Integrand size = 38, antiderivative size = 185 \[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {2}{135} (26-9 x) \sqrt {1+2 x} \sqrt {3-4 x^2}-\frac {142}{27} \sqrt {1+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )-\frac {4696 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{405 \sqrt {1+\sqrt {3}}}+\frac {76}{81} \sqrt {2 \left (53+67 \sqrt {3}\right )} \operatorname {EllipticPi}\left (-\frac {6}{73} \left (9+10 \sqrt {3}\right ),\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right ) \] Output:
-2/135*(26-9*x)*(1+2*x)^(1/2)*(-4*x^2+3)^(1/2)-142/27*(1+3^(1/2))^(1/2)*El lipticE(1/6*(3-2*x*3^(1/2))^(1/2)*6^(1/2),(3-3^(1/2))^(1/2))-4696/405*Elli pticF(1/6*(3-2*x*3^(1/2))^(1/2)*6^(1/2),(3-3^(1/2))^(1/2))/(1+3^(1/2))^(1/ 2)+76/81*(106+134*3^(1/2))^(1/2)*EllipticPi(1/6*(3-2*x*3^(1/2))^(1/2)*6^(1 /2),-54/73-60/73*3^(1/2),(3-3^(1/2))^(1/2))
Result contains complex when optimal does not.
Time = 23.16 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {2 (1+2 x) \sqrt {3-4 x^2} \left (39 (-26+9 x)-\frac {(1+2 x) \left (-\frac {13845 i \left (-1+\sqrt {3}\right ) \sqrt {\frac {-3+4 x^2}{(1+2 x)^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-1+\sqrt {3}}}{\sqrt {1+2 x}}\right )|-2-\sqrt {3}\right )}{\sqrt {1+2 x}}+\frac {3 i \left (-5543+4615 \sqrt {3}\right ) \sqrt {\frac {-3+4 x^2}{(1+2 x)^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-1+\sqrt {3}}}{\sqrt {1+2 x}}\right ),-2-\sqrt {3}\right )}{\sqrt {1+2 x}}-\frac {5 \left (2769 \sqrt {-1+\sqrt {3}} \left (-3+4 x^2\right )+5548 i (1+2 x)^{3/2} \sqrt {\frac {-3+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (1+\sqrt {3}\right ),i \text {arcsinh}\left (\frac {\sqrt {-1+\sqrt {3}}}{\sqrt {1+2 x}}\right ),-2-\sqrt {3}\right )\right )}{(1+2 x)^2}\right )}{\sqrt {-1+\sqrt {3}} \left (3-4 x^2\right )}\right )}{1755 \sqrt {9+18 x}} \] Input:
Integrate[(Sqrt[3 - 4*x^2]*(4 + 6*x - 2*x^2))/((5 - 3*x)*Sqrt[1 + 2*x]),x]
Output:
(2*(1 + 2*x)*Sqrt[3 - 4*x^2]*(39*(-26 + 9*x) - ((1 + 2*x)*(((-13845*I)*(-1 + Sqrt[3])*Sqrt[(-3 + 4*x^2)/(1 + 2*x)^2]*EllipticE[I*ArcSinh[Sqrt[-1 + S qrt[3]]/Sqrt[1 + 2*x]], -2 - Sqrt[3]])/Sqrt[1 + 2*x] + ((3*I)*(-5543 + 461 5*Sqrt[3])*Sqrt[(-3 + 4*x^2)/(1 + 2*x)^2]*EllipticF[I*ArcSinh[Sqrt[-1 + Sq rt[3]]/Sqrt[1 + 2*x]], -2 - Sqrt[3]])/Sqrt[1 + 2*x] - (5*(2769*Sqrt[-1 + S qrt[3]]*(-3 + 4*x^2) + (5548*I)*(1 + 2*x)^(3/2)*Sqrt[(-3 + 4*x^2)/(1 + 2*x )^2]*EllipticPi[(-13*(1 + Sqrt[3]))/6, I*ArcSinh[Sqrt[-1 + Sqrt[3]]/Sqrt[1 + 2*x]], -2 - Sqrt[3]]))/(1 + 2*x)^2))/(Sqrt[-1 + Sqrt[3]]*(3 - 4*x^2)))) /(1755*Sqrt[9 + 18*x])
Time = 1.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.66, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {2349, 682, 27, 600, 508, 327, 511, 321, 724, 27, 600, 508, 327, 511, 321, 730, 27, 186, 25, 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.
| \(\displaystyle \int \frac {\sqrt {3-4 x^2} \left (-2 x^2+6 x+4\right )}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \sqrt {3-4 x^2}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {1}{60} \int \frac {8 (50 x+129)}{9 \sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{135} \int \frac {50 x+129}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2}{135} \left (104 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+25 \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\right )+\frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \left (104 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {25 \sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2}{135} \left (104 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {3-4 x^2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 724 |
| \(\displaystyle \frac {76}{9} \left (-\frac {73}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {1}{9} \int -\frac {4 (3 x+5)}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \int \frac {3 x+5}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {73}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (\frac {7}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {3}{2} \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\right )-\frac {73}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (\frac {7}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )-\frac {73}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (\frac {7}{2} \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )-\frac {73}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (-\frac {7 \sqrt [4]{3} \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{2 \sqrt {3+\sqrt {3}}}-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )-\frac {73}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (-\frac {7 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )-\frac {73}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 730 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (-\frac {7 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )-\frac {73 \int \frac {3}{(5-3 x) \sqrt {2 x+1} \sqrt {3-2 \sqrt {3} x} \sqrt {2 \sqrt {3} x+3}}dx}{9 \sqrt {3}}\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (-\frac {7 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )-\frac {73 \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {3-2 \sqrt {3} x} \sqrt {2 \sqrt {3} x+3}}dx}{3 \sqrt {3}}\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {76}{9} \left (\frac {146 \int -\frac {1}{(13-3 (2 x+1)) \sqrt {-\sqrt {3} (2 x+1)+\sqrt {3}+3} \sqrt {\sqrt {3} (2 x+1)-\sqrt {3}+3}}d\sqrt {2 x+1}}{3 \sqrt {3}}+\frac {4}{9} \left (-\frac {7 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {76}{9} \left (\frac {4}{9} \left (-\frac {7 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )-\frac {146 \int \frac {1}{(13-3 (2 x+1)) \sqrt {-\sqrt {3} (2 x+1)+\sqrt {3}+3} \sqrt {\sqrt {3} (2 x+1)-\sqrt {3}+3}}d\sqrt {2 x+1}}{3 \sqrt {3}}\right )-\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {2}{135} \left (-\frac {104 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {25 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {76}{9} \left (\frac {4}{9} \left (-\frac {7 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{2 \sqrt {3+\sqrt {3}}}-\frac {1}{2} 3^{3/4} \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )\right )-\frac {146 \operatorname {EllipticPi}\left (\frac {3}{13} \left (1+\sqrt {3}\right ),\arcsin \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {3}\right )} \sqrt {2 x+1}\right ),-2-\sqrt {3}\right )}{117 \sqrt {\sqrt {3}-1}}\right )-\frac {2}{135} \sqrt {2 x+1} \sqrt {3-4 x^2} (26-9 x)\) |
Input:
Int[(Sqrt[3 - 4*x^2]*(4 + 6*x - 2*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*((-25*Sqrt[3 + Sqrt [3]]*EllipticE[ArcSin[Sqrt[3 - 2*Sqrt[3]*x]/Sqrt[6]], 3 - Sqrt[3]])/3^(1/4 ) - (104*3^(1/4)*EllipticF[ArcSin[Sqrt[3 - 2*Sqrt[3]*x]/Sqrt[6]], 3 - Sqrt [3]])/Sqrt[3 + Sqrt[3]]))/135 + (76*((4*(-1/2*(3^(3/4)*Sqrt[3 + Sqrt[3]]*E llipticE[ArcSin[Sqrt[3 - 2*Sqrt[3]*x]/Sqrt[6]], 3 - Sqrt[3]]) - (7*3^(1/4) *EllipticF[ArcSin[Sqrt[3 - 2*Sqrt[3]*x]/Sqrt[6]], 3 - Sqrt[3]])/(2*Sqrt[3 + Sqrt[3]])))/9 - (146*EllipticPi[(3*(1 + Sqrt[3]))/13, ArcSin[Sqrt[(-1 + Sqrt[3])/2]*Sqrt[1 + 2*x]], -2 - Sqrt[3]])/(117*Sqrt[-1 + Sqrt[3]])))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] && GtQ[(d*e - c*f)/d, 0]
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[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[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] :> With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/((e + f*x )*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[b/a] && GtQ[a, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(429\) vs. \(2(135)=270\).
Time = 0.35 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.32
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \left (\frac {2 x \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{15}-\frac {52 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{135}+\frac {13652 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{1215 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}+\frac {568 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (-\frac {\sqrt {3}}{2}+\frac {1}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )-\frac {\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{2}\right )}{81 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}+\frac {11096 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {\sqrt {3}}{-\frac {\sqrt {3}}{2}-\frac {5}{3}}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{729 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right )}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(430\) |
| risch | \(-\frac {2 \left (-26+9 x \right ) \left (4 x^{2}-3\right ) \sqrt {1+2 x}\, \sqrt {\left (-4 x^{2}+3\right ) \left (1+2 x \right )}}{135 \sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \sqrt {-4 x^{2}+3}}-\frac {2 \left (-\frac {6826 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{1215 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {284 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (-\frac {\sqrt {3}}{2}+\frac {1}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )-\frac {\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{2}\right )}{81 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {5548 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {\sqrt {3}}{-\frac {\sqrt {3}}{2}-\frac {5}{3}}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{729 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right ) \sqrt {\left (-4 x^{2}+3\right ) \left (1+2 x \right )}}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\) | \(453\) |
| default | \(-\frac {2 \sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}\, \left (648 x^{4}+23480 \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )+1065 \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right ) \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}-27740 \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {6 \sqrt {3}}{10+3 \sqrt {3}}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )-1548 x^{3}-4536 x^{4} \sqrt {3}+21132 \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )-22365 \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right ) \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}-1422 x^{2}+10836 x^{3} \sqrt {3}+1161 x +9954 x^{2} \sqrt {3}+702-8127 \sqrt {3}\, x -4914 \sqrt {3}\right )}{1215 \left (\sqrt {3}-1\right ) \left (8 x^{3}+4 x^{2}-6 x -3\right ) \left (10+3 \sqrt {3}\right )}\) | \(560\) |
Input:
int((-4*x^2+3)^(1/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x,method=_RETURN VERBOSE)
Output:
(-(4*x^2-3)*(1+2*x))^(1/2)/(-4*x^2+3)^(1/2)/(1+2*x)^(1/2)*(2/15*x*(-8*x^3- 4*x^2+6*x+3)^(1/2)-52/135*(-8*x^3-4*x^2+6*x+3)^(1/2)+13652/1215*((x+1/2*3^ (1/2))*3^(1/2))^(1/2)*(-(x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^(1/2 ))*3^(1/2))^(1/2)/(-8*x^3-4*x^2+6*x+3)^(1/2)*EllipticF(1/3*3^(1/2)*((x+1/2 *3^(1/2))*3^(1/2))^(1/2),(3^(1/2)/(1/2*3^(1/2)-1/2))^(1/2))+568/81*((x+1/2 *3^(1/2))*3^(1/2))^(1/2)*(-(x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^( 1/2))*3^(1/2))^(1/2)/(-8*x^3-4*x^2+6*x+3)^(1/2)*((-1/2*3^(1/2)+1/2)*Ellipt icE(1/3*3^(1/2)*((x+1/2*3^(1/2))*3^(1/2))^(1/2),(3^(1/2)/(1/2*3^(1/2)-1/2) )^(1/2))-1/2*EllipticF(1/3*3^(1/2)*((x+1/2*3^(1/2))*3^(1/2))^(1/2),(3^(1/2 )/(1/2*3^(1/2)-1/2))^(1/2)))+11096/729*((x+1/2*3^(1/2))*3^(1/2))^(1/2)*(-( x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^(1/2))*3^(1/2))^(1/2)/(-8*x^3 -4*x^2+6*x+3)^(1/2)/(-1/2*3^(1/2)-5/3)*EllipticPi(1/3*3^(1/2)*((x+1/2*3^(1 /2))*3^(1/2))^(1/2),-3^(1/2)/(-1/2*3^(1/2)-5/3),(3^(1/2)/(1/2*3^(1/2)-1/2) )^(1/2)))
\[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} \sqrt {-4 \, x^{2} + 3}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-4*x^2+3)^(1/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x, algorit hm="fricas")
Output:
integral(2*(x^2 - 3*x - 2)*sqrt(-4*x^2 + 3)*sqrt(2*x + 1)/(6*x^2 - 7*x - 5 ), x)
\[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=2 \left (\int \left (- \frac {2 \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \left (- \frac {3 x \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \frac {x^{2} \sqrt {3 - 4 x^{2}}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\, dx\right ) \] Input:
integrate((-4*x**2+3)**(1/2)*(-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
2*(Integral(-2*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x) + Integral(-3*x*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x) + Integral(x**2*sqrt(3 - 4*x**2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1)), x ))
\[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} \sqrt {-4 \, x^{2} + 3}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-4*x^2+3)^(1/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x, algorit hm="maxima")
Output:
2*integrate((x^2 - 3*x - 2)*sqrt(-4*x^2 + 3)/((3*x - 5)*sqrt(2*x + 1)), x)
\[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} \sqrt {-4 \, x^{2} + 3}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-4*x^2+3)^(1/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x, algorit hm="giac")
Output:
integrate(2*(x^2 - 3*x - 2)*sqrt(-4*x^2 + 3)/((3*x - 5)*sqrt(2*x + 1)), x)
Timed out. \[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {\sqrt {3-4\,x^2}\,\left (-2\,x^2+6\,x+4\right )}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((3 - 4*x^2)^(1/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)),x)
Output:
int(-((3 - 4*x^2)^(1/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)), x)
\[ \int \frac {\sqrt {3-4 x^2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {2 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{15}+\frac {49 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{180}-\frac {71 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{3}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{3}+\frac {3797 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{180}+\frac {71 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{12} \] Input:
int((-4*x^2+3)^(1/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
(24*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x + 49*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3) - 4260*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**3)/(24*x**4 - 28*x** 3 - 38*x**2 + 21*x + 15),x) + 3797*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)* x)/(24*x**4 - 28*x**3 - 38*x**2 + 21*x + 15),x) + 1065*int((sqrt(2*x + 1)* sqrt( - 4*x**2 + 3))/(24*x**4 - 28*x**3 - 38*x**2 + 21*x + 15),x))/180