Integrand size = 41, antiderivative size = 448 \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {1}{270} (89-36 x) \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}-\frac {52 \sqrt {2} \sqrt {1+2 x} \sqrt {2+5 x+4 x^2}}{9 \left (1+\sqrt {2} (1+2 x)\right )}+\frac {76}{27} \sqrt {\frac {193}{39}} \text {arctanh}\left (\frac {\sqrt {\frac {193}{39}} \sqrt {1+2 x}}{\sqrt {2+5 x+4 x^2}}\right )+\frac {26\ 2^{3/4} \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) E\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right )|\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{9 \sqrt {2+5 x+4 x^2}}+\frac {\left (177433-1099960 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{177660\ 2^{3/4} \sqrt {2+5 x+4 x^2}}-\frac {7334 \sqrt [4]{2} \left (347-78 \sqrt {2}\right ) \sqrt {\frac {2+5 x+4 x^2}{\left (1+\sqrt {2} (1+2 x)\right )^2}} \left (1+\sqrt {2} (1+2 x)\right ) \operatorname {EllipticPi}\left (\frac {1}{312} \left (156+347 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} \sqrt {1+2 x}\right ),\frac {1}{8} \left (4-\sqrt {2}\right )\right )}{346437 \sqrt {2+5 x+4 x^2}} \] Output:
-1/270*(89-36*x)*(1+2*x)^(1/2)*(4*x^2+5*x+2)^(1/2)-52*2^(1/2)*(1+2*x)^(1/2 )*(4*x^2+5*x+2)^(1/2)/(9+9*2^(1/2)*(1+2*x))+76/1053*7527^(1/2)*arctanh(1/3 9*7527^(1/2)*(1+2*x)^(1/2)/(4*x^2+5*x+2)^(1/2))+26/9*((4*x^2+5*x+2)/(1+2^( 1/2)*(1+2*x))^2)^(1/2)*(1+2^(1/2)*(1+2*x))*EllipticE(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)+1/355 320*(177433-1099960*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^(1/4)/(4*x^2+5*x+2)^(1/2)-7334/346437*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))*E llipticPi(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 = 25.99 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.41 \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=2 \left (\frac {1}{540} \sqrt {1+2 x} (-89+36 x) \sqrt {2+5 x+4 x^2}-\frac {(1+2 x)^{3/2} \left (\frac {30420 \left (i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right )|\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {1+2 x}}-\frac {3 \left (11801 i+10140 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )}{\sqrt {1+2 x}}+\frac {80 \left (1521 \sqrt {-\frac {2 i}{-i+\sqrt {7}}} \left (2+5 x+4 x^2\right )+7334 i (1+2 x)^{3/2} \sqrt {\frac {-3 i+\sqrt {7}+2 \left (-i+\sqrt {7}\right ) x}{\left (-i+\sqrt {7}\right ) (1+2 x)}} \sqrt {\frac {3 i+\sqrt {7}+2 \left (i+\sqrt {7}\right ) x}{\left (i+\sqrt {7}\right ) (1+2 x)}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (1+i \sqrt {7}\right ),i \text {arcsinh}\left (\frac {\sqrt {-\frac {2 i}{-i+\sqrt {7}}}}{\sqrt {1+2 x}}\right ),\frac {i-\sqrt {7}}{i+\sqrt {7}}\right )\right )}{(1+2 x)^2}\right )}{42120 \sqrt {-\frac {2 i}{-i+\sqrt {7}}} \sqrt {2+5 x+4 x^2}}\right ) \] Input:
Integrate[((4 + 6*x - 2*x^2)*Sqrt[2 + 5*x + 4*x^2])/((5 - 3*x)*Sqrt[1 + 2* x]),x]
Output:
2*((Sqrt[1 + 2*x]*(-89 + 36*x)*Sqrt[2 + 5*x + 4*x^2])/540 - ((1 + 2*x)^(3/ 2)*((30420*(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*(11801*I + 10140*Sq rt[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))]*El lipticF[I*ArcSinh[Sqrt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7] )/(I + Sqrt[7])])/Sqrt[1 + 2*x] + (80*(1521*Sqrt[(-2*I)/(-I + Sqrt[7])]*(2 + 5*x + 4*x^2) + (7334*I)*(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 + Sqrt[7])*(1 + 2*x))]*EllipticPi[(-13*(1 + I*Sqrt[7]))/6, I*Ar cSinh[Sqrt[(-2*I)/(-I + Sqrt[7])]/Sqrt[1 + 2*x]], (I - Sqrt[7])/(I + Sqrt[ 7])]))/(1 + 2*x)^2))/(42120*Sqrt[(-2*I)/(-I + Sqrt[7])]*Sqrt[2 + 5*x + 4*x ^2]))
Result contains complex when optimal does not.
Time = 1.99 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.44, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {2154, 1231, 27, 1269, 1172, 321, 327, 1274, 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.
| \(\displaystyle \int \frac {\left (-2 x^2+6 x+4\right ) \sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \sqrt {4 x^2+5 x+2}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {1}{120} \int \frac {2 (160 x+303)}{9 \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {76}{9} \int \frac {\sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{540} \int \frac {160 x+303}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {76}{9} \int \frac {\sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{540} \left (-223 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx-80 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+5 x+2}}dx\right )+\frac {76}{9} \int \frac {\sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}}dx+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \int \frac {1}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1} \sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{540} \left (-\frac {40 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}\right )+\frac {76}{9} \int \frac {\sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+5 x+2}}{(5-3 x) \sqrt {2 x+1}}dx+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 1274 |
| \(\displaystyle \frac {76}{9} \left (\frac {193}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx-\frac {1}{9} \int \frac {12 x+35}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {76}{9} \left (\frac {193}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {1}{9} \left (-29 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx-6 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+5 x+2}}dx\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {76}{9} \left (\frac {193}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \int \frac {1}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1} \sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \left (\frac {193}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {1}{9} \left (-\frac {3 i \sqrt {2 x+1} \int \frac {\sqrt {1-\frac {i \left (8 x+i \sqrt {7}+5\right )}{i-\sqrt {7}}}}{\sqrt {\frac {i \left (8 x+i \sqrt {7}+5\right )}{2 \sqrt {7}}+1}}d\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \left (\frac {193}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+5 x+2}}dx+\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 1279 |
| \(\displaystyle \frac {76}{9} \left (\frac {193 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5}}dx}{9 \sqrt {4 x^2+5 x+2}}+\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle \frac {76}{9} \left (\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {386 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \int -\frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2}}\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {76}{9} \left (\frac {386 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2}}+\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {76}{9} \left (\frac {386 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \sqrt {1+\frac {4 (2 x+1)}{1-i \sqrt {7}}} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)+i \sqrt {7}+1} \sqrt {\frac {4 (2 x+1)}{1-i \sqrt {7}}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2} \sqrt {4 (2 x+1)-i \sqrt {7}+1}}+\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {76}{9} \left (\frac {386 \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \sqrt {1+\frac {4 (2 x+1)}{1-i \sqrt {7}}} \sqrt {1+\frac {4 (2 x+1)}{1+i \sqrt {7}}} \int \frac {1}{(13-3 (2 x+1)) \sqrt {\frac {4 (2 x+1)}{1-i \sqrt {7}}+1} \sqrt {\frac {4 (2 x+1)}{1+i \sqrt {7}}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+5 x+2} \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}+\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {76}{9} \left (\frac {193 \sqrt {-1+i \sqrt {7}} \sqrt {8 x-i \sqrt {7}+5} \sqrt {8 x+i \sqrt {7}+5} \sqrt {1+\frac {4 (2 x+1)}{1-i \sqrt {7}}} \sqrt {1+\frac {4 (2 x+1)}{1+i \sqrt {7}}} \operatorname {EllipticPi}\left (-\frac {3}{52} \left (1-i \sqrt {7}\right ),\arcsin \left (\frac {2 \sqrt {2 x+1}}{\sqrt {-1+i \sqrt {7}}}\right ),\frac {i+\sqrt {7}}{i-\sqrt {7}}\right )}{117 \sqrt {4 x^2+5 x+2} \sqrt {4 (2 x+1)-i \sqrt {7}+1} \sqrt {4 (2 x+1)+i \sqrt {7}+1}}+\frac {1}{9} \left (-\frac {58 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {3 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )\right )+\frac {1}{540} \left (-\frac {446 i \sqrt {-\frac {2 x+1}{1+i \sqrt {7}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right ),-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {2 x+1}}-\frac {40 i \sqrt {2 x+1} E\left (\arcsin \left (\frac {\sqrt {-i \left (8 x+i \sqrt {7}+5\right )}}{\sqrt {2} \sqrt [4]{7}}\right )|-\frac {2 \sqrt {7}}{i-\sqrt {7}}\right )}{\sqrt {-\frac {2 x+1}{1+i \sqrt {7}}}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+5 x+2} (89-36 x)\) |
Input:
Int[((4 + 6*x - 2*x^2)*Sqrt[2 + 5*x + 4*x^2])/((5 - 3*x)*Sqrt[1 + 2*x]),x]
Output:
-1/270*((89 - 36*x)*Sqrt[1 + 2*x]*Sqrt[2 + 5*x + 4*x^2]) + (((-40*I)*Sqrt[ 1 + 2*x]*EllipticE[ArcSin[Sqrt[(-I)*(5 + I*Sqrt[7] + 8*x)]/(Sqrt[2]*7^(1/4 ))], (-2*Sqrt[7])/(I - Sqrt[7])])/Sqrt[-((1 + 2*x)/(1 + I*Sqrt[7]))] - ((4 46*I)*Sqrt[-((1 + 2*x)/(1 + I*Sqrt[7]))]*EllipticF[ArcSin[Sqrt[(-I)*(5 + I *Sqrt[7] + 8*x)]/(Sqrt[2]*7^(1/4))], (-2*Sqrt[7])/(I - Sqrt[7])])/Sqrt[1 + 2*x])/540 + (76*((((-3*I)*Sqrt[1 + 2*x]*EllipticE[ArcSin[Sqrt[(-I)*(5 + I *Sqrt[7] + 8*x)]/(Sqrt[2]*7^(1/4))], (-2*Sqrt[7])/(I - Sqrt[7])])/Sqrt[-(( 1 + 2*x)/(1 + I*Sqrt[7]))] - ((58*I)*Sqrt[-((1 + 2*x)/(1 + I*Sqrt[7]))]*El lipticF[ArcSin[Sqrt[(-I)*(5 + I*Sqrt[7] + 8*x)]/(Sqrt[2]*7^(1/4))], (-2*Sq rt[7])/(I - Sqrt[7])])/Sqrt[1 + 2*x])/9 + (193*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])])/(117*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)])))/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] && !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[(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[Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]/(((d_.) + (e_.)*(x_))*Sqrt[(f_. ) + (g_.)*(x_)]), x_Symbol] :> Simp[(c*d^2 - b*d*e + a*e^2)/e^2 Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] - Simp[1/e^2 Int[(c* d - b*e - c*e*x)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
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 = 0.73 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {\left (-89+36 x \right ) \sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}}{270}+\frac {2 \left (-\frac {54109 \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 )}{1620 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}-\frac {104 \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 )}{9 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}+\frac {29336 \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 )}{1053 \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}}\) | \(508\) |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+5 x +2\right ) \left (1+2 x \right )}\, \left (\frac {2 x \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{15}-\frac {89 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}{270}-\frac {54109 \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 )}{810 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}-\frac {208 \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 )}{9 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}+\frac {58672 \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 )}{1053 \sqrt {8 x^{3}+14 x^{2}+9 x +2}}\right )}{\sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}}\) | \(519\) |
| default | \(\frac {\sqrt {4 x^{2}+5 x +2}\, \sqrt {1+2 x}\, \left (581737 i \sqrt {7}\, \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticF}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )-586720 i \sqrt {7}\, \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticPi}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, -\frac {3}{52}-\frac {3 i \sqrt {7}}{52}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )+338377 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticF}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )+243360 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticE}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )-586720 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}\, \sqrt {\frac {i \sqrt {7}-8 x -5}{i \sqrt {7}-1}}\, \sqrt {\frac {i \sqrt {7}+8 x +5}{1+i \sqrt {7}}}\, \operatorname {EllipticPi}\left (2 \sqrt {-\frac {1+2 x}{1+i \sqrt {7}}}, -\frac {3}{52}-\frac {3 i \sqrt {7}}{52}, \sqrt {-\frac {1+i \sqrt {7}}{i \sqrt {7}-1}}\right )+44928 x^{4}-32448 x^{3}-143832 x^{2}-113724 x -27768\right )}{336960 x^{3}+589680 x^{2}+379080 x +84240}\) | \(648\) |
Input:
int((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x,method=_RET URNVERBOSE)
Output:
1/270*(-89+36*x)*(4*x^2+5*x+2)^(1/2)*(1+2*x)^(1/2)+2*(-54109/1620*(-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))-104/9*(-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)))+29336/1053*(-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)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 5 \, x + 2} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(1/2)/(5-3*x)/(1+2*x)^(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)/(6*x^2 - 7* x - 5), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=2 \left (\int \left (- \frac {2 \sqrt {4 x^{2} + 5 x + 2}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \left (- \frac {3 x \sqrt {4 x^{2} + 5 x + 2}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \frac {x^{2} \sqrt {4 x^{2} + 5 x + 2}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\, dx\right ) \] Input:
integrate((-2*x**2+6*x+4)*(4*x**2+5*x+2)**(1/2)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
2*(Integral(-2*sqrt(4*x**2 + 5*x + 2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1) ), x) + Integral(-3*x*sqrt(4*x**2 + 5*x + 2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2 *x + 1)), x) + Integral(x**2*sqrt(4*x**2 + 5*x + 2)/(3*x*sqrt(2*x + 1) - 5 *sqrt(2*x + 1)), x))
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 5 \, x + 2} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="maxima")
Output:
2*integrate(sqrt(4*x^2 + 5*x + 2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1) ), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 5 \, x + 2} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="giac")
Output:
integrate(2*sqrt(4*x^2 + 5*x + 2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1) ), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {\sqrt {4\,x^2+5\,x+2}\,\left (-2\,x^2+6\,x+4\right )}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((5*x + 4*x^2 + 2)^(1/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5 )),x)
Output:
int(-((5*x + 4*x^2 + 2)^(1/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5 )), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+5 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int \frac {\left (-2 x^{2}+6 x +4\right ) \sqrt {4 x^{2}+5 x +2}}{\left (5-3 x \right ) \sqrt {2 x +1}}d x \] Input:
int((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
int((-2*x^2+6*x+4)*(4*x^2+5*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x)