Integrand size = 24, antiderivative size = 198 \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {742 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {x \left (36783+10643 x^2\right ) \sqrt {2+3 x^2+x^4}}{1155}+\frac {1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {25}{11} x \left (2+3 x^2+x^4\right )^{5/2}-\frac {742 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {13879 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{385 \sqrt {2+3 x^2+x^4}} \]
1/693*x*(2240*x^2+7281)*(x^4+3*x^2+2)^(3/2)+25/11*x*(x^4+3*x^2+2)^(5/2)+74 2/15*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-742/15*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)* EllipticE(x/(x^2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^ 4+3*x^2+2)^(1/2)+13879/385*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticF(x/(x^ 2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^(1/2 )+1/1155*x*(10643*x^2+36783)*(x^4+3*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 8.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.63 \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {429318 x+1160065 x^3+1333551 x^5+892084 x^7+363480 x^9+82075 x^{11}+7875 x^{13}-171402 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-78420 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{3465 \sqrt {2+3 x^2+x^4}} \]
(429318*x + 1160065*x^3 + 1333551*x^5 + 892084*x^7 + 363480*x^9 + 82075*x^ 11 + 7875*x^13 - (171402*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSin h[x/Sqrt[2]], 2] - (78420*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSi nh[x/Sqrt[2]], 2])/(3465*Sqrt[2 + 3*x^2 + x^4])
Time = 0.37 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1518, 1490, 1490, 27, 1503, 1412, 1455}
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.
\(\displaystyle \int \left (5 x^2+7\right )^2 \left (x^4+3 x^2+2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1518 |
\(\displaystyle \frac {1}{11} \int \left (320 x^2+489\right ) \left (x^4+3 x^2+2\right )^{3/2}dx+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 1490 |
\(\displaystyle \frac {1}{11} \left (\frac {1}{21} \int \left (10643 x^2+15684\right ) \sqrt {x^4+3 x^2+2}dx+\frac {1}{63} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 1490 |
\(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {1}{15} \int \frac {6 \left (28567 x^2+41637\right )}{\sqrt {x^4+3 x^2+2}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (10643 x^2+36783\right )\right )+\frac {1}{63} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \int \frac {28567 x^2+41637}{\sqrt {x^4+3 x^2+2}}dx+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (10643 x^2+36783\right )\right )+\frac {1}{63} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 1503 |
\(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \left (41637 \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx+28567 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (10643 x^2+36783\right )\right )+\frac {1}{63} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 1412 |
\(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \left (28567 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx+\frac {41637 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (10643 x^2+36783\right )\right )+\frac {1}{63} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 1455 |
\(\displaystyle \frac {1}{11} \left (\frac {1}{21} \left (\frac {2}{5} \left (\frac {41637 \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4+3 x^2+2}}+28567 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )+\frac {1}{5} x \sqrt {x^4+3 x^2+2} \left (10643 x^2+36783\right )\right )+\frac {1}{63} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}\right )+\frac {25}{11} x \left (x^4+3 x^2+2\right )^{5/2}\) |
(25*x*(2 + 3*x^2 + x^4)^(5/2))/11 + ((x*(7281 + 2240*x^2)*(2 + 3*x^2 + x^4 )^(3/2))/63 + ((x*(36783 + 10643*x^2)*Sqrt[2 + 3*x^2 + x^4])/5 + (2*(28567 *((x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] - (Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/ (1 + x^2)]*EllipticE[ArcTan[x], 1/2])/Sqrt[2 + 3*x^2 + x^4]) + (41637*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(Sqrt[2]*Sqrt[2 + 3*x^2 + x^4])))/5)/21)/11
3.3.94.3.1 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[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (b - q )*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] && !(PosQ[ (b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
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[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
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 = 1.80 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {x \left (7875 x^{8}+58450 x^{6}+172380 x^{4}+258044 x^{2}+214659\right ) \sqrt {x^{4}+3 x^{2}+2}}{3465}-\frac {13879 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{385 \sqrt {x^{4}+3 x^{2}+2}}+\frac {371 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}\) | \(143\) |
default | \(\frac {11492 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{231}+\frac {258044 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{3465}+\frac {23851 x \sqrt {x^{4}+3 x^{2}+2}}{385}-\frac {13879 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{385 \sqrt {x^{4}+3 x^{2}+2}}+\frac {371 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x^{9} \sqrt {x^{4}+3 x^{2}+2}}{11}+\frac {1670 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{99}\) | \(189\) |
elliptic | \(\frac {11492 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{231}+\frac {258044 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{3465}+\frac {23851 x \sqrt {x^{4}+3 x^{2}+2}}{385}-\frac {13879 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{385 \sqrt {x^{4}+3 x^{2}+2}}+\frac {371 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x^{9} \sqrt {x^{4}+3 x^{2}+2}}{11}+\frac {1670 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{99}\) | \(189\) |
1/3465*x*(7875*x^8+58450*x^6+172380*x^4+258044*x^2+214659)*(x^4+3*x^2+2)^( 1/2)-13879/385*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2) *EllipticF(1/2*I*2^(1/2)*x,2^(1/2))+371/15*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+ 1)^(1/2)/(x^4+3*x^2+2)^(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE (1/2*I*2^(1/2)*x,2^(1/2)))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.34 \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {-171402 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 421224 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (7875 \, x^{10} + 58450 \, x^{8} + 172380 \, x^{6} + 258044 \, x^{4} + 214659 \, x^{2} + 171402\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{3465 \, x} \]
1/3465*(-171402*I*x*elliptic_e(arcsin(I/x), 2) + 421224*I*x*elliptic_f(arc sin(I/x), 2) + (7875*x^10 + 58450*x^8 + 172380*x^6 + 258044*x^4 + 214659*x ^2 + 171402)*sqrt(x^4 + 3*x^2 + 2))/x
\[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \]
\[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]
\[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]
Timed out. \[ \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\int {\left (5\,x^2+7\right )}^2\,{\left (x^4+3\,x^2+2\right )}^{3/2} \,d x \]