Integrand size = 36, antiderivative size = 295 \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\frac {246517 x \left (2+3 x^2\right )}{54432 \sqrt {2+5 x^2+3 x^4}}+\frac {15823 x \sqrt {2+5 x^2+3 x^4}}{9072}-\frac {845}{504} x^3 \sqrt {2+5 x^2+3 x^4}-\frac {305}{252} x^5 \sqrt {2+5 x^2+3 x^4}+\frac {1}{6} x^7 \sqrt {2+5 x^2+3 x^4}-\frac {246517 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{27216 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {95929 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{9072 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {31 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{16 \sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
246517/54432*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+15823/9072*x*(3*x^4+5*x^2+2 )^(1/2)-845/504*x^3*(3*x^4+5*x^2+2)^(1/2)-305/252*x^5*(3*x^4+5*x^2+2)^(1/2 )+1/6*x^7*(3*x^4+5*x^2+2)^(1/2)-246517/54432*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 )+95929/18144*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(ar ctan(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)+31/48*(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 = 10.39 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.65 \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\frac {379752 x+584340 x^3-606492 x^5-1170072 x^7-304560 x^9+54432 x^{11}-493034 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )+56587 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )-17577 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{108864 \sqrt {2+5 x^2+3 x^4}} \] Input:
Integrate[((4 - 7*x^2 + x^4)*(2 + 5*x^2 + 3*x^4)^(3/2))/(1 + 2*x^2),x]
Output:
(379752*x + 584340*x^3 - 606492*x^5 - 1170072*x^7 - 304560*x^9 + 54432*x^1 1 - (493034*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[S qrt[3/2]*x], 2/3] + (56587*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*Ellipt icF[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (17577*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/(108864*Sqrt[2 + 5* x^2 + 3*x^4])
Time = 0.78 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.37, 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.
| \(\displaystyle \int \frac {\left (x^4-7 x^2+4\right ) \left (3 x^4+5 x^2+2\right )^{3/2}}{2 x^2+1} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (-\frac {479 x^4}{16 \sqrt {3 x^4+5 x^2+2}}+\frac {671 x^2}{32 \sqrt {3 x^4+5 x^2+2}}+\frac {31}{64 \left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {993}{64 \sqrt {3 x^4+5 x^2+2}}+\frac {9 x^{10}}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {75 x^8}{4 \sqrt {3 x^4+5 x^2+2}}-\frac {473 x^6}{8 \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {404 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{567 \sqrt {3 x^4+5 x^2+2}}+\frac {27667 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{3024 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}-\frac {66046 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{8505 \sqrt {3 x^4+5 x^2+2}}+\frac {41947 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{6480 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {31 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{16 \sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {15823 \sqrt {3 x^4+5 x^2+2} x}{9072}+\frac {246517 \left (3 x^2+2\right ) x}{54432 \sqrt {3 x^4+5 x^2+2}}+\frac {1}{6} \sqrt {3 x^4+5 x^2+2} x^7-\frac {305}{252} \sqrt {3 x^4+5 x^2+2} x^5-\frac {845}{504} \sqrt {3 x^4+5 x^2+2} x^3\) |
Input:
Int[((4 - 7*x^2 + x^4)*(2 + 5*x^2 + 3*x^4)^(3/2))/(1 + 2*x^2),x]
Output:
(246517*x*(2 + 3*x^2))/(54432*Sqrt[2 + 5*x^2 + 3*x^4]) + (15823*x*Sqrt[2 + 5*x^2 + 3*x^4])/9072 - (845*x^3*Sqrt[2 + 5*x^2 + 3*x^4])/504 - (305*x^5*S qrt[2 + 5*x^2 + 3*x^4])/252 + (x^7*Sqrt[2 + 5*x^2 + 3*x^4])/6 + (41947*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(6480*Sqrt[ 2]*Sqrt[2 + 5*x^2 + 3*x^4]) - (66046*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(8505*Sqrt[2 + 5*x^2 + 3*x^4]) + (276 67*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(3024 *Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) + (404*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2 )/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(567*Sqrt[2 + 5*x^2 + 3*x^4]) + ( 31*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(16*Sqrt[3]*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 = 5.08 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.62
| method | result | size |
| risch | \(\frac {x \left (1512 x^{6}-10980 x^{4}-15210 x^{2}+15823\right ) \sqrt {3 x^{4}+5 x^{2}+2}}{9072}-\frac {436447 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{72576 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {246517 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{54432 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{64 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(183\) |
| elliptic | \(\frac {x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{6}-\frac {305 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{252}-\frac {845 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{504}+\frac {15823 x \sqrt {3 x^{4}+5 x^{2}+2}}{9072}-\frac {323273 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{217728 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {246517 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{54432 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{64 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(210\) |
| default | \(-\frac {305 x^{5} \sqrt {3 x^{4}+5 x^{2}+2}}{252}-\frac {845 x^{3} \sqrt {3 x^{4}+5 x^{2}+2}}{504}+\frac {15823 x \sqrt {3 x^{4}+5 x^{2}+2}}{9072}+\frac {466033 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{362880 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {47101 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{17010 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {x^{7} \sqrt {3 x^{4}+5 x^{2}+2}}{6}-\frac {3503 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{480 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{64 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(266\) |
Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/9072*x*(1512*x^6-10980*x^4-15210*x^2+15823)*(3*x^4+5*x^2+2)^(1/2)-436447 /72576*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))+246517/54432*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)))-31/64*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))
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{2 \, x^{2} + 1} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1),x, algorithm="fric as")
Output:
integral((3*x^8 - 16*x^6 - 21*x^4 + 6*x^2 + 8)*sqrt(3*x^4 + 5*x^2 + 2)/(2* x^2 + 1), x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\int \frac {\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {3}{2}} \left (x^{4} - 7 x^{2} + 4\right )}{2 x^{2} + 1}\, dx \] Input:
integrate((x**4-7*x**2+4)*(3*x**4+5*x**2+2)**(3/2)/(2*x**2+1),x)
Output:
Integral(((x**2 + 1)*(3*x**2 + 2))**(3/2)*(x**4 - 7*x**2 + 4)/(2*x**2 + 1) , x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{2 \, x^{2} + 1} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1),x, algorithm="maxi ma")
Output:
integrate((3*x^4 + 5*x^2 + 2)^(3/2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1), x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\int { \frac {{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{2 \, x^{2} + 1} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1),x, algorithm="giac ")
Output:
integrate((3*x^4 + 5*x^2 + 2)^(3/2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1), x)
Timed out. \[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\int \frac {\left (x^4-7\,x^2+4\right )\,{\left (3\,x^4+5\,x^2+2\right )}^{3/2}}{2\,x^2+1} \,d x \] Input:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(3/2))/(2*x^2 + 1),x)
Output:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(3/2))/(2*x^2 + 1), x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \left (2+5 x^2+3 x^4\right )^{3/2}}{1+2 x^2} \, dx=\frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}}{6}-\frac {305 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}}{252}-\frac {845 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}}{504}+\frac {15823 \sqrt {3 x^{4}+5 x^{2}+2}\, x}{9072}+\frac {56753 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{6 x^{6}+13 x^{4}+9 x^{2}+2}d x \right )}{4536}+\frac {246517 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{6 x^{6}+13 x^{4}+9 x^{2}+2}d x \right )}{9072}+\frac {170741 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{6 x^{6}+13 x^{4}+9 x^{2}+2}d x \right )}{4536} \] Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(3/2)/(2*x^2+1),x)
Output:
(1512*sqrt(3*x**4 + 5*x**2 + 2)*x**7 - 10980*sqrt(3*x**4 + 5*x**2 + 2)*x** 5 - 15210*sqrt(3*x**4 + 5*x**2 + 2)*x**3 + 15823*sqrt(3*x**4 + 5*x**2 + 2) *x + 113506*int(sqrt(3*x**4 + 5*x**2 + 2)/(6*x**6 + 13*x**4 + 9*x**2 + 2), x) + 246517*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(6*x**6 + 13*x**4 + 9*x** 2 + 2),x) + 341482*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(6*x**6 + 13*x**4 + 9*x**2 + 2),x))/9072