Integrand size = 38, antiderivative size = 302 \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {2}{135} (26-9 x) \sqrt {1+2 x} \sqrt {3+4 x^2}-\frac {602 \sqrt {1+2 x} \sqrt {3+4 x^2}}{135 (3+2 x)}+\frac {76}{27} \sqrt {\frac {127}{39}} \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {1+2 x}}{\sqrt {3+4 x^2}}\right )+\frac {602 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} E\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{135 \sqrt {3+4 x^2}}-\frac {85 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{27 \sqrt {3+4 x^2}}-\frac {889 \sqrt {2} (3+2 x) \sqrt {\frac {3+4 x^2}{(3+2 x)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{1053 \sqrt {3+4 x^2}} \] Output:
-2/135*(26-9*x)*(1+2*x)^(1/2)*(4*x^2+3)^(1/2)-602*(1+2*x)^(1/2)*(4*x^2+3)^ (1/2)/(405+270*x)+76/1053*4953^(1/2)*arctanh(1/39*4953^(1/2)*(1+2*x)^(1/2) /(4*x^2+3)^(1/2))+602/135*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*Elli pticE(sin(2*arctan(1/2*(1+2*x)^(1/2)*2^(1/2))),1/2*3^(1/2))/(4*x^2+3)^(1/2 )-85/27*2^(1/2)*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*InverseJacobiAM(2*arct an(1/2*(1+2*x)^(1/2)*2^(1/2)),1/2*3^(1/2))/(4*x^2+3)^(1/2)-889/1053*2^(1/2 )*(3+2*x)*((4*x^2+3)/(3+2*x)^2)^(1/2)*EllipticPi(sin(2*arctan(1/2*(1+2*x)^ (1/2)*2^(1/2))),361/312,1/2*3^(1/2))/(4*x^2+3)^(1/2)
Result contains complex when optimal does not.
Time = 24.00 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.17 \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {\sqrt {1+2 x} \left (78 (-26+9 x) \left (3+4 x^2\right )-\frac {(1+2 x) \left (23478 \sqrt {-\frac {i}{i+\sqrt {3}}}+\frac {93912 \sqrt {-\frac {i}{i+\sqrt {3}}}}{(1+2 x)^2}-\frac {46956 \sqrt {-\frac {i}{i+\sqrt {3}}}}{1+2 x}+\frac {11739 \left (-i+\sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}-\frac {3 \left (-2057 i+3913 \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}+\frac {48260 i \sqrt {\frac {3 i+\sqrt {3}+2 \left (-i+\sqrt {3}\right ) x}{\left (-i+\sqrt {3}\right ) (1+2 x)}} \sqrt {\frac {-3 i+\sqrt {3}+2 \left (i+\sqrt {3}\right ) x}{\left (i+\sqrt {3}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (\frac {13}{12} \left (1-i \sqrt {3}\right ),i \text {arcsinh}\left (\frac {2 \sqrt {-\frac {i}{i+\sqrt {3}}}}{\sqrt {1+2 x}}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\sqrt {1+2 x}}\right )}{\sqrt {-\frac {i}{i+\sqrt {3}}}}\right )}{5265 \sqrt {3+4 x^2}} \] Input:
Integrate[((4 + 6*x - 2*x^2)*Sqrt[3 + 4*x^2])/((5 - 3*x)*Sqrt[1 + 2*x]),x]
Output:
(Sqrt[1 + 2*x]*(78*(-26 + 9*x)*(3 + 4*x^2) - ((1 + 2*x)*(23478*Sqrt[(-I)/( I + Sqrt[3])] + (93912*Sqrt[(-I)/(I + Sqrt[3])])/(1 + 2*x)^2 - (46956*Sqrt [(-I)/(I + Sqrt[3])])/(1 + 2*x) + (11739*(-I + Sqrt[3])*Sqrt[(3*I + Sqrt[3 ] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3])*(1 + 2*x))]*Sqrt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*EllipticE[I*ArcSinh[(2*Sqrt [(-I)/(I + Sqrt[3])])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x] - (3*(-2057*I + 3913*Sqrt[3])*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3 ])*x)/((-I + Sqrt[3])*(1 + 2*x))]*Sqrt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x )/((I + Sqrt[3])*(1 + 2*x))]*EllipticF[I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3] )])/Sqrt[1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x] + ((48260* I)*Sqrt[(3*I + Sqrt[3] + 2*(-I + Sqrt[3])*x)/((-I + Sqrt[3])*(1 + 2*x))]*S qrt[(-3*I + Sqrt[3] + 2*(I + Sqrt[3])*x)/((I + Sqrt[3])*(1 + 2*x))]*Ellipt icPi[(13*(1 - I*Sqrt[3]))/12, I*ArcSinh[(2*Sqrt[(-I)/(I + Sqrt[3])])/Sqrt[ 1 + 2*x]], (I + Sqrt[3])/(I - Sqrt[3])])/Sqrt[1 + 2*x]))/Sqrt[(-I)/(I + Sq rt[3])]))/(5265*Sqrt[3 + 4*x^2])
Leaf count is larger than twice the leaf count of optimal. \(683\) vs. \(2(302)=604\).
Time = 1.09 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.26, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {2349, 682, 27, 599, 27, 724, 27, 599, 25, 729, 1511, 27, 1416, 1509, 1540, 27, 1416, 2222}
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.
| \(\displaystyle \int \frac {\left (-2 x^2+6 x+4\right ) \sqrt {4 x^2+3}}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+3}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \sqrt {4 x^2+3}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {1}{60} \int -\frac {8 (129-158 x)}{9 \sqrt {2 x+1} \sqrt {4 x^2+3}}dx+\frac {76}{9} \int \frac {\sqrt {4 x^2+3}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{135} \int \frac {129-158 x}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx+\frac {76}{9} \int \frac {\sqrt {4 x^2+3}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+3}}{(5-3 x) \sqrt {2 x+1}}dx+\frac {1}{135} \int -\frac {2 (208-79 (2 x+1))}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+3}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \int \frac {208-79 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 724 |
| \(\displaystyle \frac {76}{9} \left (\frac {127}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {1}{9} \int \frac {4 (3 x+5)}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx\right )-\frac {2}{135} \int \frac {208-79 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \left (\frac {127}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {4}{9} \int \frac {3 x+5}{\sqrt {2 x+1} \sqrt {4 x^2+3}}dx\right )-\frac {2}{135} \int \frac {208-79 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {76}{9} \left (\frac {127}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+3}}dx+\frac {2}{9} \int -\frac {3 (2 x+1)+7}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {2}{135} \int \frac {208-79 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {76}{9} \left (\frac {127}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+3}}dx-\frac {2}{9} \int \frac {3 (2 x+1)+7}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {2}{135} \int \frac {208-79 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 729 |
| \(\displaystyle -\frac {2}{135} \int \frac {208-79 (2 x+1)}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {76}{9} \left (\frac {254}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {2}{9} \int \frac {3 (2 x+1)+7}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {2}{135} \left (50 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+158 \int \frac {1-2 x}{2 \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {76}{9} \left (\frac {2}{9} \left (6 \int \frac {1-2 x}{2 \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-13 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {254}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{135} \left (50 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+79 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {76}{9} \left (\frac {2}{9} \left (3 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-13 \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {254}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2}{135} \left (79 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {25 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {76}{9} \left (\frac {2}{9} \left (3 \int \frac {1-2 x}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}-\frac {13 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {254}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {76}{9} \left (\frac {254}{9} \int \frac {1}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {2}{9} \left (3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )-\frac {13 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )\right )-\frac {2}{135} \left (\frac {25 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}+79 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle \frac {76}{9} \left (\frac {254}{9} \left (\frac {1}{19} \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {6}{19} \int \frac {2 x+3}{2 (13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {2}{9} \left (3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )-\frac {13 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )\right )-\frac {2}{135} \left (\frac {25 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}+79 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \left (\frac {254}{9} \left (\frac {1}{19} \int \frac {1}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {3}{19} \int \frac {2 x+3}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}\right )+\frac {2}{9} \left (3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )-\frac {13 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )\right )-\frac {2}{135} \left (\frac {25 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}+79 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {76}{9} \left (\frac {254}{9} \left (\frac {3}{19} \int \frac {2 x+3}{(13-3 (2 x+1)) \sqrt {(2 x+1)^2-2 (2 x+1)+4}}d\sqrt {2 x+1}+\frac {(2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{38 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {2}{9} \left (3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )-\frac {13 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )\right )-\frac {2}{135} \left (\frac {25 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}+79 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {76}{9} \left (\frac {254}{9} \left (\frac {3}{19} \left (\frac {19 \text {arctanh}\left (\frac {\sqrt {\frac {127}{39}} \sqrt {2 x+1}}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )}{2 \sqrt {4953}}-\frac {7 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticPi}\left (\frac {361}{312},2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{156 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {(2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{38 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )+\frac {2}{9} \left (3 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )-\frac {13 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{2 \sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}\right )\right )-\frac {2}{135} \left (\frac {25 (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ),\frac {3}{4}\right )}{\sqrt {2} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}+79 \left (\frac {\sqrt {2} (2 x+3) \sqrt {\frac {(2 x+1)^2-2 (2 x+1)+4}{(2 x+3)^2}} E\left (2 \arctan \left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right )|\frac {3}{4}\right )}{\sqrt {(2 x+1)^2-2 (2 x+1)+4}}-\frac {\sqrt {2 x+1} \sqrt {(2 x+1)^2-2 (2 x+1)+4}}{2 x+3}\right )\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {4 x^2+3} (26-9 x)\) |
Input:
Int[((4 + 6*x - 2*x^2)*Sqrt[3 + 4*x^2])/((5 - 3*x)*Sqrt[1 + 2*x]),x]
Output:
(-2*(26 - 9*x)*Sqrt[1 + 2*x]*Sqrt[3 + 4*x^2])/135 - (2*(79*(-((Sqrt[1 + 2* x]*Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2])/(3 + 2*x)) + (Sqrt[2]*(3 + 2*x)*Sq rt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]*EllipticE[2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2]) + (25*(3 + 2*x )*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]*EllipticF[2*ArcTan[Sqr t[1 + 2*x]/Sqrt[2]], 3/4])/(Sqrt[2]*Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2]))) /135 + (76*((2*(3*(-((Sqrt[1 + 2*x]*Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2])/( 3 + 2*x)) + (Sqrt[2]*(3 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2 *x)^2]*EllipticE[2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/Sqrt[4 - 2*(1 + 2* x) + (1 + 2*x)^2]) - (13*(3 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]*EllipticF[2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/(2*Sqrt[2]*Sqr t[4 - 2*(1 + 2*x) + (1 + 2*x)^2])))/9 + (254*(((3 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]*EllipticF[2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]] , 3/4])/(38*Sqrt[2]*Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2]) + (3*((19*ArcTanh [(Sqrt[127/39]*Sqrt[1 + 2*x])/Sqrt[4 - 2*(1 + 2*x) + (1 + 2*x)^2]])/(2*Sqr t[4953]) - (7*(3 + 2*x)*Sqrt[(4 - 2*(1 + 2*x) + (1 + 2*x)^2)/(3 + 2*x)^2]* EllipticPi[361/312, 2*ArcTan[Sqrt[1 + 2*x]/Sqrt[2]], 3/4])/(156*Sqrt[2]*Sq rt[4 - 2*(1 + 2*x) + (1 + 2*x)^2])))/19))/9))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[Sqrt[(a_) + (c_.)*(x_)^2]/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_) ]), x_Symbol] :> Simp[(c*d^2 + a*e^2)/e^2 Int[1/((d + e*x)*Sqrt[f + g*x]* Sqrt[a + c*x^2]), x], x] - Simp[1/e^2 Int[(c*d - c*e*x)/(Sqrt[f + g*x]*Sq rt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) ^2]), x_Symbol] :> Simp[2 Subst[Int[1/((d*e - c*f + f*x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a]
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)/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[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), 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] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d *x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, 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 = 0.33 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(\frac {2 \left (-26+9 x \right ) \sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}{135}+\frac {2 \left (-\frac {8374 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{405 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}-\frac {1204 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )+\frac {i \sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2}\right )}{135 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {19304 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{1053 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right ) \sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(490\) |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+3\right ) \left (1+2 x \right )}\, \left (\frac {2 x \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{15}-\frac {52 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}{135}-\frac {16748 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{405 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}-\frac {2408 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )+\frac {i \sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2}\right )}{135 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}+\frac {38608 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {1}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {\frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\right )}{1053 \sqrt {8 x^{3}+4 x^{2}+6 x +3}}\right )}{\sqrt {4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(504\) |
| default | \(\frac {2 \sqrt {4 x^{2}+3}\, \sqrt {1+2 x}\, \left (42692 i \sqrt {3}\, \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-48260 i \sqrt {3}\, \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )-89648 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+46956 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+48260 \sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x}{1+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x}{-1+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1+2 x}{-1+i \sqrt {3}}}, \frac {3}{13}-\frac {3 i \sqrt {3}}{13}, \sqrt {-\frac {-1+i \sqrt {3}}{1+i \sqrt {3}}}\right )+2808 x^{4}-6708 x^{3}-1950 x^{2}-5031 x -3042\right )}{5265 \left (8 x^{3}+4 x^{2}+6 x +3\right )}\) | \(625\) |
Input:
int((-2*x^2+6*x+4)*(4*x^2+3)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x,method=_RETURNV ERBOSE)
Output:
2/135*(-26+9*x)*(4*x^2+3)^(1/2)*(1+2*x)^(1/2)+2*(-8374/405*(1/2-1/2*I*3^(1 /2))*((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2*I*3^(1/2))/(-1/2-1/2*I*3^ (1/2)))^(1/2)*((x+1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2)))^(1/2)/(8*x^3+4*x^2+ 6*x+3)^(1/2)*EllipticF(((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2),((-1/2+1/2*I*3^ (1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2))-1204/135*(1/2-1/2*I*3^(1/2))*((x+1/2)/ (1/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)* ((x+1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2)))^(1/2)/(8*x^3+4*x^2+6*x+3)^(1/2)*( (-1/2-1/2*I*3^(1/2))*EllipticE(((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2),((-1/2+ 1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2))+1/2*I*3^(1/2)*EllipticF(((x+1/ 2)/(1/2-1/2*I*3^(1/2)))^(1/2),((-1/2+1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^ (1/2)))+19304/1053*(1/2-1/2*I*3^(1/2))*((x+1/2)/(1/2-1/2*I*3^(1/2)))^(1/2) *((x-1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2*I*3^(1/2))/(-1/2+1 /2*I*3^(1/2)))^(1/2)/(8*x^3+4*x^2+6*x+3)^(1/2)*EllipticPi(((x+1/2)/(1/2-1/ 2*I*3^(1/2)))^(1/2),3/13-3/13*I*3^(1/2),((-1/2+1/2*I*3^(1/2))/(-1/2-1/2*I* 3^(1/2)))^(1/2)))*((4*x^2+3)*(1+2*x))^(1/2)/(4*x^2+3)^(1/2)/(1+2*x)^(1/2)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 3} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+3)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algorith m="fricas")
Output:
integral(2*sqrt(4*x^2 + 3)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(6*x^2 - 7*x - 5) , x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=2 \left (\int \left (- \frac {2 \sqrt {4 x^{2} + 3}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \left (- \frac {3 x \sqrt {4 x^{2} + 3}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \frac {x^{2} \sqrt {4 x^{2} + 3}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\, dx\right ) \] Input:
integrate((-2*x**2+6*x+4)*(4*x**2+3)**(1/2)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
2*(Integral(-2*sqrt(4*x**2 + 3)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x) + Integral(-3*x*sqrt(4*x**2 + 3)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x) + Integral(x**2*sqrt(4*x**2 + 3)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x ))
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 3} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+3)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algorith m="maxima")
Output:
2*integrate(sqrt(4*x^2 + 3)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 3} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+3)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algorith m="giac")
Output:
integrate(2*sqrt(4*x^2 + 3)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {\sqrt {4\,x^2+3}\,\left (-2\,x^2+6\,x+4\right )}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((4*x^2 + 3)^(1/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)),x)
Output:
int(-((4*x^2 + 3)^(1/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {3+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int \frac {\left (-2 x^{2}+6 x +4\right ) \sqrt {4 x^{2}+3}}{\left (5-3 x \right ) \sqrt {2 x +1}}d x \] Input:
int((-2*x^2+6*x+4)*(4*x^2+3)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
int((-2*x^2+6*x+4)*(4*x^2+3)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x)