Integrand size = 24, antiderivative size = 284 \[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{7+5 x^2} \, dx=\frac {94 x \sqrt {4+3 x^2+x^4}}{125 \left (2+x^2\right )}+\frac {1}{75} x \left (11+3 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {44}{125} \sqrt {\frac {11}{35}} \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )-\frac {94 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{125 \sqrt {4+3 x^2+x^4}}+\frac {54 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{125 \sqrt {4+3 x^2+x^4}}+\frac {4114 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{13125 \sqrt {4+3 x^2+x^4}} \]
44/4375*arctan(2/35*x*385^(1/2)/(x^4+3*x^2+4)^(1/2))*385^(1/2)+94/125*x*(x ^4+3*x^2+4)^(1/2)/(x^2+2)+1/75*x*(3*x^2+11)*(x^4+3*x^2+4)^(1/2)-94/125*(x^ 2+2)*(cos(2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*E llipticE(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)+54/125*(x^2+2)*(cos(2*arctan(1/2*x*2^ (1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*EllipticF(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)+4114/13125*(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)
Result contains complex when optimal does not.
Time = 10.56 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.68 \[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{7+5 x^2} \, dx=\frac {350 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (44+45 x^2+20 x^4+3 x^6\right )-4935 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+7 \sqrt {2} \left (-241 i+705 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )-5808 i \sqrt {2} \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticPi}\left (\frac {5}{14} \left (3+i \sqrt {7}\right ),i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{26250 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
(350*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(44 + 45*x^2 + 20*x^4 + 3*x^6) - 4935*S qrt[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[Sqr t[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 7*Sqrt[2 ]*(-241*I + 705*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])]*EllipticF[I*ArcSinh[S qrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - (5808* 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*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7] )])/(26250*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])
Time = 0.71 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1530, 27, 2207, 27, 2207, 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.
| \(\displaystyle \int \frac {\left (x^4+3 x^2+4\right )^{3/2}}{5 x^2+7} \, dx\) |
| \(\Big \downarrow \) 1530 |
| \(\displaystyle \frac {1936 \int \frac {17 \left (x^2+2\right )}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx}{6375}-\frac {\int -\frac {17 \left (75 x^6+345 x^4+792 x^2+304\right )}{\sqrt {x^4+3 x^2+4}}dx}{6375}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \int \frac {75 x^6+345 x^4+792 x^2+304}{\sqrt {x^4+3 x^2+4}}dx\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \left (\frac {1}{5} \int \frac {5 \left (165 x^4+612 x^2+304\right )}{\sqrt {x^4+3 x^2+4}}dx+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \left (\int \frac {165 x^4+612 x^2+304}{\sqrt {x^4+3 x^2+4}}dx+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \left (\frac {1}{3} \int \frac {18 \left (47 x^2+14\right )}{\sqrt {x^4+3 x^2+4}}dx+55 \sqrt {x^4+3 x^2+4} x+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \left (6 \int \frac {47 x^2+14}{\sqrt {x^4+3 x^2+4}}dx+55 \sqrt {x^4+3 x^2+4} x+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \left (6 \left (108 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-94 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx\right )+55 \sqrt {x^4+3 x^2+4} x+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \left (6 \left (108 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-47 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+55 \sqrt {x^4+3 x^2+4} x+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{375} \left (6 \left (\frac {27 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-47 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+55 \sqrt {x^4+3 x^2+4} x+15 \sqrt {x^4+3 x^2+4} x^3\right )+\frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1936}{375} \int \frac {x^2+2}{\left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}dx+\frac {1}{375} \left (6 \left (\frac {27 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-47 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+55 \sqrt {x^4+3 x^2+4} x+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
| \(\Big \downarrow \) 2220 |
| \(\displaystyle \frac {1936}{375} \left (\frac {3 \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{4 \sqrt {385}}+\frac {17 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{140 \sqrt {2} \sqrt {x^4+3 x^2+4}}\right )+\frac {1}{375} \left (6 \left (\frac {27 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-47 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+55 \sqrt {x^4+3 x^2+4} x+15 \sqrt {x^4+3 x^2+4} x^3\right )\) |
(55*x*Sqrt[4 + 3*x^2 + x^4] + 15*x^3*Sqrt[4 + 3*x^2 + x^4] + 6*(-47*(-((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 ]) + (27*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2 *ArcTan[x/Sqrt[2]], 1/8])/Sqrt[4 + 3*x^2 + x^4]))/375 + (1936*((3*ArcTan[( 2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/(4*Sqrt[385]) + (17*(2 + x^2)*Sqr t[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280, 2*ArcTan[x/Sqrt[2]], 1 /8])/(140*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])))/375
3.4.61.3.1 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_Symb ol] :> Simp[-(c*d^2 - b*d*e + a*e^2)^(p + 1/2)/(e^(2*p)*(c*d^2 - a*e^2)) Int[(a*d*Rt[c/a, 2] + a*e + (c*d + a*e*Rt[c/a, 2])*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] + Simp[1/(e^(2*p)*(c*d^2 - a*e^2)) Int[(1/Sqrt [a + b*x^2 + c*x^4])*ExpandToSum[(e^(2*p)*(c*d^2 - a*e^2)*(a + b*x^2 + c*x^ 4)^(p + 1/2) + (c*d^2 - b*d*e + a*e^2)^(p + 1/2)*(a*d*Rt[c/a, 2] + a*e + (c *d + a*e*Rt[c/a, 2])*x^2))/(d + e*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[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[{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]
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 = 0.56 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(\frac {x \left (3 x^{2}+11\right ) \sqrt {x^{4}+3 x^{2}+4}}{75}+\frac {9424 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{1875 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {3008 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{125 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {1936 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{4375 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(337\) |
| default | \(\frac {x^{3} \sqrt {x^{4}+3 x^{2}+4}}{25}+\frac {11 x \sqrt {x^{4}+3 x^{2}+4}}{75}+\frac {9424 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{1875 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {3008 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{125 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {3008 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{125 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {1936 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{4375 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(418\) |
| elliptic | \(\frac {x^{3} \sqrt {x^{4}+3 x^{2}+4}}{25}+\frac {11 x \sqrt {x^{4}+3 x^{2}+4}}{75}+\frac {9424 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{1875 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {3008 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{125 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {3008 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{125 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {1936 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{4375 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(418\) |
1/75*x*(3*x^2+11)*(x^4+3*x^2+4)^(1/2)+9424/1875/(-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))-3008/125/(-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))*(Ellipt icF(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)))+1936/4375/(-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))
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{7+5 x^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{5 \, x^{2} + 7} \,d x } \]
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{7+5 x^2} \, dx=\int \frac {\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}{5 x^{2} + 7}\, dx \]
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{7+5 x^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{5 \, x^{2} + 7} \,d x } \]
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{7+5 x^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{5 \, x^{2} + 7} \,d x } \]
Timed out. \[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{7+5 x^2} \, dx=\int \frac {{\left (x^4+3\,x^2+4\right )}^{3/2}}{5\,x^2+7} \,d x \]