Integrand size = 22, antiderivative size = 207 \[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {2798 x \sqrt {4+3 x^2+x^4}}{105 \left (2+x^2\right )}+\frac {1}{105} x \left (1029+289 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}-\frac {2798 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{105 \sqrt {4+3 x^2+x^4}}+\frac {74 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{3 \sqrt {4+3 x^2+x^4}} \]
1/63*x*(35*x^2+108)*(x^4+3*x^2+4)^(3/2)+2798/105*x*(x^4+3*x^2+4)^(1/2)/(x^ 2+2)+1/105*x*(289*x^2+1029)*(x^4+3*x^2+4)^(1/2)-2798/105*(x^2+2)*(cos(2*ar ctan(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)+74/3*(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)
Result contains complex when optimal does not.
Time = 6.38 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.69 \[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {2 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (20988+28489 x^2+19068 x^4+7082 x^6+1590 x^8+175 x^{10}\right )-4197 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+3 \sqrt {2} \left (-567 i+1399 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{630 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(20988 + 28489*x^2 + 19068*x^4 + 7082*x^6 + 1590*x^8 + 175*x^10) - 4197*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 + S qrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt [7])/(3*I + Sqrt[7])] + 3*Sqrt[2]*(-567*I + 1399*Sqrt[7])*Sqrt[(-3*I + Sqr t[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[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - S qrt[7])/(3*I + Sqrt[7])])/(630*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])
Time = 0.35 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1490, 1490, 27, 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.
| \(\displaystyle \int \left (5 x^2+7\right ) \left (x^4+3 x^2+4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{21} \int \left (289 x^2+444\right ) \sqrt {x^4+3 x^2+4}dx+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{15} \int \frac {6 \left (1399 x^2+2382\right )}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (289 x^2+1029\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \int \frac {1399 x^2+2382}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (289 x^2+1029\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \left (5180 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-2798 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (289 x^2+1029\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \left (5180 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-1399 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (289 x^2+1029\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \left (\frac {1295 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-1399 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (289 x^2+1029\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{5} \left (\frac {1295 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-1399 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (289 x^2+1029\right )\right )+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}\) |
(x*(108 + 35*x^2)*(4 + 3*x^2 + x^4)^(3/2))/63 + ((x*(1029 + 289*x^2)*Sqrt[ 4 + 3*x^2 + x^4])/5 + (2*(-1399*(-((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]) + (1295*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]))/5)/21
3.4.59.3.1 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)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), 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] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
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]
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {x \left (175 x^{6}+1065 x^{4}+3187 x^{2}+5247\right ) \sqrt {x^{4}+3 x^{2}+4}}{315}+\frac {6352 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {89536 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{105 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\) | \(241\) |
| default | \(\frac {71 x^{5} \sqrt {x^{4}+3 x^{2}+4}}{21}+\frac {3187 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{315}+\frac {583 x \sqrt {x^{4}+3 x^{2}+4}}{35}+\frac {6352 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {89536 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{105 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {5 x^{7} \sqrt {x^{4}+3 x^{2}+4}}{9}\) | \(275\) |
| elliptic | \(\frac {71 x^{5} \sqrt {x^{4}+3 x^{2}+4}}{21}+\frac {3187 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{315}+\frac {583 x \sqrt {x^{4}+3 x^{2}+4}}{35}+\frac {6352 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {89536 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{105 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {5 x^{7} \sqrt {x^{4}+3 x^{2}+4}}{9}\) | \(275\) |
1/315*x*(175*x^6+1065*x^4+3187*x^2+5247)*(x^4+3*x^2+4)^(1/2)+6352/35/(-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))-89536/105/(-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 = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.64 \[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {8394 \, \sqrt {2} {\left (\sqrt {-7} x - 3 \, x\right )} \sqrt {\sqrt {-7} - 3} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) - 3 \, \sqrt {2} {\left (1607 \, \sqrt {-7} x - 11967 \, x\right )} \sqrt {\sqrt {-7} - 3} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) + 4 \, {\left (175 \, x^{8} + 1065 \, x^{6} + 3187 \, x^{4} + 5247 \, x^{2} + 8394\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}}{1260 \, x} \]
1/1260*(8394*sqrt(2)*(sqrt(-7)*x - 3*x)*sqrt(sqrt(-7) - 3)*elliptic_e(arcs in(1/2*sqrt(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) - 3*sqrt(2)*(160 7*sqrt(-7)*x - 11967*x)*sqrt(sqrt(-7) - 3)*elliptic_f(arcsin(1/2*sqrt(2)*s qrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) + 4*(175*x^8 + 1065*x^6 + 3187*x ^4 + 5247*x^2 + 8394)*sqrt(x^4 + 3*x^2 + 4))/x
\[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int \left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \cdot \left (5 x^{2} + 7\right )\, dx \]
\[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )} \,d x } \]
\[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )} \,d x } \]
Timed out. \[ \int \left (7+5 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int \left (5\,x^2+7\right )\,{\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]