Integrand size = 24, antiderivative size = 247 \[ \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {4525662 x \sqrt {4+3 x^2+x^4}}{5005 \left (2+x^2\right )}+\frac {x \left (1653701+435441 x^2\right ) \sqrt {4+3 x^2+x^4}}{5005}+\frac {x \left (53504+15365 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}}{1001}+\frac {3825}{143} x \left (4+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (4+3 x^2+x^4\right )^{5/2}-\frac {4525662 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{5005 \sqrt {4+3 x^2+x^4}}+\frac {121826 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{143 \sqrt {4+3 x^2+x^4}} \]
1/1001*x*(15365*x^2+53504)*(x^4+3*x^2+4)^(3/2)+3825/143*x*(x^4+3*x^2+4)^(5 /2)+125/13*x^3*(x^4+3*x^2+4)^(5/2)+4525662/5005*x*(x^4+3*x^2+4)^(1/2)/(x^2 +2)+1/5005*x*(435441*x^2+1653701)*(x^4+3*x^2+4)^(1/2)-4525662/5005*(x^2+2) *(cos(2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*Ellip ticE(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)+121826/143*(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 = 7.79 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.45 \[ \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {2 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (19463124+36710547 x^2+37166164 x^4+24107711 x^6+10713970 x^8+3158575 x^{10}+567000 x^{12}+48125 x^{14}\right )-2262831 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+\sqrt {2} \left (-1215823 i+2262831 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{10010 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(19463124 + 36710547*x^2 + 37166164*x^4 + 24107711*x^6 + 10713970*x^8 + 3158575*x^10 + 567000*x^12 + 48125*x^14) - 2262831*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*Ar cSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-1215823*I + 2262831*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])]*Ellipt icF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqr t[7])])/(10010*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])
Time = 0.45 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1518, 2207, 1490, 27, 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.
| \(\displaystyle \int \left (5 x^2+7\right )^3 \left (x^4+3 x^2+4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1518 |
| \(\displaystyle \frac {1}{13} \int \left (x^4+3 x^2+4\right )^{3/2} \left (3825 x^4+8055 x^2+4459\right )dx+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \left (19755 x^2+33749\right ) \left (x^4+3 x^2+4\right )^{3/2}dx+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{21} \int 9 \left (145147 x^2+243652\right ) \sqrt {x^4+3 x^2+4}dx+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {3}{7} \int \left (145147 x^2+243652\right ) \sqrt {x^4+3 x^2+4}dx+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {3}{7} \left (\frac {1}{15} \int \frac {2 \left (2262831 x^2+4002158\right )}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (435441 x^2+1653701\right )\right )+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {3}{7} \left (\frac {2}{15} \int \frac {2262831 x^2+4002158}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (435441 x^2+1653701\right )\right )+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {3}{7} \left (\frac {2}{15} \left (8527820 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-4525662 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (435441 x^2+1653701\right )\right )+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {3}{7} \left (\frac {2}{15} \left (8527820 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-2262831 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (435441 x^2+1653701\right )\right )+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {3}{7} \left (\frac {2}{15} \left (\frac {2131955 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-2262831 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (435441 x^2+1653701\right )\right )+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {3}{7} \left (\frac {2}{15} \left (\frac {2131955 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-2262831 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (435441 x^2+1653701\right )\right )+\frac {1}{7} x \left (15365 x^2+53504\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {3825}{11} x \left (x^4+3 x^2+4\right )^{5/2}\right )+\frac {125}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3\) |
(125*x^3*(4 + 3*x^2 + x^4)^(5/2))/13 + ((3825*x*(4 + 3*x^2 + x^4)^(5/2))/1 1 + ((x*(53504 + 15365*x^2)*(4 + 3*x^2 + x^4)^(3/2))/7 + (3*((x*(1653701 + 435441*x^2)*Sqrt[4 + 3*x^2 + x^4])/15 + (2*(-2262831*(-((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]) + (2131955* 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]))/15))/7)/11)/13
3.4.57.3.1 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]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Simp[e^q*x^(2*q - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c*(4*p + 2*q + 1)) Int[(a + b*x^2 + c*x^4)^p*Expand ToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2* p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1]
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]
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.02
| method | result | size |
| risch | \(\frac {x \left (48125 x^{10}+422625 x^{8}+1698200 x^{6}+3928870 x^{4}+5528301 x^{2}+4865781\right ) \sqrt {x^{4}+3 x^{2}+4}}{5005}+\frac {32017264 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {144821184 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\) | \(251\) |
| default | \(\frac {71434 x^{5} \sqrt {x^{4}+3 x^{2}+4}}{91}+\frac {5528301 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{5005}+\frac {4865781 x \sqrt {x^{4}+3 x^{2}+4}}{5005}+\frac {32017264 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {144821184 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {125 x^{11} \sqrt {x^{4}+3 x^{2}+4}}{13}+\frac {12075 x^{9} \sqrt {x^{4}+3 x^{2}+4}}{143}+\frac {48520 x^{7} \sqrt {x^{4}+3 x^{2}+4}}{143}\) | \(309\) |
| elliptic | \(\frac {71434 x^{5} \sqrt {x^{4}+3 x^{2}+4}}{91}+\frac {5528301 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{5005}+\frac {4865781 x \sqrt {x^{4}+3 x^{2}+4}}{5005}+\frac {32017264 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {144821184 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{5005 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {125 x^{11} \sqrt {x^{4}+3 x^{2}+4}}{13}+\frac {12075 x^{9} \sqrt {x^{4}+3 x^{2}+4}}{143}+\frac {48520 x^{7} \sqrt {x^{4}+3 x^{2}+4}}{143}\) | \(309\) |
1/5005*x*(48125*x^10+422625*x^8+1698200*x^6+3928870*x^4+5528301*x^2+486578 1)*(x^4+3*x^2+4)^(1/2)+32017264/5005/(-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))-144821 184/5005/(-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 = 143, normalized size of antiderivative = 0.58 \[ \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {4525662 \, \sqrt {2} {\left (\sqrt {-7} x - 3 \, x\right )} \sqrt {\sqrt {-7} - 3} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) - \sqrt {2} {\left (2524583 \, \sqrt {-7} x - 19580223 \, x\right )} \sqrt {\sqrt {-7} - 3} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) + 4 \, {\left (48125 \, x^{12} + 422625 \, x^{10} + 1698200 \, x^{8} + 3928870 \, x^{6} + 5528301 \, x^{4} + 4865781 \, x^{2} + 4525662\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}}{20020 \, x} \]
1/20020*(4525662*sqrt(2)*(sqrt(-7)*x - 3*x)*sqrt(sqrt(-7) - 3)*elliptic_e( arcsin(1/2*sqrt(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) - sqrt(2)*(2 524583*sqrt(-7)*x - 19580223*x)*sqrt(sqrt(-7) - 3)*elliptic_f(arcsin(1/2*s qrt(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) + 4*(48125*x^12 + 422625 *x^10 + 1698200*x^8 + 3928870*x^6 + 5528301*x^4 + 4865781*x^2 + 4525662)*s qrt(x^4 + 3*x^2 + 4))/x
\[ \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int \left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{3}\, dx \]
\[ \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3} \,d x } \]
\[ \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3} \,d x } \]
Timed out. \[ \int \left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int {\left (5\,x^2+7\right )}^3\,{\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]