Integrand size = 24, antiderivative size = 305 \[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\frac {1}{75} x \sqrt {4+3 x^2+x^4}+\frac {4 x \sqrt {4+3 x^2+x^4}}{175 \left (2+x^2\right )}+\frac {22 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac {13}{350} \sqrt {\frac {11}{35}} \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )-\frac {4 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{175 \sqrt {4+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{175 \sqrt {4+3 x^2+x^4}}+\frac {2431 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{36750 \sqrt {2} \sqrt {4+3 x^2+x^4}} \]
13/12250*arctan(2/35*x*385^(1/2)/(x^4+3*x^2+4)^(1/2))*385^(1/2)+1/75*x*(x^ 4+3*x^2+4)^(1/2)+4/175*x*(x^4+3*x^2+4)^(1/2)/(x^2+2)+22/175*x*(x^4+3*x^2+4 )^(1/2)/(5*x^2+7)+2431/73500*(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)-4/175*(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)+4/175*(x^2+2)*(cos(2*arct an(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*EllipticF(sin(2*a rctan(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 = 10.48 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.01 \[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\frac {\frac {175 x \left (23+7 x^2\right ) \left (4+3 x^2+x^4\right )}{7+5 x^2}-i \sqrt {6+2 i \sqrt {7}} \sqrt {1-\frac {2 i x^2}{-3 i+\sqrt {7}}} \sqrt {1+\frac {2 i x^2}{3 i+\sqrt {7}}} \left (105 \left (3-i \sqrt {7}\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+7 \left (158+15 i \sqrt {7}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+429 \operatorname {EllipticPi}\left (\frac {5}{14} \left (3+i \sqrt {7}\right ),i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )\right )}{18375 \sqrt {4+3 x^2+x^4}} \]
((175*x*(23 + 7*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2) - I*Sqrt[6 + (2*I)*Sqr t[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + S qrt[7])]*(105*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt [7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 7*(158 + (15*I)*Sqrt[7])*Elli pticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + S qrt[7])] + 429*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])]))/(18375*Sqrt[4 + 3*x ^2 + x^4])
Time = 0.72 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1556, 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.
| \(\displaystyle \int \frac {\left (x^4+3 x^2+4\right )^{3/2}}{\left (5 x^2+7\right )^2} \, dx\) |
| \(\Big \downarrow \) 1556 |
| \(\displaystyle \int \left (\frac {x^4}{25 \sqrt {x^4+3 x^2+4}}+\frac {16 x^2}{125 \sqrt {x^4+3 x^2+4}}+\frac {88}{625 \left (5 x^2+7\right ) \sqrt {x^4+3 x^2+4}}+\frac {1936}{625 \left (5 x^2+7\right )^2 \sqrt {x^4+3 x^2+4}}+\frac {152}{625 \sqrt {x^4+3 x^2+4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {13}{350} \sqrt {\frac {11}{35}} \arctan \left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )+\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{175 \sqrt {x^4+3 x^2+4}}-\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{175 \sqrt {x^4+3 x^2+4}}+\frac {187 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{13125 \sqrt {x^4+3 x^2+4}}+\frac {6919 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (-\frac {9}{280},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{183750 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {4 \sqrt {x^4+3 x^2+4} x}{175 \left (x^2+2\right )}+\frac {22 \sqrt {x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )}+\frac {1}{75} \sqrt {x^4+3 x^2+4} x\) |
(x*Sqrt[4 + 3*x^2 + x^4])/75 + (4*x*Sqrt[4 + 3*x^2 + x^4])/(175*(2 + x^2)) + (22*x*Sqrt[4 + 3*x^2 + x^4])/(175*(7 + 5*x^2)) + (13*Sqrt[11/35]*ArcTan [(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/350 - (4*Sqrt[2]*(2 + x^2)*Sqrt [(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(175* Sqrt[4 + 3*x^2 + x^4]) + (4*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(175*Sqrt[4 + 3*x^2 + x^4]) + (6919*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280, 2* ArcTan[x/Sqrt[2]], 1/8])/(183750*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) + (187*Sqr t[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280, 2*Ar cTan[x/Sqrt[2]], 1/8])/(13125*Sqrt[4 + 3*x^2 + x^4])
3.4.62.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Module[{aa, bb, cc}, Int[ExpandIntegrand[1/Sqrt[aa + bb*x^2 + c c*x^4], (d + e*x^2)^q*(aa + bb*x^2 + cc*x^4)^(p + 1/2), x] /. {aa -> a, bb -> b, cc -> c}, 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] && ILtQ[q, 0] && IntegerQ[p + 1/2]
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {\sqrt {x^{4}+3 x^{2}+4}\, x \left (7 x^{2}+23\right )}{525 x^{2}+735}+\frac {232 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{375 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {128 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{175 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {286 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{6125 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(346\) |
| default | \(\frac {22 x \sqrt {x^{4}+3 x^{2}+4}}{175 \left (5 x^{2}+7\right )}+\frac {x \sqrt {x^{4}+3 x^{2}+4}}{75}+\frac {232 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{375 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {128 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{175 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {128 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{175 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {286 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{6125 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(425\) |
| elliptic | \(\frac {22 x \sqrt {x^{4}+3 x^{2}+4}}{175 \left (5 x^{2}+7\right )}+\frac {x \sqrt {x^{4}+3 x^{2}+4}}{75}+\frac {232 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{375 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {128 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{175 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {128 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{175 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {286 \sqrt {1+\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1+\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \Pi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{6125 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}\) | \(425\) |
1/105*(x^4+3*x^2+4)^(1/2)*x*(7*x^2+23)/(5*x^2+7)+232/375/(-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))-128/175/(-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))-Ellipt icE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2)))+286/6125/(-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))
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int \frac {\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \]
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
\[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int { \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx=\int \frac {{\left (x^4+3\,x^2+4\right )}^{3/2}}{{\left (5\,x^2+7\right )}^2} \,d x \]