Integrand size = 24, antiderivative size = 181 \[ \int \frac {\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=-\frac {x \left (2323+949 x^2\right )}{28 \sqrt {4+3 x^2+x^4}}+\frac {4449 x \sqrt {4+3 x^2+x^4}}{28 \left (2+x^2\right )}-\frac {4449 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{14 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {973 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{4 \sqrt {2} \sqrt {4+3 x^2+x^4}} \]
-1/28*x*(949*x^2+2323)/(x^4+3*x^2+4)^(1/2)+4449/28*x*(x^4+3*x^2+4)^(1/2)/( x^2+2)-4449/28*(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)+973/8*(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(s in(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 = 10.29 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.81 \[ \int \frac {\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\frac {-4 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (2323+949 x^2\right )-4449 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+\sqrt {2} \left (3899 i+4449 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{112 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
(-4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(2323 + 949*x^2) - 4449*Sqrt[2]*(3*I + S qrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sq rt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(3899*I + 4449* Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + S qrt[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])])/(112*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])
Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1517, 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.
| \(\displaystyle \int \frac {\left (5 x^2+7\right )^3}{\left (x^4+3 x^2+4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {1}{28} \int \frac {4449 x^2+4724}{\sqrt {x^4+3 x^2+4}}dx-\frac {x \left (949 x^2+2323\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{28} \left (13622 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-8898 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx\right )-\frac {x \left (949 x^2+2323\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{28} \left (13622 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-4449 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )-\frac {x \left (949 x^2+2323\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{28} \left (\frac {6811 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+4}}-4449 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )-\frac {x \left (949 x^2+2323\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{28} \left (\frac {6811 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+4}}-4449 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )-\frac {x \left (949 x^2+2323\right )}{28 \sqrt {x^4+3 x^2+4}}\) |
-1/28*(x*(2323 + 949*x^2))/Sqrt[4 + 3*x^2 + x^4] + (-4449*(-((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]) + (6811 *(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2 ]], 1/8])/(Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]))/28
3.4.73.3.1 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]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(-\frac {x \left (949 x^{2}+2323\right )}{28 \sqrt {x^{4}+3 x^{2}+4}}+\frac {4724 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {35592 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\) | \(231\) |
| elliptic | \(-\frac {2 \left (\frac {949}{56} x^{3}+\frac {2323}{56} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}+\frac {4724 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {35592 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\) | \(232\) |
| default | \(-\frac {686 \left (\frac {1}{56} x +\frac {3}{56} x^{3}\right )}{\sqrt {x^{4}+3 x^{2}+4}}+\frac {4724 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {35592 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{7 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}-\frac {250 \left (-\frac {1}{14} x^{3}-\frac {6}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {1050 \left (\frac {3}{14} x^{3}+\frac {4}{7} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}-\frac {1470 \left (-\frac {1}{7} x^{3}-\frac {3}{14} x \right )}{\sqrt {x^{4}+3 x^{2}+4}}\) | \(301\) |
-1/28*x*(949*x^2+2323)/(x^4+3*x^2+4)^(1/2)+4724/7/(-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)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^ (1/2))-35592/7/(-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))*(Ellip ticF(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 = 177, normalized size of antiderivative = 0.98 \[ \int \frac {\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=-\frac {4449 \, \sqrt {2} {\left (3 \, x^{5} + 9 \, x^{3} - \sqrt {-7} {\left (x^{5} + 3 \, x^{3} + 4 \, x\right )} + 12 \, x\right )} \sqrt {\sqrt {-7} - 3} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) - 2 \, \sqrt {2} {\left (8445 \, x^{5} + 25335 \, x^{3} - 1634 \, \sqrt {-7} {\left (x^{5} + 3 \, x^{3} + 4 \, x\right )} + 33780 \, x\right )} \sqrt {\sqrt {-7} - 3} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) - 16 \, {\left (875 \, x^{4} + 2756 \, x^{2} + 4449\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}}{112 \, {\left (x^{5} + 3 \, x^{3} + 4 \, x\right )}} \]
-1/112*(4449*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)*(8445*x^5 + 25335*x^3 - 1634*sqrt(-7)*(x^5 + 3*x^3 + 4*x) + 33780*x)*sqrt(sqrt(-7) - 3)*elliptic_f(arcsin(1/2*sqrt(2) *sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) - 16*(875*x^4 + 2756*x^2 + 444 9)*sqrt(x^4 + 3*x^2 + 4))/(x^5 + 3*x^3 + 4*x)
\[ \int \frac {\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {\left (5 x^{2} + 7\right )^{3}}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {{\left (5\,x^2+7\right )}^3}{{\left (x^4+3\,x^2+4\right )}^{3/2}} \,d x \]