Integrand size = 36, antiderivative size = 195 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=-\frac {x \left (689+1013 x^2\right )}{81 \left (2+5 x^2+3 x^4\right )^{3/2}}-\frac {5803 x \left (2+3 x^2\right )}{162 \sqrt {2+5 x^2+3 x^4}}+\frac {x \left (15689+17553 x^2\right )}{162 \sqrt {2+5 x^2+3 x^4}}+\frac {5803 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{81 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {2473 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{27 \sqrt {2} \sqrt {2+5 x^2+3 x^4}} \] Output:
-1/81*x*(1013*x^2+689)/(3*x^4+5*x^2+2)^(3/2)-5803/162*x*(3*x^2+2)/(3*x^4+5 *x^2+2)^(1/2)+1/162*x*(17553*x^2+15689)/(3*x^4+5*x^2+2)^(1/2)+5803/162*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)-2473/54*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1 /2)*InverseJacobiAM(arctan(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\frac {3 x \left (10000+37175 x^2+44944 x^4+17553 x^6\right )+5803 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-857 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \left (2+5 x^2+3 x^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{162 \left (2+5 x^2+3 x^4\right )^{3/2}} \] Input:
Integrate[((1 + 2*x^2)^3*(4 - 7*x^2 + x^4))/(2 + 5*x^2 + 3*x^4)^(5/2),x]
Output:
(3*x*(10000 + 37175*x^2 + 44944*x^4 + 17553*x^6) + (5803*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*(2 + 5*x^2 + 3*x^4)*EllipticE[I*ArcSinh[Sqrt[3/2]* x], 2/3] - (857*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])/(162*(2 + 5*x^2 + 3*x^4)^(3/2))
Time = 0.44 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2206, 27, 2206, 27, 1503, 1413, 1456}
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.
| \(\displaystyle \int \frac {\left (2 x^2+1\right )^3 \left (x^4-7 x^2+4\right )}{\left (3 x^4+5 x^2+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {1}{6} \int -\frac {2 \left (216 x^6-1548 x^4-1845 x^2+851\right )}{27 \left (3 x^4+5 x^2+2\right )^{3/2}}dx-\frac {x \left (1013 x^2+689\right )}{81 \left (3 x^4+5 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{81} \int \frac {216 x^6-1548 x^4-1845 x^2+851}{\left (3 x^4+5 x^2+2\right )^{3/2}}dx-\frac {x \left (1013 x^2+689\right )}{81 \left (3 x^4+5 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{81} \left (\frac {x \left (17553 x^2+15689\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {1}{2} \int \frac {3 \left (5803 x^2+4946\right )}{\sqrt {3 x^4+5 x^2+2}}dx\right )-\frac {x \left (1013 x^2+689\right )}{81 \left (3 x^4+5 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{81} \left (\frac {x \left (17553 x^2+15689\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \int \frac {5803 x^2+4946}{\sqrt {3 x^4+5 x^2+2}}dx\right )-\frac {x \left (1013 x^2+689\right )}{81 \left (3 x^4+5 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{81} \left (\frac {x \left (17553 x^2+15689\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \left (4946 \int \frac {1}{\sqrt {3 x^4+5 x^2+2}}dx+5803 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx\right )\right )-\frac {x \left (1013 x^2+689\right )}{81 \left (3 x^4+5 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {1}{81} \left (\frac {x \left (17553 x^2+15689\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \left (5803 \int \frac {x^2}{\sqrt {3 x^4+5 x^2+2}}dx+\frac {2473 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}\right )\right )-\frac {x \left (1013 x^2+689\right )}{81 \left (3 x^4+5 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {1}{81} \left (\frac {x \left (17553 x^2+15689\right )}{2 \sqrt {3 x^4+5 x^2+2}}-\frac {3}{2} \left (\frac {2473 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+5803 \left (\frac {x \left (3 x^2+2\right )}{3 \sqrt {3 x^4+5 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{3 \sqrt {3 x^4+5 x^2+2}}\right )\right )\right )-\frac {x \left (1013 x^2+689\right )}{81 \left (3 x^4+5 x^2+2\right )^{3/2}}\) |
Input:
Int[((1 + 2*x^2)^3*(4 - 7*x^2 + x^4))/(2 + 5*x^2 + 3*x^4)^(5/2),x]
Output:
-1/81*(x*(689 + 1013*x^2))/(2 + 5*x^2 + 3*x^4)^(3/2) + ((x*(15689 + 17553* x^2))/(2*Sqrt[2 + 5*x^2 + 3*x^4]) - (3*(5803*((x*(2 + 3*x^2))/(3*Sqrt[2 + 5*x^2 + 3*x^4]) - (Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE [ArcTan[x], -1/2])/(3*Sqrt[2 + 5*x^2 + 3*x^4])) + (2473*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]))/2)/81
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q )*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 16.48 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\frac {x \left (17553 x^{6}+44944 x^{4}+37175 x^{2}+10000\right )}{54 \left (3 x^{4}+5 x^{2}+2\right )^{\frac {3}{2}}}+\frac {2473 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{54 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {5803 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{162 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(135\) |
| elliptic | \(\frac {\left (-\frac {1013}{729} x^{3}-\frac {689}{729} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {6 \left (-\frac {5851}{324} x^{3}-\frac {15689}{972} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {2473 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{54 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {5803 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{162 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(162\) |
| default | \(\frac {4 \left (\frac {5}{18} x^{3}+\frac {13}{54} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {24 \left (\frac {115}{12} x^{3}+\frac {145}{18} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {5803 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{162 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {2473 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{54 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {17 \left (-\frac {2}{9} x^{3}-\frac {5}{27} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {102 \left (-\frac {97}{12} x^{3}-\frac {245}{36} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {7 \left (\frac {5}{27} x^{3}+\frac {4}{27} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {42 \left (\frac {20}{3} x^{3}+\frac {101}{18} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {46 \left (-\frac {13}{81} x^{3}-\frac {10}{81} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}+\frac {-\frac {4462}{3} x^{3}-\frac {11270}{9} x}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {44 \left (\frac {35}{243} x^{3}+\frac {26}{243} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}+\frac {\frac {10120}{9} x^{3}+\frac {25652}{27} x}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {8 \left (-\frac {97}{729} x^{3}-\frac {70}{729} x \right ) \sqrt {3 x^{4}+5 x^{2}+2}}{\left (x^{4}+\frac {5}{3} x^{2}+\frac {2}{3}\right )^{2}}-\frac {48 \left (-\frac {529}{162} x^{3}-\frac {1355}{486} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\) | \(473\) |
Input:
int((2*x^2+1)^3*(x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(5/2),x,method=_RETURNVERBOS E)
Output:
1/54*x*(17553*x^6+44944*x^4+37175*x^2+10000)/(3*x^4+5*x^2+2)^(3/2)+2473/54 *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))-5803/162*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*(El lipticF(I*x,1/2*6^(1/2))-EllipticE(I*x,1/2*6^(1/2)))
Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.77 \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\frac {11606 \, \sqrt {3} \sqrt {-\frac {2}{3}} {\left (9 \, x^{9} + 30 \, x^{7} + 37 \, x^{5} + 20 \, x^{3} + 4 \, x\right )} E(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) - 33863 \, \sqrt {3} \sqrt {-\frac {2}{3}} {\left (9 \, x^{9} + 30 \, x^{7} + 37 \, x^{5} + 20 \, x^{3} + 4 \, x\right )} F(\arcsin \left (\frac {\sqrt {-\frac {2}{3}}}{x}\right )\,|\,\frac {3}{2}) + 6 \, {\left (216 \, x^{8} - 19629 \, x^{6} - 51593 \, x^{4} - 43030 \, x^{2} - 11606\right )} \sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}{486 \, {\left (9 \, x^{9} + 30 \, x^{7} + 37 \, x^{5} + 20 \, x^{3} + 4 \, x\right )}} \] Input:
integrate((2*x^2+1)^3*(x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(5/2),x, algorithm="fr icas")
Output:
1/486*(11606*sqrt(3)*sqrt(-2/3)*(9*x^9 + 30*x^7 + 37*x^5 + 20*x^3 + 4*x)*e lliptic_e(arcsin(sqrt(-2/3)/x), 3/2) - 33863*sqrt(3)*sqrt(-2/3)*(9*x^9 + 3 0*x^7 + 37*x^5 + 20*x^3 + 4*x)*elliptic_f(arcsin(sqrt(-2/3)/x), 3/2) + 6*( 216*x^8 - 19629*x^6 - 51593*x^4 - 43030*x^2 - 11606)*sqrt(3*x^4 + 5*x^2 + 2))/(9*x^9 + 30*x^7 + 37*x^5 + 20*x^3 + 4*x)
\[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int \frac {\left (2 x^{2} + 1\right )^{3} \left (x^{4} - 7 x^{2} + 4\right )}{\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((2*x**2+1)**3*(x**4-7*x**2+4)/(3*x**4+5*x**2+2)**(5/2),x)
Output:
Integral((2*x**2 + 1)**3*(x**4 - 7*x**2 + 4)/((x**2 + 1)*(3*x**2 + 2))**(5 /2), x)
\[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (x^{4} - 7 \, x^{2} + 4\right )} {\left (2 \, x^{2} + 1\right )}^{3}}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*x^2+1)^3*(x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(5/2),x, algorithm="ma xima")
Output:
integrate((x^4 - 7*x^2 + 4)*(2*x^2 + 1)^3/(3*x^4 + 5*x^2 + 2)^(5/2), x)
\[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (x^{4} - 7 \, x^{2} + 4\right )} {\left (2 \, x^{2} + 1\right )}^{3}}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*x^2+1)^3*(x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(5/2),x, algorithm="gi ac")
Output:
integrate((x^4 - 7*x^2 + 4)*(2*x^2 + 1)^3/(3*x^4 + 5*x^2 + 2)^(5/2), x)
Timed out. \[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\int \frac {{\left (2\,x^2+1\right )}^3\,\left (x^4-7\,x^2+4\right )}{{\left (3\,x^4+5\,x^2+2\right )}^{5/2}} \,d x \] Input:
int(((2*x^2 + 1)^3*(x^4 - 7*x^2 + 4))/(5*x^2 + 3*x^4 + 2)^(5/2),x)
Output:
int(((2*x^2 + 1)^3*(x^4 - 7*x^2 + 4))/(5*x^2 + 3*x^4 + 2)^(5/2), x)
\[ \int \frac {\left (1+2 x^2\right )^3 \left (4-7 x^2+x^4\right )}{\left (2+5 x^2+3 x^4\right )^{5/2}} \, dx=\frac {2160 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{7}+26280 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{5}+36700 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}+20643 \sqrt {3 x^{4}+5 x^{2}+2}\, x -342414 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{8}-1141380 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{6}-1407702 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{4}-760920 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{2}-152184 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right )+472635 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{8}+1575450 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{6}+1943055 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{4}+1050300 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right ) x^{2}+210060 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{27 x^{12}+135 x^{10}+279 x^{8}+305 x^{6}+186 x^{4}+60 x^{2}+8}d x \right )}{7290 x^{8}+24300 x^{6}+29970 x^{4}+16200 x^{2}+3240} \] Input:
int((2*x^2+1)^3*(x^4-7*x^2+4)/(3*x^4+5*x^2+2)^(5/2),x)
Output:
(2160*sqrt(3*x**4 + 5*x**2 + 2)*x**7 + 26280*sqrt(3*x**4 + 5*x**2 + 2)*x** 5 + 36700*sqrt(3*x**4 + 5*x**2 + 2)*x**3 + 20643*sqrt(3*x**4 + 5*x**2 + 2) *x - 342414*int(sqrt(3*x**4 + 5*x**2 + 2)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**8 - 1141380*int(sqrt(3*x**4 + 5*x**2 + 2)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x* *2 + 8),x)*x**6 - 1407702*int(sqrt(3*x**4 + 5*x**2 + 2)/(27*x**12 + 135*x* *10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**4 - 760920*int(s qrt(3*x**4 + 5*x**2 + 2)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186 *x**4 + 60*x**2 + 8),x)*x**2 - 152184*int(sqrt(3*x**4 + 5*x**2 + 2)/(27*x* *12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x) + 47263 5*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**8 + 1575450*int((sqrt(3*x**4 + 5* x**2 + 2)*x**4)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 6 0*x**2 + 8),x)*x**6 + 1943055*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(27*x** 12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**4 + 1 050300*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(27*x**12 + 135*x**10 + 279*x* *8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x)*x**2 + 210060*int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(27*x**12 + 135*x**10 + 279*x**8 + 305*x**6 + 186*x**4 + 60*x**2 + 8),x))/(810*(9*x**8 + 30*x**6 + 37*x**4 + 20*x**2 + 4))