Integrand size = 35, antiderivative size = 505 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2-\frac {i c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]
1/4*b^2*c^2*d*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-3/2*c^2*d*e*x*(a+b*arcs in(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-d*e*(-c^2*x^2+1)*(a+b*arcsin(c *x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/x-5/4*b^2*c*d*e*arcsin(c*x)*(c*d*x +d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+3/2*b*c^3*d*e*x^2*(a+b*arcsi n(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-I*c*d*e*(a+b*a rcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-1/2*c*d* e*(a+b*arcsin(c*x))^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/b/(-c^2*x^2+1)^(1/2 )+2*b*c*d*e*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(c*d*x+d) ^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-I*b^2*c*d*e*polylog(2,(I*c*x+(- c^2*x^2+1)^(1/2))^2)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+b *c*d*e*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)^(1/ 2)
Time = 3.07 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.07 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {-8 a^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}-4 a^2 c^2 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}-4 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3+12 a^2 c d^{3/2} e^{3/2} x \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x} \cos (2 \arcsin (c x))+16 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x} \log (c x)-8 i b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \sin (2 \arcsin (c x))-2 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) \left (8 a \sqrt {1-c^2 x^2}+b c x \cos (2 \arcsin (c x))-8 b c x \log \left (1-e^{2 i \arcsin (c x)}\right )+2 a c x \sin (2 \arcsin (c x))\right )-2 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 \left (6 a c x+4 i b c x+4 b \sqrt {1-c^2 x^2}+b c x \sin (2 \arcsin (c x))\right )}{8 x \sqrt {1-c^2 x^2}} \]
(-8*a^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - 4*a^2*c^2* d*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - 4*b^2*c*d*e*x* Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 + 12*a^2*c*d^(3/2)*e^(3/2)*x *Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*S qrt[e]*(-1 + c^2*x^2))] - 2*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Co s[2*ArcSin[c*x]] + 16*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Log[c*x] - (8*I)*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, E^((2*I)*A rcSin[c*x])] + b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sin[2*ArcSin[c* x]] - 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(8*a*Sqrt[1 - c^ 2*x^2] + b*c*x*Cos[2*ArcSin[c*x]] - 8*b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*a*c*x*Sin[2*ArcSin[c*x]]) - 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*A rcSin[c*x]^2*(6*a*c*x + (4*I)*b*c*x + 4*b*Sqrt[1 - c^2*x^2] + b*c*x*Sin[2* ArcSin[c*x]]))/(8*x*Sqrt[1 - c^2*x^2])
Time = 1.92 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.63, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {5238, 5200, 5156, 5138, 262, 223, 5152, 5188, 211, 223, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 5238 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^2}dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-3 c^2 \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx+2 b c \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-3 c^2 \left (\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-b c \int x (a+b \arcsin (c x))dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+2 b c \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-3 c^2 \left (-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+2 b c \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-3 c^2 \left (-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+2 b c \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-3 c^2 \left (\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5188 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \int \sqrt {1-c^2 x^2}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (\int \frac {a+b \arcsin (c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (\int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (-\int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (2 b c \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x}-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )}{\sqrt {1-c^2 x^2}}\) |
(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-(((1 - c^2*x^2)^(3/2)*(a + b*ArcSin [c*x])^2)/x) - 3*c^2*((x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (a + b*ArcSin[c*x])^3/(6*b*c) - b*c*((x^2*(a + b*ArcSin[c*x]))/2 - (b*c*(-1/2* (x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)) + 2*b*c*(((1 - c^2*x ^2)*(a + b*ArcSin[c*x]))/2 - ((I/2)*(a + b*ArcSin[c*x])^2)/b - (b*c*((x*Sq rt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/2 - (2*I)*((I/2)*(a + b*ArcSin[c* x])*Log[1 - E^((2*I)*ArcSin[c*x])] + (b*PolyLog[2, E^((2*I)*ArcSin[c*x])]) /4))))/Sqrt[1 - c^2*x^2]
3.6.85.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^ m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcSin[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcSin[c*x])/x), x], x] - Simp[b*c*(d^p/(2 *p)) Int[(1 - c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2*x^2) ^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f} , x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar t[q]) Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n , x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] & & EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
\[\int \frac {\left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{x^{2}}d x\]
\[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
integral(-(a^2*c^2*d*e*x^2 - a^2*d*e + (b^2*c^2*d*e*x^2 - b^2*d*e)*arcsin( c*x)^2 + 2*(a*b*c^2*d*e*x^2 - a*b*d*e)*arcsin(c*x))*sqrt(c*d*x + d)*sqrt(- c*e*x + e)/x^2, x)
Timed out. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}}{x^2} \,d x \]