Integrand size = 41, antiderivative size = 884 \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (-c_1 c_2 c_4+c_0 c_3 c_4+c_1 c_2 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5-c_0 c_3 \sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right )}{\left (c_1 c_6-\frac {(c_0+x c_1) c_3 c_6}{c_2+x c_3}-c_0 c_7+\frac {(c_0+x c_1) c_2 c_7}{c_2+x c_3}\right ) \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right )}+\frac {\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \left (-c_1 c_2 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_1 c_2 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}-c_0 c_3 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right ) \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}+\frac {\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt {c_3 c_6-c_2 c_7}}{\sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}}\right ) \left (-c_1 c_2 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt {c_1 c_6-c_0 c_7}-c_1 c_2 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}+c_0 c_3 c_4 c_5 \sqrt {c_3 c_6-c_2 c_7}\right )}{2 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} \left (-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right ) \sqrt {-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7}}} \]
(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-_C1*_C2*_C4+_C0*_C3*_C4+ _C1*_C2*((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5-_C0*_C3*((_C1*x+_C0)/(_C3*x+_C 2))^(1/2)*_C5)/(_C1*_C6-(_C1*x+_C0)*_C3*_C6/(_C3*x+_C2)-_C0*_C7+(_C1*x+_C0 )*_C2*_C7/(_C3*x+_C2))/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^2*_C7-_C3*_C4 ^2*_C6)+1/2*arctan((_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)*(-_C2*_ C7+_C3*_C6)^(1/2)/(-_C3*_C4*_C6+_C2*_C4*_C7-_C5*(-_C0*_C7+_C1*_C6)^(1/2)*( -_C2*_C7+_C3*_C6)^(1/2))^(1/2))*(-_C1*_C2*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)+_ C0*_C3*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)+_C1*_C2*_C4*_C5*(-_C2*_C7+_C3*_C6)^( 1/2)-_C0*_C3*_C4*_C5*(-_C2*_C7+_C3*_C6)^(1/2))/(-_C0*_C7+_C1*_C6)^(1/2)/(- _C2*_C7+_C3*_C6)^(1/2)/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^2*_C7-_C3*_C4 ^2*_C6)/(-_C3*_C4*_C6+_C2*_C4*_C7-_C5*(-_C0*_C7+_C1*_C6)^(1/2)*(-_C2*_C7+_ C3*_C6)^(1/2))^(1/2)+1/2*arctan((_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^ (1/2)*(-_C2*_C7+_C3*_C6)^(1/2)/(-_C3*_C4*_C6+_C2*_C4*_C7+_C5*(-_C0*_C7+_C1 *_C6)^(1/2)*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2))*(-_C1*_C2*_C5^2*(-_C0*_C7+_C1 *_C6)^(1/2)+_C0*_C3*_C5^2*(-_C0*_C7+_C1*_C6)^(1/2)-_C1*_C2*_C4*_C5*(-_C2*_ C7+_C3*_C6)^(1/2)+_C0*_C3*_C4*_C5*(-_C2*_C7+_C3*_C6)^(1/2))/(-_C0*_C7+_C1* _C6)^(1/2)/(-_C2*_C7+_C3*_C6)^(1/2)/(-_C0*_C5^2*_C7+_C1*_C5^2*_C6+_C2*_C4^ 2*_C7-_C3*_C4^2*_C6)/(-_C3*_C4*_C6+_C2*_C4*_C7+_C5*(-_C0*_C7+_C1*_C6)^(1/2 )*(-_C2*_C7+_C3*_C6)^(1/2))^(1/2)
Time = 1.66 (sec) , antiderivative size = 821, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\frac {(c_1 c_2-c_0 c_3) \left (\frac {(c_2+x c_3) \left (c_4-\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_1 c_6-c_0 c_7) (-c_3 c_6+c_2 c_7)}{(c_1 c_2-c_0 c_3) (c_6+x c_7)}+c_4 c_5 \sqrt {c_1 c_6-c_0 c_7} (c_3 c_6-c_2 c_7){}^{3/4} \left (-\frac {\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}-\frac {\text {arctanh}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right )-\frac {1}{2} c_5 \sqrt {c_1 c_6-c_0 c_7} \sqrt [4]{c_3 c_6-c_2 c_7} \left (\arctan \left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}-c_4 \sqrt {c_3 c_6-c_2 c_7}}-\text {arctanh}\left (\frac {\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \sqrt [4]{c_3 c_6-c_2 c_7}}{\sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}}\right ) \sqrt {c_5 \sqrt {c_1 c_6-c_0 c_7}+c_4 \sqrt {c_3 c_6-c_2 c_7}}\right )\right )}{(-c_1 c_6+c_0 c_7) (c_3 c_6-c_2 c_7) \left (c_5{}^2 (-c_1 c_6+c_0 c_7)+c_4{}^2 (c_3 c_6-c_2 c_7)\right )} \]
((C[1]*C[2] - C[0]*C[3])*(((C[2] + x*C[3])*(C[4] - Sqrt[(C[0] + x*C[1])/(C [2] + x*C[3])]*C[5])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5 ]]*(C[1]*C[6] - C[0]*C[7])*(-(C[3]*C[6]) + C[2]*C[7]))/((C[1]*C[2] - C[0]* C[3])*(C[6] + x*C[7])) + C[4]*C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*(C[3]*C[6] - C[2]*C[7])^(3/4)*(-(ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C [3])]*C[5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0] *C[7]] - C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0 ]*C[7]] - C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) - ArcTanh[(Sqrt[C[4] + Sqrt[( C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[ C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]/Sqrt [C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) - ( C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*(C[3]*C[6] - C[2]*C[7])^(1/4)*(ArcTan[(Sq rt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[3]*C[6] - C[2]*C[ 7])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] - C[4]*Sqrt[C[3]*C[6] - C [2]*C[7]]]]*Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] - C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]] - ArcTanh[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C [5]]*(C[3]*C[6] - C[2]*C[7])^(1/4))/Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]]]))/2))/((-(C[1]*C[6]) + C[0]*C[7])*(C[ 3]*C[6] - C[2]*C[7])*(C[5]^2*(-(C[1]*C[6]) + C[0]*C[7]) + C[4]^2*(C[3]*...
Time = 1.44 (sec) , antiderivative size = 732, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {7268, 561, 27, 1492, 27, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {1}{\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4} (c_7 x+c_6){}^2} \, dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle 2 (c_1 c_2-c_0 c_3) \int \frac {\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} \left (c_1 c_6-c_0 c_7-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{c_2+x c_3}\right ){}^2}d\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \int \frac {\frac {c_0+x c_1}{c_2+x c_3}-c_4}{c_5 \left (-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right ){}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \int \frac {\frac {c_0+x c_1}{c_2+x c_3}-c_4}{\left (-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right ){}^2}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_5{}^2}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (\frac {c_5{}^4 \int -\frac {2 (c_1 c_6-c_0 c_7) \left (\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{c_2+x c_3}-2 c_4 (c_3 c_6-c_2 c_7)\right )}{c_5{}^2 \left (-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{8 (c_1 c_6-c_0 c_7) (c_3 c_6-c_2 c_7) \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \int \frac {\left (\frac {c_0+x c_1}{c_2+x c_3}-2 c_4\right ) (c_3 c_6-c_2 c_7)}{-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{4 (c_3 c_6-c_2 c_7) \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \int \frac {\frac {c_0+x c_1}{c_2+x c_3}-2 c_4}{-\frac {(c_3 c_6-c_2 c_7) (c_0+x c_1){}^2}{(c_2+x c_3){}^2 c_5{}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_0+x c_1)}{(c_2+x c_3) c_5{}^2}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \left (\frac {1}{2} \left (1+\frac {\sqrt {c_3 c_6-c_2 c_7} c_4}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) \int \frac {1}{\frac {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}{c_5{}^2}-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{(c_2+x c_3) c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}+\frac {1}{2} \left (1-\frac {c_4 \sqrt {c_3 c_6-c_2 c_7}}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) \int \frac {1}{\frac {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}{c_5{}^2}-\frac {(c_0+x c_1) (c_3 c_6-c_2 c_7)}{(c_2+x c_3) c_5{}^2}}d\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 (c_1 c_2-c_0 c_3) \left (-\frac {c_5{}^2 \left (\frac {\left (1+\frac {\sqrt {c_3 c_6-c_2 c_7} c_4}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) c_5{}^2 \text {arctanh}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{2 \sqrt {c_3 c_6-c_2 c_7} \sqrt {-\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}+\frac {\left (1-\frac {c_4 \sqrt {c_3 c_6-c_2 c_7}}{c_5 \sqrt {c_1 c_6-c_0 c_7}}\right ) c_5{}^2 \text {arctanh}\left (\frac {\sqrt {c_3 c_6-c_2 c_7} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{2 \sqrt {c_3 c_6-c_2 c_7} \sqrt {\sqrt {c_1 c_6-c_0 c_7} \sqrt {c_3 c_6-c_2 c_7} c_5+c_3 c_4 c_6-c_2 c_4 c_7}}\right )}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right )}-\frac {c_5{}^2 \left (\frac {c_1 x+c_0}{c_3 x+c_2}-2 c_4\right ) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 \left (c_3 c_6 c_4{}^2-c_1 c_5{}^2 c_6-\left (c_2 c_4{}^2-c_0 c_5{}^2\right ) c_7\right ) \left (-\frac {(c_3 c_6-c_2 c_7) (c_1 x+c_0){}^2}{c_5{}^2 (c_3 x+c_2){}^2}+\frac {2 c_4 (c_3 c_6-c_2 c_7) (c_1 x+c_0)}{c_5{}^2 (c_3 x+c_2)}+c_1 c_6-c_0 c_7-\frac {c_4{}^2 (c_3 c_6-c_2 c_7)}{c_5{}^2}\right )}\right )}{c_5{}^2}\) |
(4*(C[1]*C[2] - C[0]*C[3])*(-1/4*(((C[0] + x*C[1])/(C[2] + x*C[3]) - 2*C[4 ])*C[5]^2*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/((C[3]* C[4]^2*C[6] - C[1]*C[5]^2*C[6] - (C[2]*C[4]^2 - C[0]*C[5]^2)*C[7])*(C[1]*C [6] - C[0]*C[7] - ((C[0] + x*C[1])^2*(C[3]*C[6] - C[2]*C[7]))/((C[2] + x*C [3])^2*C[5]^2) + (2*(C[0] + x*C[1])*C[4]*(C[3]*C[6] - C[2]*C[7]))/((C[2] + x*C[3])*C[5]^2) - (C[4]^2*(C[3]*C[6] - C[2]*C[7]))/C[5]^2)) - (C[5]^2*((A rcTanh[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]* C[6] - C[2]*C[7]])/Sqrt[C[3]*C[4]*C[6] - C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C [6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*C[5]^2*(1 + (C[4]*Sqrt[C[3] *C[6] - C[2]*C[7]])/(C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]])))/(2*Sqrt[C[3]*C[6] - C[2]*C[7]]*Sqrt[C[3]*C[4]*C[6] - C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) + (ArcTanh[(Sqrt[C[4] + Sqrt[(C[ 0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[C[3] *C[4]*C[6] - C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C [6] - C[2]*C[7]]]]*C[5]^2*(1 - (C[4]*Sqrt[C[3]*C[6] - C[2]*C[7]])/(C[5]*Sq rt[C[1]*C[6] - C[0]*C[7]])))/(2*Sqrt[C[3]*C[6] - C[2]*C[7]]*Sqrt[C[3]*C[4] *C[6] - C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]])))/(4*(C[3]*C[4]^2*C[6] - C[1]*C[5]^2*C[6] - (C[2]*C[4]^2 - C [0]*C[5]^2)*C[7]))))/C[5]^2
3.32.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
\[\int \frac {1}{\sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}\, \left (\textit {\_C7} x +\textit {\_C6} \right )^{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 56059 vs. \(2 (579) = 1158\).
Time = 45.16 (sec) , antiderivative size = 56059, normalized size of antiderivative = 63.42 \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Too large to display} \]
integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2, x, algorithm="fricas")
Timed out. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Exception raised: RuntimeError} \]
integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2, x, algorithm="maxima")
Exception generated. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\text {Exception raised: TypeError} \]
integrate(1/(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(_C7*x+_C6)^2, x, algorithm="giac")
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Timed out. \[ \int \frac {1}{\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx=\int \frac {1}{\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}\,{\left (_{\mathrm {C6}}+_{\mathrm {C7}}\,x\right )}^2} \,d x \]