Integrand size = 36, antiderivative size = 253 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=-\frac {x \left (49+53 x^2\right )}{\left (2+5 x^2+3 x^4\right )^{3/2}}-\frac {395 x \left (2+3 x^2\right )}{2 \sqrt {2+5 x^2+3 x^4}}+\frac {x \left (889+1185 x^2\right )}{2 \sqrt {2+5 x^2+3 x^4}}+\frac {395 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {915 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {496 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
-x*(53*x^2+49)/(3*x^4+5*x^2+2)^(3/2)-395/2*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/ 2)+1/2*x*(1185*x^2+889)/(3*x^4+5*x^2+2)^(1/2)+395/2*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)-915/2*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(a rctan(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)+496/3*(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 = 10.42 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.95 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\frac {-294 x-318 x^3+2667 x \left (2+5 x^2+3 x^4\right )+3555 x^3 \left (2+5 x^2+3 x^4\right )+1185 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-99 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )-248 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{6 \left (2+5 x^2+3 x^4\right )^{3/2}} \] Input:
Integrate[(4 - 7*x^2 + x^4)/((1 + 2*x^2)*(2 + 5*x^2 + 3*x^4)^(5/2)),x]
Output:
(-294*x - 318*x^3 + 2667*x*(2 + 5*x^2 + 3*x^4) + 3555*x^3*(2 + 5*x^2 + 3*x ^4) + (1185*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*(2 + 5*x^2 + 3*x^4)*E llipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (99*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[ 2 + 3*x^2]*(2 + 5*x^2 + 3*x^4)*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (2 48*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*(2 + 5*x^2 + 3*x^4)*EllipticPi [4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/(6*(2 + 5*x^2 + 3*x^4)^(3/2))
Time = 0.78 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.81, 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.
| \(\displaystyle \int \frac {x^4-7 x^2+4}{\left (2 x^2+1\right ) \left (3 x^4+5 x^2+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (-\frac {87}{\left (x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {124}{\left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {75}{\left (3 x^2+2\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {12}{\left (x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}-\frac {246}{\left (3 x^2+2\right )^2 \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {186 \sqrt {2} \sqrt {\frac {3 x^2+2}{x^2+1}} \left (x^2+1\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}-\frac {42 \sqrt {2} \left (3 x^2+2\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}-\frac {459 \left (3 x^2+2\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {160 \sqrt {2} \left (3 x^2+2\right ) E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {75 \left (3 x^2+2\right ) E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {496 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {471 x \left (x^2+1\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {123 x \left (x^2+1\right )}{\left (3 x^2+2\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {157 x \left (3 x^2+2\right )}{2 \sqrt {3 x^4+5 x^2+2}}+\frac {4 x \left (3 x^2+2\right )}{\sqrt {3 x^4+5 x^2+2} \left (x^2+1\right )}\) |
Input:
Int[(4 - 7*x^2 + x^4)/((1 + 2*x^2)*(2 + 5*x^2 + 3*x^4)^(5/2)),x]
Output:
(471*x*(1 + x^2))/(2*Sqrt[2 + 5*x^2 + 3*x^4]) - (123*x*(1 + x^2))/((2 + 3* x^2)*Sqrt[2 + 5*x^2 + 3*x^4]) - (157*x*(2 + 3*x^2))/(2*Sqrt[2 + 5*x^2 + 3* x^4]) + (4*x*(2 + 3*x^2))/((1 + x^2)*Sqrt[2 + 5*x^2 + 3*x^4]) + (75*(2 + 3 *x^2)*EllipticE[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqr t[2 + 5*x^2 + 3*x^4]) + (160*Sqrt[2]*(2 + 3*x^2)*EllipticE[ArcTan[x], -1/2 ])/(Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - (459*(2 + 3*x^2 )*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - (42*Sqrt[2]*(2 + 3*x^2)*EllipticF[ArcTan[x], -1/2])/(S qrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - (186*Sqrt[2]*(1 + x^ 2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/Sqrt[2 + 5*x^2 + 3*x^4] + (496*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(Sqr t[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 = 1.98 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.78
| method | result | size |
| elliptic | \(\frac {\left (-\frac {53}{9} x^{3}-\frac {49}{9} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {6 \left (-\frac {395}{4} x^{3}-\frac {889}{12} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {74 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {395 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {124 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\) | \(197\) |
| default | \(-\frac {15 \left (\frac {5}{18} x^{3}+\frac {13}{54} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{4 \left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}+\frac {\frac {1725}{8} x^{3}+\frac {725}{4} x}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {3695 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{24 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {1919 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{24 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {\left (-\frac {2}{9} x^{3}-\frac {5}{27} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{2 \left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {3 \left (-\frac {97}{12} x^{3}-\frac {245}{36} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {31 \left (-\frac {11}{18} x^{3}-\frac {31}{54} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{4 \left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {93 \left (-\frac {91}{12} x^{3}-\frac {47}{9} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {2821 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{24 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {124 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\) | \(378\) |
Input:
int((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
(-53/9*x^3-49/9*x)*(3*x^4+5*x^2+2)^(1/2)/(x^4+5/3*x^2+2/3)^2-6*(-395/4*x^3 -889/12*x)/(3*x^4+5*x^2+2)^(1/2)+74*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))+395/2*I*(x^2+1)^(1/2)*(6*x^2+4) ^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticE(I*x,1/2*6^(1/2))-124*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))
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}} {\left (2 \, x^{2} + 1\right )}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(5/2),x, algorithm="fric as")
Output:
integral(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(54*x^14 + 297*x^12 + 6 93*x^10 + 889*x^8 + 677*x^6 + 306*x^4 + 76*x^2 + 8), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int \frac {x^{4} - 7 x^{2} + 4}{\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {5}{2}} \cdot \left (2 x^{2} + 1\right )}\, dx \] Input:
integrate((x**4-7*x**2+4)/(2*x**2+1)/(3*x**4+5*x**2+2)**(5/2),x)
Output:
Integral((x**4 - 7*x**2 + 4)/(((x**2 + 1)*(3*x**2 + 2))**(5/2)*(2*x**2 + 1 )), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}} {\left (2 \, x^{2} + 1\right )}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(5/2),x, algorithm="maxi ma")
Output:
integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(5/2)*(2*x^2 + 1)), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}} {\left (2 \, x^{2} + 1\right )}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(5/2),x, algorithm="giac ")
Output:
integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(5/2)*(2*x^2 + 1)), x)
Timed out. \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int \frac {x^4-7\,x^2+4}{\left (2\,x^2+1\right )\,{\left (3\,x^4+5\,x^2+2\right )}^{5/2}} \,d x \] Input:
int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)*(5*x^2 + 3*x^4 + 2)^(5/2)),x)
Output:
int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)*(5*x^2 + 3*x^4 + 2)^(5/2)), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=4 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{54 x^{14}+297 x^{12}+693 x^{10}+889 x^{8}+677 x^{6}+306 x^{4}+76 x^{2}+8}d x \right )+\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{54 x^{14}+297 x^{12}+693 x^{10}+889 x^{8}+677 x^{6}+306 x^{4}+76 x^{2}+8}d x -7 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{54 x^{14}+297 x^{12}+693 x^{10}+889 x^{8}+677 x^{6}+306 x^{4}+76 x^{2}+8}d x \right ) \] Input:
int((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(5/2),x)
Output:
4*int(sqrt(3*x**4 + 5*x**2 + 2)/(54*x**14 + 297*x**12 + 693*x**10 + 889*x* *8 + 677*x**6 + 306*x**4 + 76*x**2 + 8),x) + int((sqrt(3*x**4 + 5*x**2 + 2 )*x**4)/(54*x**14 + 297*x**12 + 693*x**10 + 889*x**8 + 677*x**6 + 306*x**4 + 76*x**2 + 8),x) - 7*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(54*x**14 + 29 7*x**12 + 693*x**10 + 889*x**8 + 677*x**6 + 306*x**4 + 76*x**2 + 8),x)