+2}}{350 \left (5 x^{2}+7\right )^{2}}+\frac {563 x \sqrt {-x^{4}+x^{2}+2}}{9800 \left (5 x^{2}+7\right )}-\frac {1251 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{49000 \sqrt {-x^{4}+x^{2}+2}}+\frac {191 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{19600 \sqrt {-x^{4}+x^{2}+2}}+\frac {9879 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{343000 \sqrt {-x^{4}+x^{2}+2}}\) | \(189\) |
| elliptic | \(-\frac {17 x \sqrt {-x^{4}+x^{2}+2}}{350 \left (5 x^{2}+7\right )^{2}}+\frac {563 x \sqrt {-x^{4}+x^{2}+2}}{9800 \left (5 x^{2}+7\right )}-\frac {1251 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{49000 \sqrt {-x^{4}+x^{2}+2}}+\frac {191 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{19600 \sqrt {-x^{4}+x^{2}+2}}+\frac {9879 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{343000 \sqrt {-x^{4}+x^{2}+2}}\) | \(189\) |
| risch | \(-\frac {\left (x^{4}-x^{2}-2\right ) x \left (563 x^{2}+693\right )}{1960 \left (5 x^{2}+7\right )^{2} \sqrt {-x^{4}+x^{2}+2}}-\frac {221 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{14000 \sqrt {-x^{4}+x^{2}+2}}-\frac {191 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{19600 \sqrt {-x^{4}+x^{2}+2}}+\frac {9879 \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}\, \sqrt {x^{2}+1}\, \Pi \left (\frac {x \sqrt {2}}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{343000 \sqrt {-x^{4}+x^{2}+2}}\) | \(198\) |
-17/350*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)^2+563/9800*x*(-x^4+x^2+2)^(1/2)/(5* x^2+7)-1251/49000*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2 )*EllipticF(1/2*x*2^(1/2),I*2^(1/2))+191/19600*2^(1/2)*(-2*x^2+4)^(1/2)*(x ^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticE(1/2*x*2^(1/2),I*2^(1/2))+9879/343 000*2^(1/2)*(1-1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticPi( 1/2*x*2^(1/2),-10/7,I*2^(1/2))
\[ \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\int { \frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]
\[ \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\int \frac {\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{3}}\, dx \]
\[ \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\int { \frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]
\[ \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\int { \frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx=\int \frac {{\left (-x^4+x^2+2\right )}^{3/2}}{{\left (5\,x^2+7\right )}^3} \,d x \]