Integrand size = 41, antiderivative size = 354 \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {1}{270} (83-36 x) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}-\frac {173 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {-2-7 x-4 x^2} E\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{54 \sqrt {-1-2 x} \sqrt {2+7 x+4 x^2}}-\frac {52513 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{810 \sqrt {3+\sqrt {17}} \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}}+\frac {271168 \sqrt {-1-2 x} \sqrt {-2-7 x-4 x^2} \operatorname {EllipticPi}\left (-\frac {3 \left (51-61 \sqrt {17}\right )}{1784},\arcsin \left (\frac {\sqrt {17+\sqrt {17} (7+8 x)}}{\sqrt {34}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{81 \sqrt {3+\sqrt {17}} \left (61+3 \sqrt {17}\right ) \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Output:
-1/270*(83-36*x)*(1+2*x)^(1/2)*(4*x^2+7*x+2)^(1/2)-173/54*(3+17^(1/2))^(1/ 2)*(1+2*x)^(1/2)*(-4*x^2-7*x-2)^(1/2)*EllipticE(1/34*(17+17^(1/2)*(7+8*x)) ^(1/2)*34^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(-1-2*x)^(1/2)/(4*x^2+7*x+2)^(1 /2)-52513/810*(-1-2*x)^(1/2)*(-4*x^2-7*x-2)^(1/2)*EllipticF(1/34*(17+17^(1 /2)*(7+8*x))^(1/2)*34^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(3+17^(1/2))^(1/2)/ (1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2)+271168/81*(-1-2*x)^(1/2)*(-4*x^2-7*x-2)^ (1/2)*EllipticPi(1/34*(17+17^(1/2)*(7+8*x))^(1/2)*34^(1/2),-153/1784+183/1 784*17^(1/2),1/2*(17-3*17^(1/2))^(1/2))/(3+17^(1/2))^(1/2)/(61+3*17^(1/2)) /(1+2*x)^(1/2)/(4*x^2+7*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 23.31 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.24 \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {-141414-505089 x-312702 x^2-624 x^3+11232 x^4-33735 i \sqrt {3+\sqrt {17}} (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right )|-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )+\frac {3 i \left (32152+11245 \sqrt {17}\right ) (1+2 x)^{3/2} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),-\frac {13}{4}+\frac {3 \sqrt {17}}{4}\right )}{\sqrt {3+\sqrt {17}}}-\frac {677920 i \sqrt {1+2 x} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (-3+\sqrt {17}\right ),i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}}}-\frac {1355840 i x \sqrt {1+2 x} \sqrt {\frac {2+7 x+4 x^2}{(1+2 x)^2}} \operatorname {EllipticPi}\left (-\frac {13}{6} \left (-3+\sqrt {17}\right ),i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{-3+\sqrt {17}}}}{\sqrt {1+2 x}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}}}}{10530 \sqrt {1+2 x} \sqrt {2+7 x+4 x^2}} \] Input:
Integrate[((4 + 6*x - 2*x^2)*Sqrt[2 + 7*x + 4*x^2])/((5 - 3*x)*Sqrt[1 + 2* x]),x]
Output:
(-141414 - 505089*x - 312702*x^2 - 624*x^3 + 11232*x^4 - (33735*I)*Sqrt[3 + Sqrt[17]]*(1 + 2*x)^(3/2)*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*EllipticE[ I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], -13/4 + (3*Sqrt[17])/4] + ((3*I)*(32152 + 11245*Sqrt[17])*(1 + 2*x)^(3/2)*Sqrt[(2 + 7*x + 4*x^2)/( 1 + 2*x)^2]*EllipticF[I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], -1 3/4 + (3*Sqrt[17])/4])/Sqrt[3 + Sqrt[17]] - ((677920*I)*Sqrt[1 + 2*x]*Sqrt [(2 + 7*x + 4*x^2)/(1 + 2*x)^2]*EllipticPi[(-13*(-3 + Sqrt[17]))/6, I*ArcS inh[Sqrt[2/(-3 + Sqrt[17])]/Sqrt[1 + 2*x]], (-13 + 3*Sqrt[17])/4])/Sqrt[3 + Sqrt[17]] - ((1355840*I)*x*Sqrt[1 + 2*x]*Sqrt[(2 + 7*x + 4*x^2)/(1 + 2*x )^2]*EllipticPi[(-13*(-3 + Sqrt[17]))/6, I*ArcSinh[Sqrt[2/(-3 + Sqrt[17])] /Sqrt[1 + 2*x]], (-13 + 3*Sqrt[17])/4])/Sqrt[3 + Sqrt[17]])/(10530*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2])
Time = 1.92 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.69, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {2154, 1231, 27, 1269, 1172, 321, 327, 1274, 1269, 1172, 321, 327, 1279, 186, 25, 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.
| \(\displaystyle \int \frac {\left (-2 x^2+6 x+4\right ) \sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \sqrt {4 x^2+7 x+2}}{\sqrt {2 x+1}}dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {1}{120} \int \frac {2 (840 x+191)}{9 \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {76}{9} \int \frac {\sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{540} \int \frac {840 x+191}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {76}{9} \int \frac {\sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{540} \left (229 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-420 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+7 x+2}}dx\right )+\frac {76}{9} \int \frac {\sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}}dx+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \int \frac {1}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}} \sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )+\frac {76}{9} \int \frac {\sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \int \frac {\sqrt {4 x^2+7 x+2}}{(5-3 x) \sqrt {2 x+1}}dx+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 1274 |
| \(\displaystyle \frac {76}{9} \left (\frac {223}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-\frac {1}{9} \int \frac {12 x+41}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {76}{9} \left (\frac {223}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {1}{9} \left (-35 \int \frac {1}{\sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx-6 \int \frac {\sqrt {2 x+1}}{\sqrt {4 x^2+7 x+2}}dx\right )\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {76}{9} \left (\frac {223}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {1}{9} \left (-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \int \frac {1}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}} \sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {76}{9} \left (\frac {1}{9} \left (-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} \int \frac {\sqrt {1-\frac {8 x+\sqrt {17}+7}{3+\sqrt {17}}}}{\sqrt {1-\frac {8 x+\sqrt {17}+7}{2 \sqrt {17}}}}d\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}\right )+\frac {223}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {76}{9} \left (\frac {223}{9} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}dx+\frac {1}{9} \left (-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 1279 |
| \(\displaystyle \frac {76}{9} \left (\frac {223 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \int \frac {1}{(5-3 x) \sqrt {2 x+1} \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7}}dx}{9 \sqrt {4 x^2+7 x+2}}+\frac {1}{9} \left (-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {76}{9} \left (\frac {1}{9} \left (-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {446 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \int -\frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-\sqrt {17}+3} \sqrt {4 (2 x+1)+\sqrt {17}+3}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+7 x+2}}\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {76}{9} \left (\frac {446 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)-\sqrt {17}+3} \sqrt {4 (2 x+1)+\sqrt {17}+3}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+7 x+2}}+\frac {1}{9} \left (-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {76}{9} \left (\frac {446 \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \sqrt {\frac {4 (2 x+1)}{3-\sqrt {17}}+1} \int \frac {1}{(13-3 (2 x+1)) \sqrt {4 (2 x+1)+\sqrt {17}+3} \sqrt {\frac {4 (2 x+1)}{3-\sqrt {17}}+1}}d\sqrt {2 x+1}}{9 \sqrt {4 x^2+7 x+2} \sqrt {4 (2 x+1)-\sqrt {17}+3}}+\frac {1}{9} \left (-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )\right )+\frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {1}{540} \left (\frac {458 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {210 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )+\frac {76}{9} \left (\frac {223 \sqrt {13-3 \sqrt {17}} \sqrt {8 x-\sqrt {17}+7} \sqrt {8 x+\sqrt {17}+7} \sqrt {\frac {4 (2 x+1)}{3-\sqrt {17}}+1} \operatorname {EllipticPi}\left (-\frac {3}{52} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 \sqrt {2 x+1}}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{234 \sqrt {4 x^2+7 x+2} \sqrt {4 (2 x+1)-\sqrt {17}+3}}+\frac {1}{9} \left (-\frac {70 \sqrt {-2 x-1} \sqrt {-4 x^2-7 x-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right ),\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2}}-\frac {3 \sqrt {3+\sqrt {17}} \sqrt {2 x+1} \sqrt {-4 x^2-7 x-2} E\left (\arcsin \left (\frac {\sqrt {8 x+\sqrt {17}+7}}{\sqrt {2} \sqrt [4]{17}}\right )|\frac {1}{4} \left (17-3 \sqrt {17}\right )\right )}{\sqrt {-2 x-1} \sqrt {4 x^2+7 x+2}}\right )\right )-\frac {1}{270} \sqrt {2 x+1} \sqrt {4 x^2+7 x+2} (83-36 x)\) |
Input:
Int[((4 + 6*x - 2*x^2)*Sqrt[2 + 7*x + 4*x^2])/((5 - 3*x)*Sqrt[1 + 2*x]),x]
Output:
-1/270*((83 - 36*x)*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2]) + ((-210*Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]*Sqrt[-2 - 7*x - 4*x^2]*EllipticE[ArcSin[Sqrt[7 + Sqrt[17] + 8*x]/(Sqrt[2]*17^(1/4))], (17 - 3*Sqrt[17])/4])/(Sqrt[-1 - 2*x] *Sqrt[2 + 7*x + 4*x^2]) + (458*Sqrt[-1 - 2*x]*Sqrt[-2 - 7*x - 4*x^2]*Ellip ticF[ArcSin[Sqrt[7 + Sqrt[17] + 8*x]/(Sqrt[2]*17^(1/4))], (17 - 3*Sqrt[17] )/4])/(Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2]))/540 + (76* (((-3*Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]*Sqrt[-2 - 7*x - 4*x^2]*EllipticE[Ar cSin[Sqrt[7 + Sqrt[17] + 8*x]/(Sqrt[2]*17^(1/4))], (17 - 3*Sqrt[17])/4])/( Sqrt[-1 - 2*x]*Sqrt[2 + 7*x + 4*x^2]) - (70*Sqrt[-1 - 2*x]*Sqrt[-2 - 7*x - 4*x^2]*EllipticF[ArcSin[Sqrt[7 + Sqrt[17] + 8*x]/(Sqrt[2]*17^(1/4))], (17 - 3*Sqrt[17])/4])/(Sqrt[3 + Sqrt[17]]*Sqrt[1 + 2*x]*Sqrt[2 + 7*x + 4*x^2] ))/9 + (223*Sqrt[13 - 3*Sqrt[17]]*Sqrt[7 - Sqrt[17] + 8*x]*Sqrt[7 + Sqrt[1 7] + 8*x]*Sqrt[1 + (4*(1 + 2*x))/(3 - Sqrt[17])]*EllipticPi[(-3*(3 - Sqrt[ 17]))/52, ArcSin[(2*Sqrt[1 + 2*x])/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17] )/4])/(234*Sqrt[2 + 7*x + 4*x^2]*Sqrt[3 - Sqrt[17] + 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] && 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[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 = 0.72 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.40
| method | result | size |
| risch | \(\frac {\left (-83+36 x \right ) \sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}{270}+\frac {2 \left (-\frac {62893 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{1620 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}-\frac {346 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \left (\left (\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticE}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )+\left (-\frac {7}{8}+\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )\right )}{27 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {33896 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{1053 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}\right ) \sqrt {\left (4 x^{2}+7 x +2\right ) \left (1+2 x \right )}}{\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}\) | \(497\) |
| elliptic | \(\frac {\sqrt {\left (4 x^{2}+7 x +2\right ) \left (1+2 x \right )}\, \left (\frac {2 x \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{15}-\frac {83 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}{270}-\frac {62893 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{810 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}-\frac {692 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \left (\left (\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticE}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )+\left (-\frac {7}{8}+\frac {\sqrt {17}}{8}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )\right )}{27 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}+\frac {67792 \left (-\frac {3}{8}-\frac {\sqrt {17}}{8}\right ) \sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {-\frac {x +\frac {7}{8}-\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \sqrt {\frac {x +\frac {7}{8}+\frac {\sqrt {17}}{8}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {x +\frac {1}{2}}{\frac {3}{8}+\frac {\sqrt {17}}{8}}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, i \sqrt {\frac {\frac {3}{8}+\frac {\sqrt {17}}{8}}{-\frac {3}{8}+\frac {\sqrt {17}}{8}}}\right )}{1053 \sqrt {8 x^{3}+18 x^{2}+11 x +2}}\right )}{\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}}\) | \(508\) |
| default | \(\frac {\sqrt {4 x^{2}+7 x +2}\, \sqrt {1+2 x}\, \left (359424 \sqrt {17}\, x^{4}-19968 \sqrt {17}\, x^{3}+1078272 x^{4}+682669 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}-677920 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticPi}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right ) \sqrt {17}-1370304 \sqrt {17}\, x^{2}-59904 x^{3}+2317887 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticF}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )-2033760 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticPi}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, -\frac {9}{52}-\frac {3 \sqrt {17}}{52}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )-269880 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}\, \sqrt {\left (-8 x -7+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}\, \sqrt {\left (8 x +7+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}\, \operatorname {EllipticE}\left (\frac {2 \sqrt {-\left (1+2 x \right ) \left (3+\sqrt {17}\right )}}{3+\sqrt {17}}, \frac {i \sqrt {\left (3+\sqrt {17}\right ) \left (-3+\sqrt {17}\right )}}{-3+\sqrt {17}}\right )-1049568 \sqrt {17}\, x -4110912 x^{2}-207168 \sqrt {17}-3148704 x -621504\right )}{42120 \left (3+\sqrt {17}\right )^{2} \left (-3+\sqrt {17}\right ) \left (8 x^{3}+18 x^{2}+11 x +2\right )}\) | \(603\) |
Input:
int((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x,method=_RET URNVERBOSE)
Output:
1/270*(-83+36*x)*(4*x^2+7*x+2)^(1/2)*(1+2*x)^(1/2)+2*(-62893/1620*(-3/8-1/ 8*17^(1/2))*(-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2)*(-(x+7/8-1/8*17^(1/2))/(-3 /8+1/8*17^(1/2)))^(1/2)*((x+7/8+1/8*17^(1/2))/(3/8+1/8*17^(1/2)))^(1/2)/(8 *x^3+18*x^2+11*x+2)^(1/2)*EllipticF((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2),I* ((3/8+1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2))-346/27*(-3/8-1/8*17^(1/2)) *(-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2)*(-(x+7/8-1/8*17^(1/2))/(-3/8+1/8*17^( 1/2)))^(1/2)*((x+7/8+1/8*17^(1/2))/(3/8+1/8*17^(1/2)))^(1/2)/(8*x^3+18*x^2 +11*x+2)^(1/2)*((3/8-1/8*17^(1/2))*EllipticE((-(x+1/2)/(3/8+1/8*17^(1/2))) ^(1/2),I*((3/8+1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2))+(-7/8+1/8*17^(1/2 ))*EllipticF((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2),I*((3/8+1/8*17^(1/2))/(-3 /8+1/8*17^(1/2)))^(1/2)))+33896/1053*(-3/8-1/8*17^(1/2))*(-(x+1/2)/(3/8+1/ 8*17^(1/2)))^(1/2)*(-(x+7/8-1/8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)*((x+7 /8+1/8*17^(1/2))/(3/8+1/8*17^(1/2)))^(1/2)/(8*x^3+18*x^2+11*x+2)^(1/2)*Ell ipticPi((-(x+1/2)/(3/8+1/8*17^(1/2)))^(1/2),-9/52-3/52*17^(1/2),I*((3/8+1/ 8*17^(1/2))/(-3/8+1/8*17^(1/2)))^(1/2)))*((4*x^2+7*x+2)*(1+2*x))^(1/2)/(4* x^2+7*x+2)^(1/2)/(1+2*x)^(1/2)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 7 \, x + 2} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="fricas")
Output:
integral(2*sqrt(4*x^2 + 7*x + 2)*(x^2 - 3*x - 2)*sqrt(2*x + 1)/(6*x^2 - 7* x - 5), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=2 \left (\int \left (- \frac {2 \sqrt {4 x^{2} + 7 x + 2}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \left (- \frac {3 x \sqrt {4 x^{2} + 7 x + 2}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\right )\, dx + \int \frac {x^{2} \sqrt {4 x^{2} + 7 x + 2}}{3 x \sqrt {2 x + 1} - 5 \sqrt {2 x + 1}}\, dx\right ) \] Input:
integrate((-2*x**2+6*x+4)*(4*x**2+7*x+2)**(1/2)/(5-3*x)/(1+2*x)**(1/2),x)
Output:
2*(Integral(-2*sqrt(4*x**2 + 7*x + 2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2*x + 1) ), x) + Integral(-3*x*sqrt(4*x**2 + 7*x + 2)/(3*x*sqrt(2*x + 1) - 5*sqrt(2 *x + 1)), x) + Integral(x**2*sqrt(4*x**2 + 7*x + 2)/(3*x*sqrt(2*x + 1) - 5 *sqrt(2*x + 1)), x))
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 7 \, x + 2} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="maxima")
Output:
2*integrate(sqrt(4*x^2 + 7*x + 2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1) ), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, \sqrt {4 \, x^{2} + 7 \, x + 2} {\left (x^{2} - 3 \, x - 2\right )}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x, algo rithm="giac")
Output:
integrate(2*sqrt(4*x^2 + 7*x + 2)*(x^2 - 3*x - 2)/((3*x - 5)*sqrt(2*x + 1) ), x)
Timed out. \[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {\left (-2\,x^2+6\,x+4\right )\,\sqrt {4\,x^2+7\,x+2}}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:
int(-((6*x - 2*x^2 + 4)*(7*x + 4*x^2 + 2)^(1/2))/((2*x + 1)^(1/2)*(3*x - 5 )),x)
Output:
int(-((6*x - 2*x^2 + 4)*(7*x + 4*x^2 + 2)^(1/2))/((2*x + 1)^(1/2)*(3*x - 5 )), x)
\[ \int \frac {\left (4+6 x-2 x^2\right ) \sqrt {2+7 x+4 x^2}}{(5-3 x) \sqrt {1+2 x}} \, dx=\int \frac {\left (-2 x^{2}+6 x +4\right ) \sqrt {4 x^{2}+7 x +2}}{\left (5-3 x \right ) \sqrt {2 x +1}}d x \] Input:
int((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x)
Output:
int((-2*x^2+6*x+4)*(4*x^2+7*x+2)^(1/2)/(5-3*x)/(1+2*x)^(1/2),x)