Integrand size = 24, antiderivative size = 189 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\frac {7679 x \left (2+x^2\right )}{2 \sqrt {2+3 x^2+x^4}}-\frac {x \left (115+179 x^2\right )}{2 \sqrt {2+3 x^2+x^4}}+\frac {5000}{3} x \sqrt {2+3 x^2+x^4}+625 x^3 \sqrt {2+3 x^2+x^4}-\frac {7679 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {2+3 x^2+x^4}}+\frac {15383 \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{3 \sqrt {2} \sqrt {2+3 x^2+x^4}} \]
7679/2*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-1/2*x*(179*x^2+115)/(x^4+3*x^2+2)^(1/ 2)-7679/2*(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)+15383/6*(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)+5000/3*x*(x^4+3*x^2+2)^(1/2)+625 *x^3*(x^4+3*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.58 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\frac {19655 x+36963 x^3+21250 x^5+3750 x^7-23037 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-7729 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{6 \sqrt {2+3 x^2+x^4}} \]
(19655*x + 36963*x^3 + 21250*x^5 + 3750*x^7 - (23037*I)*Sqrt[1 + x^2]*Sqrt [2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (7729*I)*Sqrt[1 + x^2]*Sqrt [2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(6*Sqrt[2 + 3*x^2 + x^4])
Time = 0.42 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1517, 25, 2207, 27, 2207, 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.
| \(\displaystyle \int \frac {\left (5 x^2+7\right )^5}{\left (x^4+3 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle -\frac {1}{2} \int -\frac {6250 x^6+25000 x^4+35179 x^2+16922}{\sqrt {x^4+3 x^2+2}}dx-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \frac {6250 x^6+25000 x^4+35179 x^2+16922}{\sqrt {x^4+3 x^2+2}}dx-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int \frac {5 \left (10000 x^4+27679 x^2+16922\right )}{\sqrt {x^4+3 x^2+2}}dx+1250 \sqrt {x^4+3 x^2+2} x^3\right )-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {10000 x^4+27679 x^2+16922}{\sqrt {x^4+3 x^2+2}}dx+1250 \sqrt {x^4+3 x^2+2} x^3\right )-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {23037 x^2+30766}{\sqrt {x^4+3 x^2+2}}dx+\frac {10000}{3} \sqrt {x^4+3 x^2+2} x+1250 \sqrt {x^4+3 x^2+2} x^3\right )-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (30766 \int \frac {1}{\sqrt {x^4+3 x^2+2}}dx+23037 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx\right )+\frac {10000}{3} \sqrt {x^4+3 x^2+2} x+1250 \sqrt {x^4+3 x^2+2} x^3\right )-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
| \(\Big \downarrow \) 1412 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (23037 \int \frac {x^2}{\sqrt {x^4+3 x^2+2}}dx+\frac {15383 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}}\right )+\frac {10000}{3} \sqrt {x^4+3 x^2+2} x+1250 \sqrt {x^4+3 x^2+2} x^3\right )-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
| \(\Big \downarrow \) 1455 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {15383 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{\sqrt {x^4+3 x^2+2}}+23037 \left (\frac {x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}-\frac {\sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}\right )\right )+\frac {10000}{3} \sqrt {x^4+3 x^2+2} x+1250 \sqrt {x^4+3 x^2+2} x^3\right )-\frac {x \left (179 x^2+115\right )}{2 \sqrt {x^4+3 x^2+2}}\) |
-1/2*(x*(115 + 179*x^2))/Sqrt[2 + 3*x^2 + x^4] + ((10000*x*Sqrt[2 + 3*x^2 + x^4])/3 + 1250*x^3*Sqrt[2 + 3*x^2 + x^4] + (23037*((x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] - (Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[Ar cTan[x], 1/2])/Sqrt[2 + 3*x^2 + x^4]) + (15383*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/Sqrt[2 + 3*x^2 + x^4])/3)/2
3.4.7.3.1 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)/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] :> 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]
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 = 8.96 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {x \left (3750 x^{6}+21250 x^{4}+36963 x^{2}+19655\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}-\frac {15383 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {7679 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{4 \sqrt {x^{4}+3 x^{2}+2}}\) | \(138\) |
| elliptic | \(-\frac {2 \left (\frac {179}{4} x^{3}+\frac {115}{4} x \right )}{\sqrt {x^{4}+3 x^{2}+2}}+625 x^{3} \sqrt {x^{4}+3 x^{2}+2}+\frac {5000 x \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {15383 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {7679 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{4 \sqrt {x^{4}+3 x^{2}+2}}\) | \(161\) |
| default | \(-\frac {33614 \left (-\frac {3}{4} x^{3}-\frac {5}{4} x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {15383 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {7679 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{4 \sqrt {x^{4}+3 x^{2}+2}}-\frac {6250 \left (\frac {17}{2} x^{3}+9 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}+625 x^{3} \sqrt {x^{4}+3 x^{2}+2}+\frac {5000 x \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {43750 \left (-\frac {9}{2} x^{3}-5 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {122500 \left (\frac {5}{2} x^{3}+3 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {171500 \left (-\frac {3}{2} x^{3}-2 x \right )}{\sqrt {x^{4}+3 x^{2}+2}}-\frac {120050 \left (x^{3}+\frac {3}{2} x \right )}{\sqrt {x^{4}+3 x^{2}+2}}\) | \(274\) |
1/6*x*(3750*x^6+21250*x^4+36963*x^2+19655)/(x^4+3*x^2+2)^(1/2)-15383/6*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))+7679/4*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 = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.53 \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=-\frac {23037 \, {\left (i \, x^{5} + 3 i \, x^{3} + 2 i \, x\right )} E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 53803 \, {\left (-i \, x^{5} - 3 i \, x^{3} - 2 i \, x\right )} F(\arcsin \left (\frac {i}{x}\right )\,|\,2) - 2 \, {\left (1875 \, x^{8} + 10625 \, x^{6} + 30000 \, x^{4} + 44383 \, x^{2} + 23037\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{6 \, {\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} \]
-1/6*(23037*(I*x^5 + 3*I*x^3 + 2*I*x)*elliptic_e(arcsin(I/x), 2) + 53803*( -I*x^5 - 3*I*x^3 - 2*I*x)*elliptic_f(arcsin(I/x), 2) - 2*(1875*x^8 + 10625 *x^6 + 30000*x^4 + 44383*x^2 + 23037)*sqrt(x^4 + 3*x^2 + 2))/(x^5 + 3*x^3 + 2*x)
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {\left (5 x^{2} + 7\right )^{5}}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (5 \, x^{2} + 7\right )}^{5}}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (7+5 x^2\right )^5}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx=\int \frac {{\left (5\,x^2+7\right )}^5}{{\left (x^4+3\,x^2+2\right )}^{3/2}} \,d x \]