Integrand size = 36, antiderivative size = 302 \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\frac {61199 x \left (2+3 x^2\right )}{20160 \sqrt {2+5 x^2+3 x^4}}-\frac {265}{336} x \sqrt {2+5 x^2+3 x^4}-\frac {32}{35} x^3 \sqrt {2+5 x^2+3 x^4}+\frac {3}{28} x^5 \sqrt {2+5 x^2+3 x^4}+\frac {31 x \sqrt {2+5 x^2+3 x^4}}{32 \left (1+2 x^2\right )}-\frac {61199 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{10080 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {2719 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{672 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {371 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{32 \sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
61199/20160*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)-265/336*x*(3*x^4+5*x^2+2)^(1 /2)-32/35*x^3*(3*x^4+5*x^2+2)^(1/2)+3/28*x^5*(3*x^4+5*x^2+2)^(1/2)+31*x*(3 *x^4+5*x^2+2)^(1/2)/(64*x^2+32)-61199/20160*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^ 2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2) +2719/1344*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(arcta n(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)+371/96*(x^2+1)*EllipticPi(x*6^(1 /2)/(6*x^2+4)^(1/2),-1/3,1/3*3^(1/2))*3^(1/2)/((x^2+1)/(3*x^2+2))^(1/2)/(3 *x^4+5*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.52 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.75 \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\frac {12 x \left (1210-13719 x^2-51613 x^4-52596 x^6-13752 x^8+2160 x^{10}\right )-122398 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-48847 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )-38955 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{40320 \left (1+2 x^2\right ) \sqrt {2+5 x^2+3 x^4}} \] Input:
Integrate[((4 - 7*x^2 + x^4)*(2 + 5*x^2 + 3*x^4)^(3/2))/(1 + 2*x^2)^2,x]
Output:
(12*x*(1210 - 13719*x^2 - 51613*x^4 - 52596*x^6 - 13752*x^8 + 2160*x^10) - (122398*I)*Sqrt[3]*Sqrt[1 + x^2]*(1 + 2*x^2)*Sqrt[2 + 3*x^2]*EllipticE[I* ArcSinh[Sqrt[3/2]*x], 2/3] - (48847*I)*Sqrt[3]*Sqrt[1 + x^2]*(1 + 2*x^2)*S qrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (38955*I)*Sqrt[3]* Sqrt[1 + x^2]*(1 + 2*x^2)*Sqrt[2 + 3*x^2]*EllipticPi[4/3, I*ArcSinh[Sqrt[3 /2]*x], 2/3])/(40320*(1 + 2*x^2)*Sqrt[2 + 5*x^2 + 3*x^4])
Time = 0.88 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2258, 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.
| \(\displaystyle \int \frac {\left (x^4-7 x^2+4\right ) \left (3 x^4+5 x^2+2\right )^{3/2}}{\left (2 x^2+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (-\frac {389 x^4}{16 \sqrt {3 x^4+5 x^2+2}}-\frac {45 x^2}{16 \sqrt {3 x^4+5 x^2+2}}+\frac {29}{8 \left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {31}{64 \left (2 x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}+\frac {761}{64 \sqrt {3 x^4+5 x^2+2}}+\frac {9 x^8}{4 \sqrt {3 x^4+5 x^2+2}}-\frac {21 x^6}{2 \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {14 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{9 \sqrt {3 x^4+5 x^2+2}}+\frac {14429 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{2016 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {4297 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{945 \sqrt {3 x^4+5 x^2+2}}-\frac {13103 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{864 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}-\frac {31 \sqrt {3} \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{32 \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {29 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{2 \sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {31 \sqrt {3 x^4+5 x^2+2} x}{32 \left (2 x^2+1\right )}-\frac {265}{336} \sqrt {3 x^4+5 x^2+2} x+\frac {61199 \left (3 x^2+2\right ) x}{20160 \sqrt {3 x^4+5 x^2+2}}+\frac {3}{28} \sqrt {3 x^4+5 x^2+2} x^5-\frac {32}{35} \sqrt {3 x^4+5 x^2+2} x^3\) |
Input:
Int[((4 - 7*x^2 + x^4)*(2 + 5*x^2 + 3*x^4)^(3/2))/(1 + 2*x^2)^2,x]
Output:
(61199*x*(2 + 3*x^2))/(20160*Sqrt[2 + 5*x^2 + 3*x^4]) - (265*x*Sqrt[2 + 5* x^2 + 3*x^4])/336 - (32*x^3*Sqrt[2 + 5*x^2 + 3*x^4])/35 + (3*x^5*Sqrt[2 + 5*x^2 + 3*x^4])/28 + (31*x*Sqrt[2 + 5*x^2 + 3*x^4])/(32*(1 + 2*x^2)) - (13 103*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(864 *Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) + (4297*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^ 2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(945*Sqrt[2 + 5*x^2 + 3*x^4]) + (14429*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/( 2016*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) - (14*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3* x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(9*Sqrt[2 + 5*x^2 + 3*x^4]) + (29*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(2*Sqrt[3]*Sqrt[ (1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - (31*Sqrt[3]*(1 + x^2)*El lipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(32*Sqrt[(1 + x^2)/(2 + 3*x^2)]* Sqrt[2 + 5*x^2 + 3*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e *x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Time = 6.89 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x \left (720 x^{6}-5784 x^{4}-8372 x^{2}+605\right )}{6720 x^{2}+3360}-\frac {34249 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{5376 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {61199 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{20160 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {371 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{128 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(192\) |
| elliptic | \(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{32 \left (2 x^{2}+1\right )}+\frac {3 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{28}-\frac {32 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{35}-\frac {265 x \sqrt {3 x^{4}+5 x^{2}+2}}{336}-\frac {268939 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{80640 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {61199 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{20160 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {371 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{128 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(217\) |
| default | \(\frac {3 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{28}-\frac {32 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{35}-\frac {265 x \sqrt {3 x^{4}+5 x^{2}+2}}{336}-\frac {94553 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{26880 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {23 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{126 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{32 \left (2 x^{2}+1\right )}-\frac {913 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{320 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {371 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{128 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(273\) |
Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^2,x,method=_RETURNVERBOS E)
Output:
1/3360*(3*x^4+5*x^2+2)^(1/2)*x*(720*x^6-5784*x^4-8372*x^2+605)/(2*x^2+1)-3 4249/5376*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticF( I*x,1/2*6^(1/2))+61199/20160*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+ 2)^(1/2)*(EllipticF(I*x,1/2*6^(1/2))-EllipticE(I*x,1/2*6^(1/2)))-371/128*I *(x^2+1)^(1/2)*(1+3/2*x^2)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticPi(I*x,2,1/ 2*I*(-3)^(1/2)*2^(1/2))
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^2,x, algorithm="fr icas")
Output:
integral((3*x^8 - 16*x^6 - 21*x^4 + 6*x^2 + 8)*sqrt(3*x^4 + 5*x^2 + 2)/(4* x^4 + 4*x^2 + 1), x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\int \frac {\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {3}{2}} \left (x^{4} - 7 x^{2} + 4\right )}{\left (2 x^{2} + 1\right )^{2}}\, dx \] Input:
integrate((x**4-7*x**2+4)*(3*x**4+5*x**2+2)**(3/2)/(2*x**2+1)**2,x)
Output:
Integral(((x**2 + 1)*(3*x**2 + 2))**(3/2)*(x**4 - 7*x**2 + 4)/(2*x**2 + 1) **2, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^2,x, algorithm="ma xima")
Output:
integrate((3*x^4 + 5*x^2 + 2)^(3/2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^2, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^2,x, algorithm="gi ac")
Output:
integrate((3*x^4 + 5*x^2 + 2)^(3/2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^2, x)
Timed out. \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\int \frac {\left (x^4-7\,x^2+4\right )\,{\left (3\,x^4+5\,x^2+2\right )}^{3/2}}{{\left (2\,x^2+1\right )}^2} \,d x \] Input:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(3/2))/(2*x^2 + 1)^2,x)
Output:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(3/2))/(2*x^2 + 1)^2, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{\left (1+2 x^2\right )^2} \, dx=\frac {180 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}-1446 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}-2093 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}+8308 \sqrt {3 x^{4}+5 x^{2}+2}\, x -6352 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}-3176 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )-36682 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}-18341 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )+12780 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}+6390 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )}{1680 x^{2}+840} \] Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1)^2,x)
Output:
(180*sqrt(3*x**4 + 5*x**2 + 2)*x**7 - 1446*sqrt(3*x**4 + 5*x**2 + 2)*x**5 - 2093*sqrt(3*x**4 + 5*x**2 + 2)*x**3 + 8308*sqrt(3*x**4 + 5*x**2 + 2)*x - 6352*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x)*x**2 - 3176*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31 *x**4 + 13*x**2 + 2),x) - 36682*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(12*x **8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x)*x**2 - 18341*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x) + 12780*i nt((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x)*x**2 + 6390*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(12*x**8 + 32*x* *6 + 31*x**4 + 13*x**2 + 2),x))/(840*(2*x**2 + 1))