Integrand size = 24, antiderivative size = 226 \[ \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {175346 x \sqrt {4+3 x^2+x^4}}{1155 \left (2+x^2\right )}+\frac {x \left (64533+18253 x^2\right ) \sqrt {4+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (6831+2240 x^2\right ) \left (4+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (4+3 x^2+x^4\right )^{5/2}-\frac {175346 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{1155 \sqrt {4+3 x^2+x^4}}+\frac {4628 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{33 \sqrt {4+3 x^2+x^4}} \]
1/693*x*(2240*x^2+6831)*(x^4+3*x^2+4)^(3/2)+25/11*x*(x^4+3*x^2+4)^(5/2)+17 5346/1155*x*(x^4+3*x^2+4)^(1/2)/(x^2+2)+1/1155*x*(18253*x^2+64533)*(x^4+3* x^2+4)^(1/2)-175346/1155*(x^2+2)*(cos(2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/co s(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)+4628/33*( 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 = 7.14 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.57 \[ \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {2 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (1824876+2932753 x^2+2435811 x^4+1229714 x^6+408480 x^8+82075 x^{10}+7875 x^{12}\right )-263019 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+3 \sqrt {2} \left (-34209 i+87673 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{6930 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]
(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(1824876 + 2932753*x^2 + 2435811*x^4 + 12 29714*x^6 + 408480*x^8 + 82075*x^10 + 7875*x^12) - 263019*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])] + 3*Sqrt[2]*(-34209*I + 8 7673*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[Sqrt[(-2*I)/ (-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(6930*Sqrt[(-I)/(- 3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])
Time = 0.38 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1518, 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.
| \(\displaystyle \int \left (5 x^2+7\right )^2 \left (x^4+3 x^2+4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1518 |
| \(\displaystyle \frac {1}{11} \int \left (320 x^2+439\right ) \left (x^4+3 x^2+4\right )^{3/2}dx+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{21} \int \left (18253 x^2+27768\right ) \sqrt {x^4+3 x^2+4}dx+\frac {1}{63} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {1}{15} \int \frac {6 \left (87673 x^2+148614\right )}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (18253 x^2+64533\right )\right )+\frac {1}{63} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \int \frac {87673 x^2+148614}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (18253 x^2+64533\right )\right )+\frac {1}{63} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \left (323960 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-175346 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (18253 x^2+64533\right )\right )+\frac {1}{63} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \left (323960 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-87673 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (18253 x^2+64533\right )\right )+\frac {1}{63} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \left (\frac {80990 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-87673 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (18253 x^2+64533\right )\right )+\frac {1}{63} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \left (\frac {80990 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-87673 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+4} \left (18253 x^2+64533\right )\right )+\frac {1}{63} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+4\right )^{5/2}\) |
(25*x*(4 + 3*x^2 + x^4)^(5/2))/11 + ((x*(6831 + 2240*x^2)*(4 + 3*x^2 + x^4 )^(3/2))/63 + ((x*(64533 + 18253*x^2)*Sqrt[4 + 3*x^2 + x^4])/5 + (2*(-8767 3*(-((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]) + (80990*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)/11
3.4.58.3.1 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]
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\frac {x \left (7875 x^{8}+58450 x^{6}+201630 x^{4}+391024 x^{2}+456219\right ) \sqrt {x^{4}+3 x^{2}+4}}{3465}+\frac {396304 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{385 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {5611072 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{1155 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\) | \(246\) |
| default | \(\frac {1222 x^{5} \sqrt {x^{4}+3 x^{2}+4}}{21}+\frac {391024 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{3465}+\frac {50691 x \sqrt {x^{4}+3 x^{2}+4}}{385}+\frac {396304 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{385 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {5611072 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{1155 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {25 x^{9} \sqrt {x^{4}+3 x^{2}+4}}{11}+\frac {1670 x^{7} \sqrt {x^{4}+3 x^{2}+4}}{99}\) | \(292\) |
| elliptic | \(\frac {1222 x^{5} \sqrt {x^{4}+3 x^{2}+4}}{21}+\frac {391024 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{3465}+\frac {50691 x \sqrt {x^{4}+3 x^{2}+4}}{385}+\frac {396304 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{385 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {5611072 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{1155 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}+\frac {25 x^{9} \sqrt {x^{4}+3 x^{2}+4}}{11}+\frac {1670 x^{7} \sqrt {x^{4}+3 x^{2}+4}}{99}\) | \(292\) |
1/3465*x*(7875*x^8+58450*x^6+201630*x^4+391024*x^2+456219)*(x^4+3*x^2+4)^( 1/2)+396304/385/(-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))-5611072/1155/(-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 = 138, normalized size of antiderivative = 0.61 \[ \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {526038 \, \sqrt {2} {\left (\sqrt {-7} x - 3 \, x\right )} \sqrt {\sqrt {-7} - 3} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) - 3 \, \sqrt {2} {\left (101039 \, \sqrt {-7} x - 748959 \, x\right )} \sqrt {\sqrt {-7} - 3} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) + 4 \, {\left (7875 \, x^{10} + 58450 \, x^{8} + 201630 \, x^{6} + 391024 \, x^{4} + 456219 \, x^{2} + 526038\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}}{13860 \, x} \]
1/13860*(526038*sqrt(2)*(sqrt(-7)*x - 3*x)*sqrt(sqrt(-7) - 3)*elliptic_e(a rcsin(1/2*sqrt(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) - 3*sqrt(2)*( 101039*sqrt(-7)*x - 748959*x)*sqrt(sqrt(-7) - 3)*elliptic_f(arcsin(1/2*sqr t(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) + 4*(7875*x^10 + 58450*x^8 + 201630*x^6 + 391024*x^4 + 456219*x^2 + 526038)*sqrt(x^4 + 3*x^2 + 4))/x
\[ \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int \left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \]
\[ \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]
\[ \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]
Timed out. \[ \int \left (7+5 x^2\right )^2 \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int {\left (5\,x^2+7\right )}^2\,{\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]