Integrand size = 41, antiderivative size = 512 \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {\sqrt {1+2 x} (39998722199+8990086284 x) \sqrt {2+5 x+4 x^2}}{175134960}-\frac {\sqrt {1+2 x} (4620643+1253336 x) \left (2+5 x+4 x^2\right )^{3/2}}{972972}-\frac {(205-132 x) \sqrt {1+2 x} \left (2+5 x+4 x^2\right )^{5/2}}{2574}-\frac {31634927315 \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}}{5837832 \sqrt {2} \left (1+\sqrt {2} (1+2 x)\right )}+\frac {2830924 \sqrt {\frac {193}{39}} \text {arctanh}\left (\frac {\sqrt {\frac {193}{39}} \sqrt {1+2 x}}{\sqrt {2+5 x+4 x^2}}\right )}{2187}+\frac {31634927315 \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) E\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right )|\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{11675664 \sqrt [4]{2} \sqrt {2+5 x+4 x^2}}+\frac {\left (7455591967279-47304438629395 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{16462686240\ 2^{3/4} \sqrt {2+5 x+4 x^2}}-\frac {273184166 \sqrt [4]{2} \left (347-78 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticPi}\left (\frac {1}{312} \left (156+347 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{28061397 \sqrt {2+5 x+4 x^2}} \] Output:
-1/175134960*(1+2*x)^(1/2)*(39998722199+8990086284*x)*(4*x^2+5*x+2)^(1/2)- 1/972972*(1+2*x)^(1/2)*(4620643+1253336*x)*(4*x^2+5*x+2)^(3/2)-1/2574*(205 -132*x)*(1+2*x)^(1/2)*(4*x^2+5*x+2)^(5/2)-31634927315/11675664*(1+2*x)^(1/ 2)*(4*x^2+5*x+2)^(1/2)*2^(1/2)/(1+2^(1/2)*(1+2*x))+2830924/85293*7527^(1/2 )*arctanh(1/39*7527^(1/2)*(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2))+31634927315/2 3351328*((4*x^2+5*x+2)/(1+2^(1/2)*(1+2*x))^2)^(1/2)*(1+2^(1/2)*(1+2*x))*El lipticE(sin(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)+1/32925372480*(7455591967279-47304438629395*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))*Inverse JacobiAM(2*arctan(2^(1/4)*(1+2*x)^(1/2)),1/4*(8-2*2^(1/2))^(1/2))*2^(1/4)/ (4*x^2+5*x+2)^(1/2)-273184166/28061397*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*arc tan(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 = 28.47 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.25 \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {\sqrt {1+2 x} \left (24 \left (2+5 x+4 x^2\right ) \left (-41717946479-13842904824 x-4847115960 x^2-1092097440 x^3+136080000 x^4+143700480 x^5\right )-\frac {(1+2 x) \left (\frac {1423571729175 \left (i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right )|\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {1+2 x}}+\frac {5694286916700 \sqrt {-\frac {2 i}{-i+\sqrt {7}}} \left (2+5 x+4 x^2\right )-3 \left (546266138719 i+474523909725 \sqrt {7}\right ) (1+2 x)^{3/2} \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )+26925031400960 i (1+2 x)^{3/2} \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (1+i \sqrt {7}\right ),i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{(1+2 x)^2}\right )}{\sqrt {\frac {i}{2 i-2 \sqrt {7}}}}\right )}{4203239040 \sqrt {2+5 x+4 x^2}} \] Input:
Integrate[((4 + 6*x - 2*x^2)*(2 + 5*x + 4*x^2)^(5/2))/((5 - 3*x)*Sqrt[1 + 2*x]),x]
Output:
(Sqrt[1 + 2*x]*(24*(2 + 5*x + 4*x^2)*(-41717946479 - 13842904824*x - 48471 15960*x^2 - 1092097440*x^3 + 136080000*x^4 + 143700480*x^5) - ((1 + 2*x)*( (1423571729175*(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 + S qrt[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] + (5694286916700*Sqr t[(-2*I)/(-I + Sqrt[7])]*(2 + 5*x + 4*x^2) - 3*(546266138719*I + 474523909 725*Sqrt[7])*(1 + 2*x)^(3/2)*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 + S qrt[7])*(1 + 2*x))]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqrt[7])] + (26925031400960*I)*(1 + 2*x)^(3/2 )*Sqrt[(-3*I + Sqrt[7] + 2*(-I + Sqrt[7])*x)/((-I + Sqrt[7])*(1 + 2*x))]*S qrt[(3*I + Sqrt[7] + 2*(I + Sqrt[7])*x)/((I + Sqrt[7])*(1 + 2*x))]*Ellipti cPi[(-13*(1 + I*Sqrt[7]))/6, I*ArcSinh[Sqrt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqrt[7])])/(1 + 2*x)^2))/Sqrt[I/(2*I - 2*Sqrt[ 7])]))/(4203239040*Sqrt[2 + 5*x + 4*x^2])
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.
| \(\displaystyle \int \frac {\left (-2 x^2+6 x+4\right ) \left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \left (4 x^2+5 x+2\right )^{5/2}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {5 \int \frac {2 (344 x+747) \left (4 x^2+5 x+2\right )^{3/2}}{9 \sqrt {2 x+1}}dx}{1144}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \int \frac {(344 x+747) \left (4 x^2+5 x+2\right )^{3/2}}{\sqrt {2 x+1}}dx}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {5 \left (\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}-\frac {1}{168} \int -\frac {4 (14478 x+17503) \sqrt {4 x^2+5 x+2}}{\sqrt {2 x+1}}dx\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \left (\frac {1}{42} \int \frac {(14478 x+17503) \sqrt {4 x^2+5 x+2}}{\sqrt {2 x+1}}dx+\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {5 \left (\frac {1}{42} \left (\frac {1}{30} \sqrt {2 x+1} (86868 x+153313) \sqrt {4 x^2+5 x+2}-\frac {1}{120} \int -\frac {2 (350060 x+373071)}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx\right )+\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \left (\frac {1}{42} \left (\frac {1}{60} \int \frac {350060 x+373071}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (86868 x+153313)\right )+\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {5 \left (\frac {1}{42} \left (\frac {1}{60} \left (198041 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+175030 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+5 x+2}}dx\right )+\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (86868 x+153313)\right )+\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {5 \left (\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}+\frac {1}{42} \left (\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (86868 x+153313)+\frac {1}{60} \left (\frac {396082 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \int \frac {1}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1} \sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {2 x+1}}+\frac {87515 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )\right )}{5148}-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {5 \left (\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}+\frac {1}{42} \left (\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (86868 x+153313)+\frac {1}{60} \left (\frac {87515 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}+\frac {396082 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}\right )\right )\right )}{5148}+\frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {5 \left (\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}+\frac {1}{42} \left (\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (86868 x+153313)+\frac {1}{60} \left (\frac {396082 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}+\frac {87515 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )\right )}{5148}-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
| \(\Big \downarrow \) 1292 |
| \(\displaystyle \frac {76}{9} \int \frac {\left (4 x^2+5 x+2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {5 \left (\frac {1}{63} \sqrt {2 x+1} (2408 x+6637) \left (4 x^2+5 x+2\right )^{3/2}+\frac {1}{42} \left (\frac {1}{30} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (86868 x+153313)+\frac {1}{60} \left (\frac {396082 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}+\frac {87515 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )\right )}{5148}-\frac {(205-132 x) \sqrt {2 x+1} \left (4 x^2+5 x+2\right )^{5/2}}{2574}\) |
Input:
Int[((4 + 6*x - 2*x^2)*(2 + 5*x + 4*x^2)^(5/2))/((5 - 3*x)*Sqrt[1 + 2*x]), x]
Output:
$Aborted
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]*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[((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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* (a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, 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 = 1.55 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {\left (143700480 x^{5}+136080000 x^{4}-1092097440 x^{3}-4847115960 x^{2}-13842904824 x -41717946479\right ) \sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}}{175134960}+\frac {2 \left (-\frac {16202045815339 \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 )}{1050809760 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}-\frac {31634927315 \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 )}{5837832 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}+\frac {1092736664 \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 )}{85293 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}\right ) \sqrt {\left (4 x^{2}+5 x +2\right ) \left (1+2 x \right )}}{\sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}}\) | \(528\) |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+5 x +2\right ) \left (1+2 x \right )}\, \left (\frac {32 x^{5} \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{39}+\frac {1000 x^{4} \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{1287}-\frac {216686 x^{3} \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{34749}-\frac {13464211 x^{2} \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{486486}-\frac {27466081 x \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{347490}-\frac {3792540589 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{15921360}-\frac {16202045815339 \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 )}{525404880 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}-\frac {31634927315 \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 )}{2918916 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}+\frac {2185473328 \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 )}{85293 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}\right )}{\sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}}\) | \(607\) |
| default | \(\frac {\sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}\, \left (13795246080 x^{8}+37205360640 x^{7}+13354902356989 i \sqrt {7}\, \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticF}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )-13462515700480 i \sqrt {7}\, \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticPi}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, -\frac {3}{52}-\frac {3 i \sqrt {7}}{52}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )-66460262400 x^{6}-630650050560 x^{5}+7660615440289 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticF}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )+5694286916700 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticE}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )-13462515700480 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticPi}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, -\frac {3}{52}-\frac {3 i \sqrt {7}}{52}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )-2257914947904 x^{4}-6880229734656 x^{3}-8619979512504 x^{2}-4837767935508 x -1001230715496\right )}{16812956160 x^{3}+29422673280 x^{2}+18914575680 x +4203239040}\) | \(668\) |
Input:
int((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x,method=_RET URNVERBOSE)
Output:
1/175134960*(143700480*x^5+136080000*x^4-1092097440*x^3-4847115960*x^2-138 42904824*x-41717946479)*(4*x^2+5*x+2)^(1/2)*(1+2*x)^(1/2)+2*(-162020458153 39/1050809760*(-1/8-1/8*I*7^(1/2))*((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/2)*(( x+5/8-1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8+1/8*I*7^(1/2))/(1/ 8+1/8*I*7^(1/2)))^(1/2)/(8*x^3+14*x^2+9*x+2)^(1/2)*EllipticF(((x+1/2)/(-1/ 8-1/8*I*7^(1/2)))^(1/2),((1/8+1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2))-3 1634927315/5837832*(-1/8-1/8*I*7^(1/2))*((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/ 2)*((x+5/8-1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8+1/8*I*7^(1/2) )/(1/8+1/8*I*7^(1/2)))^(1/2)/(8*x^3+14*x^2+9*x+2)^(1/2)*((1/8-1/8*I*7^(1/2 ))*EllipticE(((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/2),((1/8+1/8*I*7^(1/2))/(1/ 8-1/8*I*7^(1/2)))^(1/2))+(-5/8+1/8*I*7^(1/2))*EllipticF(((x+1/2)/(-1/8-1/8 *I*7^(1/2)))^(1/2),((1/8+1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)))+10927 36664/85293*(-1/8-1/8*I*7^(1/2))*((x+1/2)/(-1/8-1/8*I*7^(1/2)))^(1/2)*((x+ 5/8-1/8*I*7^(1/2))/(1/8-1/8*I*7^(1/2)))^(1/2)*((x+5/8+1/8*I*7^(1/2))/(1/8+ 1/8*I*7^(1/2)))^(1/2)/(8*x^3+14*x^2+9*x+2)^(1/2)*EllipticPi(((x+1/2)/(-1/8 -1/8*I*7^(1/2)))^(1/2),-3/52-3/52*I*7^(1/2),((1/8+1/8*I*7^(1/2))/(1/8-1/8* I*7^(1/2)))^(1/2)))*((4*x^2+5*x+2)*(1+2*x))^(1/2)/(4*x^2+5*x+2)^(1/2)/(1+2 *x)^(1/2)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="fricas")
Output:
integral(2*(16*x^6 - 8*x^5 - 111*x^4 - 183*x^3 - 138*x^2 - 52*x - 8)*sqrt( 4*x^2 + 5*x + 2)*sqrt(2*x + 1)/(6*x^2 - 7*x - 5), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\text {Timed out} \] Input:
integrate((-2*x**2+6*x+4)*(4*x**2+5*x+2)**(5/2)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
Timed out
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="maxima")
Output:
2*integrate((4*x^2 + 5*x + 2)^(5/2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (4 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="giac")
Output:
integrate(2*(4*x^2 + 5*x + 2)^(5/2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {{\left (4\,x^2+5\,x+2\right )}^{5/2}\,\left (-2\,x^2+6\,x+4\right )}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((5*x + 4*x^2 + 2)^(5/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5 )),x)
Output:
int(-((5*x + 4*x^2 + 2)^(5/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5 )), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \left (2+5 x+4 x^2\right )^{5/2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int \frac {\left (-2 x^{2}+6 x +4\right ) \left (4 x^{2}+5 x +2\right )^{\frac {5}{2}}}{\left (5-3 x \right ) \sqrt {2 x +1}}d x \] Input:
int((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
int((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(5/2)/(5-3*x)/(1+2*x)^(1/2),x)