Integrand size = 41, antiderivative size = 414 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=\frac {\sqrt {2} \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}}{3 \left (1+\sqrt {2} (1+2 x)\right )}+\frac {76 \text {arctanh}\left (\frac {\sqrt {\frac {193}{39}} \sqrt {1+2 x}}{\sqrt {2+5 x+4 x^2}}\right )}{3 \sqrt {7527}}-\frac {\sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) E\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right )|\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{3 \sqrt [4]{2} \sqrt {2+5 x+4 x^2}}+\frac {\left (25+111 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{1974 \sqrt [4]{2} \sqrt {2+5 x+4 x^2}}-\frac {38 \sqrt [4]{2} \left (347-78 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticPi}\left (\frac {1}{312} \left (156+347 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{38493 \sqrt {2+5 x+4 x^2}} \] Output:
2^(1/2)*(1+2*x)^(1/2)*(4*x^2+5*x+2)^(1/2)/(3+3*2^(1/2)*(1+2*x))+76/22581*7 527^(1/2)*arctanh(1/39*7527^(1/2)*(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2))-1/6*( (4*x^2+5*x+2)/(1+2^(1/2)*(1+2*x))^2)^(1/2)*(1+2^(1/2)*(1+2*x))*EllipticE(s in(2*arctan(2^(1/4)*(1+2*x)^(1/2))),1/4*(8-2*2^(1/2))^(1/2))*2^(3/4)/(4*x^ 2+5*x+2)^(1/2)+1/3948*(25+111*2^(1/2))*((4*x^2+5*x+2)/(1+2^(1/2)*(1+2*x))^ 2)^(1/2)*(1+2^(1/2)*(1+2*x))*InverseJacobiAM(2*arctan(2^(1/4)*(1+2*x)^(1/2 )),1/4*(8-2*2^(1/2))^(1/2))*2^(3/4)/(4*x^2+5*x+2)^(1/2)-38/38493*2^(1/4)*( 347-78*2^(1/2))*((4*x^2+5*x+2)/(1+2^(1/2)*(1+2*x))^2)^(1/2)*(1+2^(1/2)*(1+ 2*x))*EllipticPi(sin(2*arctan(2^(1/4)*(1+2*x)^(1/2))),1/2+347/312*2^(1/2), 1/4*(8-2*2^(1/2))^(1/2))/(4*x^2+5*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 27.09 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.43 \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=\frac {(3+6 x)^{3/2} \left (\frac {312 \sqrt {-\frac {i}{-i+\sqrt {7}}} \left (2+5 x+4 x^2\right )}{(1+2 x)^2}+\frac {39 \left (i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right )|\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {\frac {1}{2}+x}}-\frac {3 \left (i+13 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {\frac {1}{2}+x}}-\frac {608 i \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (1+i \sqrt {7}\right ),i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {\frac {1}{2}+x}}\right )}{1404 \sqrt {-\frac {6 i}{-i+\sqrt {7}}} \sqrt {4+10 x+8 x^2}} \] Input:
Integrate[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*Sqrt[2 + 5*x + 4*x^2] ),x]
Output:
((3 + 6*x)^(3/2)*((312*Sqrt[(-I)/(-I + Sqrt[7])]*(2 + 5*x + 4*x^2))/(1 + 2 *x)^2 + (39*(I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] + 2*(-I + Sqrt[7])*x)/((-I + Sqrt[7])*(1 + 2*x))]*Sqrt[(3*I + Sqrt[7] + 2*(I + Sqrt[7])*x)/((I + Sqrt [7])*(1 + 2*x))]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqrt[7])])/Sqrt[1/2 + x] - (3*(I + 13*Sqrt[7])*S qrt[(-3*I + Sqrt[7] + 2*(-I + Sqrt[7])*x)/((-I + Sqrt[7])*(1 + 2*x))]*Sqrt [(3*I + Sqrt[7] + 2*(I + Sqrt[7])*x)/((I + Sqrt[7])*(1 + 2*x))]*EllipticF[ I*ArcSinh[Sqrt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + S qrt[7])])/Sqrt[1/2 + x] - ((608*I)*Sqrt[(-3*I + Sqrt[7] + 2*(-I + Sqrt[7]) *x)/((-I + Sqrt[7])*(1 + 2*x))]*Sqrt[(3*I + Sqrt[7] + 2*(I + Sqrt[7])*x)/( (I + Sqrt[7])*(1 + 2*x))]*EllipticPi[(-13*(1 + I*Sqrt[7]))/6, I*ArcSinh[Sq rt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqrt[7])])/Sq rt[1/2 + x]))/(1404*Sqrt[(-6*I)/(-I + Sqrt[7])]*Sqrt[4 + 10*x + 8*x^2])
Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.268, Rules used = {2154, 1269, 1172, 321, 327, 1279, 187, 25, 413, 413, 412}
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.
| \(\displaystyle \int \frac {-2 x^2+6 x+4}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}} \, dx\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\int \frac {\frac {2 x}{3}-\frac {8}{9}}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {11}{9} \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {1}{3} \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+5 x+2}}dx\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \int \frac {1}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1} \sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
| \(\Big \downarrow \) 1279 |
| \(\displaystyle \frac {76 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5}}dx}{9 \sqrt {4 x^2+5 x+2}}-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle -\frac {152 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \int -\frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2}}-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {152 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2}}-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {152 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \sqrt {1+\frac {4 (2 x+1)}{1-i \sqrt {7}}} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)+i \sqrt {7}+1} \sqrt {\frac {4 (2 x+1)}{1-i \sqrt {7}}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2} \sqrt {4 (2 x+1)-i \sqrt {7}+1}}-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {152 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \sqrt {1+\frac {4 (2 x+1)}{1-i \sqrt {7}}} \sqrt {1+\frac {4 (2 x+1)}{1+i \sqrt {7}}} \int \frac {1}{(13-3 (2 x+1)) \sqrt {\frac {4 (2 x+1)}{1-i \sqrt {7}}+1} \sqrt {\frac {4 (2 x+1)}{1+i \sqrt {7}}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2} \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {76 \sqrt {-1+i \sqrt {7}} \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \sqrt {1+\frac {4 (2 x+1)}{1-i \sqrt {7}}} \sqrt {1+\frac {4 (2 x+1)}{1+i \sqrt {7}}} \operatorname {EllipticPi}\left (-\frac {3}{52} \left (1-i \sqrt {7}\right ),\arcsin \left (\frac {2 \sqrt {2 x+1}}{\sqrt {-1+i \sqrt {7}}}\right ),\frac {i+\sqrt {7}}{i-\sqrt {7}}\right )}{117 \sqrt {4 x^2+5 x+2} \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}-\frac {22 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{9 \sqrt {2 x+1}}+\frac {i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{6 \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\) |
Input:
Int[(4 + 6*x - 2*x^2)/((5 - 3*x)*Sqrt[1 + 2*x]*Sqrt[2 + 5*x + 4*x^2]),x]
Output:
((I/6)*Sqrt[1 + 2*x]*EllipticE[ArcSin[Sqrt[(-I)*(5 + I*Sqrt[7] + 8*x)]/(Sq rt[2]*7^(1/4))], (-2*Sqrt[7])/(I - Sqrt[7])])/Sqrt[-((1 + 2*x)/(1 + I*Sqrt [7]))] - (((22*I)/9)*Sqrt[-((1 + 2*x)/(1 + I*Sqrt[7]))]*EllipticF[ArcSin[S qrt[(-I)*(5 + I*Sqrt[7] + 8*x)]/(Sqrt[2]*7^(1/4))], (-2*Sqrt[7])/(I - Sqrt [7])])/Sqrt[1 + 2*x] + (76*Sqrt[-1 + I*Sqrt[7]]*Sqrt[5 - I*Sqrt[7] + 8*x]* Sqrt[5 + I*Sqrt[7] + 8*x]*Sqrt[1 + (4*(1 + 2*x))/(1 - I*Sqrt[7])]*Sqrt[1 + (4*(1 + 2*x))/(1 + I*Sqrt[7])]*EllipticPi[(-3*(1 - I*Sqrt[7]))/52, ArcSin [(2*Sqrt[1 + 2*x])/Sqrt[-1 + I*Sqrt[7]]], (I + Sqrt[7])/(I - Sqrt[7])])/(1 17*Sqrt[2 + 5*x + 4*x^2]*Sqrt[1 - I*Sqrt[7] + 4*(1 + 2*x)]*Sqrt[1 + I*Sqrt [7] + 4*(1 + 2*x)])
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && !SimplerQ[e + f*x, c + d*x] && !SimplerQ[g + h*x, c + d*x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]) Int[1/((d + e*x )*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ {a, b, c, d, e, f, g}, x]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(-\frac {\sqrt {1+2 x}\, \sqrt {4 x^{2}+5 x +2}\, \left (1+i \sqrt {7}\right ) \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \left (39 i \operatorname {EllipticF}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right ) \sqrt {7}-39 i \operatorname {EllipticE}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right ) \sqrt {7}-611 \operatorname {EllipticF}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )+39 \operatorname {EllipticE}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )+608 \operatorname {EllipticPi}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, -\frac {3}{52}-\frac {3 i \sqrt {7}}{52}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )\right )}{936 \left (8 x^{3}+14 x^{2}+9 x +2\right )}\) | \(361\) |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+5 x +2\right ) \left (1+2 x \right )}\, \left (-\frac {16 \left (-\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}-\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}+\frac {i \sqrt {7}}{8}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )}{9 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}+\frac {4 \left (-\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}-\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}+\frac {i \sqrt {7}}{8}}}\, \left (\left (\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )+\left (-\frac {5}{8}+\frac {i \sqrt {7}}{8}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )\right )}{3 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}+\frac {304 \left (-\frac {1}{8}-\frac {i \sqrt {7}}{8}\right ) \sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}-\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\, \sqrt {\frac {x +\frac {5}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}+\frac {i \sqrt {7}}{8}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}}{-\frac {1}{8}-\frac {i \sqrt {7}}{8}}}, -\frac {3}{52}-\frac {3 i \sqrt {7}}{52}, \sqrt {\frac {\frac {1}{8}+\frac {i \sqrt {7}}{8}}{\frac {1}{8}-\frac {i \sqrt {7}}{8}}}\right )}{117 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}\right )}{\sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}}\) | \(480\) |
Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2),x,method=_RET URNVERBOSE)
Output:
-1/936*(1+2*x)^(1/2)*(4*x^2+5*x+2)^(1/2)*(1+I*7^(1/2))*(-(1+2*x)/(1+I*7^(1 /2)))^(1/2)*((I*7^(1/2)-8*x-5)/(I*7^(1/2)-1))^(1/2)*((I*7^(1/2)+8*x+5)/(1+ I*7^(1/2)))^(1/2)*(39*I*EllipticF(2*(-(1+2*x)/(1+I*7^(1/2)))^(1/2),(-(1+I* 7^(1/2))/(I*7^(1/2)-1))^(1/2))*7^(1/2)-39*I*EllipticE(2*(-(1+2*x)/(1+I*7^( 1/2)))^(1/2),(-(1+I*7^(1/2))/(I*7^(1/2)-1))^(1/2))*7^(1/2)-611*EllipticF(2 *(-(1+2*x)/(1+I*7^(1/2)))^(1/2),(-(1+I*7^(1/2))/(I*7^(1/2)-1))^(1/2))+39*E llipticE(2*(-(1+2*x)/(1+I*7^(1/2)))^(1/2),(-(1+I*7^(1/2))/(I*7^(1/2)-1))^( 1/2))+608*EllipticPi(2*(-(1+2*x)/(1+I*7^(1/2)))^(1/2),-3/52-3/52*I*7^(1/2) ,(-(1+I*7^(1/2))/(I*7^(1/2)-1))^(1/2)))/(8*x^3+14*x^2+9*x+2)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {4 \, x^{2} + 5 \, x + 2} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2),x, algo rithm="fricas")
Output:
integral(2*sqrt(4*x^2 + 5*x + 2)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(24*x^4 + 2 *x^3 - 43*x^2 - 39*x - 10), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=2 \left (\int \left (- \frac {3 x}{3 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 5 \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2}}\right )\, dx + \int \frac {x^{2}}{3 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 5 \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2}}\, dx + \int \left (- \frac {2}{3 x \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2} - 5 \sqrt {2 x + 1} \sqrt {4 x^{2} + 5 x + 2}}\right )\, dx\right ) \] Input:
integrate((-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2)/(4*x**2+5*x+2)**(1/2),x)
Output:
2*(Integral(-3*x/(3*x*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 5*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2)), x) + Integral(x**2/(3*x*sqrt(2*x + 1)*sqrt(4*x **2 + 5*x + 2) - 5*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2)), x) + Integral(-2 /(3*x*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2) - 5*sqrt(2*x + 1)*sqrt(4*x**2 + 5*x + 2)), x))
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {4 \, x^{2} + 5 \, x + 2} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2),x, algo rithm="maxima")
Output:
2*integrate((x^2 - 3*x - 2)/(sqrt(4*x^2 + 5*x + 2)*(3*x - 5)*sqrt(2*x + 1) ), x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )}}{\sqrt {4 \, x^{2} + 5 \, x + 2} {\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2),x, algo rithm="giac")
Output:
integrate(2*(x^2 - 3*x - 2)/(sqrt(4*x^2 + 5*x + 2)*(3*x - 5)*sqrt(2*x + 1) ), x)
Timed out. \[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=\int -\frac {-2\,x^2+6\,x+4}{\sqrt {2\,x+1}\,\left (3\,x-5\right )\,\sqrt {4\,x^2+5\,x+2}} \,d x \] Input:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(5*x + 4*x^2 + 2)^(1/2)) ,x)
Output:
int(-(6*x - 2*x^2 + 4)/((2*x + 1)^(1/2)*(3*x - 5)*(5*x + 4*x^2 + 2)^(1/2)) , x)
\[ \int \frac {4+6 x-2 x^2}{(5-3 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}} \, dx=\int \frac {-2 x^{2}+6 x +4}{\left (5-3 x \right ) \sqrt {2 x +1}\, \sqrt {4 x^{2}+5 x +2}}d x \] Input:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2),x)
Output:
int((-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2),x)