Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\frac {x \left (99493+45779 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {5000}{3} x \sqrt {4+3 x^2+x^4}+625 x^3 \sqrt {4+3 x^2+x^4}-\frac {220779 x \sqrt {4+3 x^2+x^4}}{28 \left (2+x^2\right )}+\frac {220779 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{14 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {130729 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{12 \sqrt {2} \sqrt {4+3 x^2+x^4}} \]
1/28*x*(45779*x^2+99493)/(x^4+3*x^2+4)^(1/2)+5000/3*x*(x^4+3*x^2+4)^(1/2)+ 625*x^3*(x^4+3*x^2+4)^(1/2)-220779/28*x*(x^4+3*x^2+4)^(1/2)/(x^2+2)+220779 /28*(x^2+2)*(cos(2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1 /2)))*EllipticE(sin(2*arctan(1/2*x*2^(1/2))),1/4*2^(1/2))*2^(1/2)*((x^4+3* x^2+4)/(x^2+2)^2)^(1/2)/(x^4+3*x^2+4)^(1/2)-130729/24*(x^2+2)*(cos(2*arcta n(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*EllipticF(sin(2*ar ctan(1/2*x*2^(1/2))),1/4*2^(1/2))*((x^4+3*x^2+4)/(x^2+2)^2)^(1/2)*2^(1/2)/ (x^4+3*x^2+4)^(1/2)
Result contains complex when optimal does not.
Time = 10.36 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.55 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\frac {4 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (858479+767337 x^2+297500 x^4+52500 x^6\right )+662337 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )-\sqrt {2} \left (975947 i+662337 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{336 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(858479 + 767337*x^2 + 297500*x^4 + 52500 *x^6) + 662337*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/( -3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*Ellipti cE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt [7])] - Sqrt[2]*(975947*I + 662337*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x ^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*El lipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(336*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])
Time = 0.44 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1517, 2207, 27, 2207, 25, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {\left (5 x^2+7\right )^5}{\left (x^4+3 x^2+4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {1}{28} \int \frac {87500 x^6+350000 x^4+269221 x^2+18156}{\sqrt {x^4+3 x^2+4}}dx+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{5} \int \frac {5 \left (140000 x^4+59221 x^2+18156\right )}{\sqrt {x^4+3 x^2+4}}dx+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\int \frac {140000 x^4+59221 x^2+18156}{\sqrt {x^4+3 x^2+4}}dx+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{3} \int -\frac {662337 x^2+505532}{\sqrt {x^4+3 x^2+4}}dx+\frac {140000}{3} \sqrt {x^4+3 x^2+4} x+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{28} \left (-\frac {1}{3} \int \frac {662337 x^2+505532}{\sqrt {x^4+3 x^2+4}}dx+\frac {140000}{3} \sqrt {x^4+3 x^2+4} x+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{3} \left (1324674 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx-1830206 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {140000}{3} \sqrt {x^4+3 x^2+4} x+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{3} \left (662337 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx-1830206 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {140000}{3} \sqrt {x^4+3 x^2+4} x+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{3} \left (662337 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx-\frac {915103 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+4}}\right )+\frac {140000}{3} \sqrt {x^4+3 x^2+4} x+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{28} \left (\frac {1}{3} \left (662337 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )-\frac {915103 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+4}}\right )+\frac {140000}{3} \sqrt {x^4+3 x^2+4} x+17500 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {x \left (45779 x^2+99493\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
(x*(99493 + 45779*x^2))/(28*Sqrt[4 + 3*x^2 + x^4]) + ((140000*x*Sqrt[4 + 3 *x^2 + x^4])/3 + 17500*x^3*Sqrt[4 + 3*x^2 + x^4] + (662337*(-((x*Sqrt[4 + 3*x^2 + x^4])/(2 + x^2)) + (Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/Sqrt[4 + 3*x^2 + x^4]) - (915 103*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqr t[2]], 1/8])/(Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]))/3)/28
3.4.71.3.1 Defintions of rubi rules used
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[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*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[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], 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] && IGtQ[q, 1] && LtQ[p, -1]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p + 1)) Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 *n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) *x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && !LtQ[p, -1]
Result contains complex when optimal does not.
Time = 2.40 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {x \left (52500 x^{6}+297500 x^{4}+767337 x^{2}+858479\right )}{84 \sqrt {x^{4}+3 x^{2}+4}}-\frac {505532 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{21 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {1766232 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\) | \(241\) |
| elliptic | \(-\frac {2 \left (-\frac {45779}{56} x^{3}-\frac {99493}{56} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}+625 x^{3} \sqrt {x^{4}+3 x^{2}+4}+\frac {5000 x \sqrt {x^{4}+3 x^{2}+4}}{3}-\frac {505532 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{21 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {1766232 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\) | \(264\) |
| default | \(-\frac {33614 \left (\frac {1}{56} x +\frac {3}{56} x^{3}\right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {505532 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{21 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {1766232 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}-\frac {6250 \left (\frac {31}{14} x^{3}+\frac {18}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}+625 x^{3} \sqrt {x^{4}+3 x^{2}+4}+\frac {5000 x \sqrt {x^{4}+3 x^{2}+4}}{3}-\frac {43750 \left (-\frac {9}{14} x^{3}+\frac {2}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {122500 \left (-\frac {1}{14} x^{3}-\frac {6}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {171500 \left (\frac {3}{14} x^{3}+\frac {4}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {120050 \left (-\frac {1}{7} x^{3}-\frac {3}{14} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}\) | \(379\) |
1/84*x*(52500*x^6+297500*x^4+767337*x^2+858479)/(x^4+3*x^2+4)^(1/2)-505532 /21/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8 *I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2)*EllipticF(1/4*x*(-6+2*I*7^(1/2) )^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))+1766232/7/(-6+2*I*7^(1/2))^(1/2)*(1-(-3 /8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2 +4)^(1/2)/(3+I*7^(1/2))*(EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I *7^(1/2))^(1/2))-EllipticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2) )^(1/2)))
Time = 0.09 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.85 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\frac {662337 \, \sqrt {2} {\left (3 \, x^{5} + 9 \, x^{3} - \sqrt {-7} {\left (x^{5} + 3 \, x^{3} + 4 \, x\right )} + 12 \, x\right )} \sqrt {\sqrt {-7} - 3} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) - 2 \, \sqrt {2} {\left (1183080 \, x^{5} + 3549240 \, x^{3} - 267977 \, \sqrt {-7} {\left (x^{5} + 3 \, x^{3} + 4 \, x\right )} + 4732320 \, x\right )} \sqrt {\sqrt {-7} - 3} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) + 16 \, {\left (13125 \, x^{8} + 74375 \, x^{6} + 26250 \, x^{4} - 282133 \, x^{2} - 662337\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}}{336 \, {\left (x^{5} + 3 \, x^{3} + 4 \, x\right )}} \]
1/336*(662337*sqrt(2)*(3*x^5 + 9*x^3 - sqrt(-7)*(x^5 + 3*x^3 + 4*x) + 12*x )*sqrt(sqrt(-7) - 3)*elliptic_e(arcsin(1/2*sqrt(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) - 2*sqrt(2)*(1183080*x^5 + 3549240*x^3 - 267977*sqrt(- 7)*(x^5 + 3*x^3 + 4*x) + 4732320*x)*sqrt(sqrt(-7) - 3)*elliptic_f(arcsin(1 /2*sqrt(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) + 16*(13125*x^8 + 74 375*x^6 + 26250*x^4 - 282133*x^2 - 662337)*sqrt(x^4 + 3*x^2 + 4))/(x^5 + 3 *x^3 + 4*x)
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {\left (5 x^{2} + 7\right )^{5}}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (7+5 x^2\right )^5}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {{\left (5\,x^2+7\right )}^5}{{\left (x^4+3\,x^2+4\right )}^{3/2}} \,d x \]