Integrand size = 24, antiderivative size = 284 \[ \int \frac {1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx=-\frac {x \left (13+4 x^2\right )}{308 \sqrt {4+3 x^2+x^4}}+\frac {x \sqrt {4+3 x^2+x^4}}{77 \left (2+x^2\right )}+\frac {25}{176} \sqrt {\frac {5}{77}} \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{77 \sqrt {4+3 x^2+x^4}}-\frac {\left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{12 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {425 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{3696 \sqrt {2} \sqrt {4+3 x^2+x^4}} \]
25/13552*arctan(2/35*x*385^(1/2)/(x^4+3*x^2+4)^(1/2))*385^(1/2)-1/308*x*(4 *x^2+13)/(x^4+3*x^2+4)^(1/2)+1/77*x*(x^4+3*x^2+4)^(1/2)/(x^2+2)-1/24*(x^2+ 2)*(cos(2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*Ell ipticF(sin(2*arctan(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)+425/7392*(x^2+2)*(cos(2*arctan(1/2*x*2^ (1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*EllipticPi(sin(2*arctan(1/2* x*2^(1/2))),-9/280,1/4*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)-1/77*(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)
Result contains complex when optimal does not.
Time = 10.43 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx=\frac {-26 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x-8 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x^3-2 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+\sqrt {2} \left (7 i+2 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )-25 i \sqrt {2} \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticPi}\left (\frac {5}{14} \left (3+i \sqrt {7}\right ),i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{616 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
(-26*Sqrt[(-I)/(-3*I + Sqrt[7])]*x - 8*Sqrt[(-I)/(-3*I + Sqrt[7])]*x^3 - 2 *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])]*EllipticE[I*ArcSinh[S qrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2 ]*(7*I + 2*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sq rt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[( -2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - (25*I)*Sqrt [2]*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])]*EllipticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSi nh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(61 6*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])
Time = 0.59 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1547, 27, 2206, 27, 1511, 27, 1416, 1509, 2220}
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.
| \(\displaystyle \int \frac {1}{\left (5 x^2+7\right ) \left (x^4+3 x^2+4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1547 |
| \(\displaystyle \frac {125}{66} \int \frac {x^2+2}{2 \left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx-\frac {1}{66} \int \frac {25 x^4+90 x^2+124}{2 \left (x^4+3 x^2+4\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {125}{132} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx-\frac {1}{132} \int \frac {25 x^4+90 x^2+124}{\left (x^4+3 x^2+4\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{132} \left (-\frac {1}{28} \int \frac {8 \left (89-6 x^2\right )}{\sqrt {x^4+3 x^2+4}}dx-\frac {3 x \left (4 x^2+13\right )}{7 \sqrt {x^4+3 x^2+4}}\right )+\frac {125}{132} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{132} \left (-\frac {2}{7} \int \frac {89-6 x^2}{\sqrt {x^4+3 x^2+4}}dx-\frac {3 x \left (4 x^2+13\right )}{7 \sqrt {x^4+3 x^2+4}}\right )+\frac {125}{132} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{132} \left (-\frac {2}{7} \left (77 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx+12 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx\right )-\frac {3 x \left (4 x^2+13\right )}{7 \sqrt {x^4+3 x^2+4}}\right )+\frac {125}{132} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{132} \left (-\frac {2}{7} \left (77 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx+6 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )-\frac {3 x \left (4 x^2+13\right )}{7 \sqrt {x^4+3 x^2+4}}\right )+\frac {125}{132} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{132} \left (-\frac {2}{7} \left (6 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx+\frac {77 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+4}}\right )-\frac {3 x \left (4 x^2+13\right )}{7 \sqrt {x^4+3 x^2+4}}\right )+\frac {125}{132} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {125}{132} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{132} \left (-\frac {2}{7} \left (\frac {77 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+4}}+6 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )-\frac {3 x \left (4 x^2+13\right )}{7 \sqrt {x^4+3 x^2+4}}\right )\) |
| \(\Big \downarrow \) 2220 |
| \(\displaystyle \frac {125}{132} \left (\frac {3 \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{4 \sqrt {385}}+\frac {17 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{140 \sqrt {2} \sqrt {x^4+3 x^2+4}}\right )+\frac {1}{132} \left (-\frac {2}{7} \left (\frac {77 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{2 \sqrt {2} \sqrt {x^4+3 x^2+4}}+6 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )-\frac {3 x \left (4 x^2+13\right )}{7 \sqrt {x^4+3 x^2+4}}\right )\) |
((-3*x*(13 + 4*x^2))/(7*Sqrt[4 + 3*x^2 + x^4]) - (2*(6*(-((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]) + (77*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1 /8])/(2*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])))/7)/132 + (125*((3*ArcTan[(2*Sqrt[ 11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/(4*Sqrt[385]) + (17*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280, 2*ArcTan[x/Sqrt[2]], 1/8])/(1 40*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])))/132
3.4.77.3.1 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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_.) + (e_.)*(x_)^2), x_Sym bol] :> Simp[-(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(e^(2*p)*(Rt[c/a, 2]*d - e) ) Int[(1 + Rt[c/a, 2]*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] + Simp[(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(Rt[c/a, 2]*d - e) Int[(a + b*x^ 2 + c*x^4)^p*ExpandToSum[((Rt[c/a, 2]*d - e)*(c*d^2 - b*d*e + a*e^2)^(-p - 1/2) + ((1 + Rt[c/a, 2]*x^2)*(a + b*x^2 + c*x^4)^(-p - 1/2))/e^(2*p))/(d + e*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N eQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0] && NeQ[c*d^2 - a*e^2, 0] & & PosQ[c/a]
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]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ -b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*El lipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] & & EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[-b + c*(d/e) + a*(e/d)]
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(-\frac {x \left (4 x^{2}+13\right )}{308 \sqrt {x^{4}+3 x^{2}+4}}-\frac {\sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {32 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {25 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{308 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(337\) |
| default | \(-\frac {2 \left (\frac {1}{154} x^{3}+\frac {13}{616} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {\sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {32 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {32 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {25 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{308 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(409\) |
| elliptic | \(-\frac {2 \left (\frac {1}{154} x^{3}+\frac {13}{616} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {\sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {32 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {32 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{77 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {25 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{308 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(409\) |
-1/308*x*(4*x^2+13)/(x^4+3*x^2+4)^(1/2)-1/77/(-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) )-32/77/(-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)))+25/308/(-3/8+1/8*I*7^(1/2)) ^(1/2)*(1+3/8*x^2-1/8*I*x^2*7^(1/2))^(1/2)*(1+3/8*x^2+1/8*I*x^2*7^(1/2))^( 1/2)/(x^4+3*x^2+4)^(1/2)*EllipticPi((-3/8+1/8*I*7^(1/2))^(1/2)*x,-5/7/(-3/ 8+1/8*I*7^(1/2)),(-3/8-1/8*I*7^(1/2))^(1/2)/(-3/8+1/8*I*7^(1/2))^(1/2))
\[ \int \frac {1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}} \,d x } \]
\[ \int \frac {1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \cdot \left (5 x^{2} + 7\right )}\, dx \]
\[ \int \frac {1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}} \,d x } \]
\[ \int \frac {1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (5\,x^2+7\right )\,{\left (x^4+3\,x^2+4\right )}^{3/2}} \,d x \]