Integrand size = 27, antiderivative size = 146 \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d} \]
2*b^2*(-c^2*x^2+1)/c^2/(-c^2*d*x^2+d)^(1/2)+2*a*b*x*(-c^2*x^2+1)^(1/2)/c/( -c^2*d*x^2+d)^(1/2)+2*b^2*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d )^(1/2)-(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2/d
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.59 \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\left (-1+c^2 x^2\right ) (a+b \arcsin (c x))^2+2 b \sqrt {1-c^2 x^2} \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )}{c^2 \sqrt {d-c^2 d x^2}} \]
((-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2 + 2*b*Sqrt[1 - c^2*x^2]*(a*c*x + b*S qrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]))/(c^2*Sqrt[d - c^2*d*x^2])
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {5182, 2009}
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.
\(\displaystyle \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\) |
-((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d)) + (2*b*Sqrt[1 - c^2 *x^2]*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/(c*Sqrt[d - c^2*d *x^2])
3.3.38.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.16
method | result | size |
default | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(316\) |
parts | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(316\) |
-a^2/c^2/d*(-c^2*d*x^2+d)^(1/2)+b^2*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2- I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))/c^2/d/(c^2*x ^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arc sin(c*x)^2-2-2*I*arcsin(c*x))/c^2/d/(c^2*x^2-1))+2*a*b*(-1/2*(-d*(c^2*x^2- 1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)/c^2/d/(c^2* x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(ar csin(c*x)-I)/c^2/d/(c^2*x^2-1))
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {2 \, \sqrt {-c^{2} d x^{2} + d} {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1} + {\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \]
-(2*sqrt(-c^2*d*x^2 + d)*(b^2*c*x*arcsin(c*x) + a*b*c*x)*sqrt(-c^2*x^2 + 1 ) + ((a^2 - 2*b^2)*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arcsin(c*x)^2 - a^2 + 2*b ^2 + 2*(a*b*c^2*x^2 - a*b)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^4*d*x^2 - c^2*d)
Exception generated. \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.89 \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=2 \, b^{2} {\left (\frac {x \arcsin \left (c x\right )}{c \sqrt {d}} + \frac {\sqrt {-c^{2} x^{2} + 1}}{c^{2} \sqrt {d}}\right )} + \frac {2 \, a b x}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} \arcsin \left (c x\right )^{2}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b \arcsin \left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2}}{c^{2} d} \]
2*b^2*(x*arcsin(c*x)/(c*sqrt(d)) + sqrt(-c^2*x^2 + 1)/(c^2*sqrt(d))) + 2*a *b*x/(c*sqrt(d)) - sqrt(-c^2*d*x^2 + d)*b^2*arcsin(c*x)^2/(c^2*d) - 2*sqrt (-c^2*d*x^2 + d)*a*b*arcsin(c*x)/(c^2*d) - sqrt(-c^2*d*x^2 + d)*a^2/(c^2*d )
\[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]