Integrand size = 29, antiderivative size = 126 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \arcsin (c x))}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 b c \sqrt {d-c^2 d x^2}} \]
-g*(-c^2*x^2+1)*(a+b*arcsin(c*x))/c^2/(-c^2*d*x^2+d)^(1/2)+b*g*x*(-c^2*x^2 +1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+1/2*f*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1 /2)/b/c/(-c^2*d*x^2+d)^(1/2)
Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.37 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 \sqrt {d} g \left (-a+a c^2 x^2+b c x \sqrt {1-c^2 x^2}\right )+2 b \sqrt {d} g \left (-1+c^2 x^2\right ) \arcsin (c x)+b c \sqrt {d} f \sqrt {1-c^2 x^2} \arcsin (c x)^2-2 a c f \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{2 c^2 \sqrt {d} \sqrt {d-c^2 d x^2}} \]
(2*Sqrt[d]*g*(-a + a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2]) + 2*b*Sqrt[d]*g*(- 1 + c^2*x^2)*ArcSin[c*x] + b*c*Sqrt[d]*f*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - 2*a*c*f*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*c^2*Sqrt[d]*Sqrt[d - c^2*d*x^2])
Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5276, 5262, 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.
\(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 5276 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 5262 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \left (\frac {f (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {g x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}\right )dx}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}+\frac {f (a+b \arcsin (c x))^2}{2 b c}+\frac {b g x}{c}\right )}{\sqrt {d-c^2 d x^2}}\) |
(Sqrt[1 - c^2*x^2]*((b*g*x)/c - (g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/ c^2 + (f*(a + b*ArcSin[c*x])^2)/(2*b*c)))/Sqrt[d - c^2*d*x^2]
3.1.46.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & & EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ [n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ p] Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ [{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege rQ[p - 1/2] && !GtQ[d, 0]
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.96
method | result | size |
default | \(\frac {a f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(247\) |
parts | \(\frac {a f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(247\) |
a*f/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-a*g/c^2/d*( -c^2*d*x^2+d)^(1/2)+b*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d/ (c^2*x^2-1)*arcsin(c*x)^2*f-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)*g*(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)*g*(arcsin(c*x)-I)/c^2/d/(c^2*x ^2-1))
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(a*g*x + a*f + (b*g*x + b*f)*arcsin(c*x))/( c^2*d*x^2 - d), x)
Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b f \arcsin \left (c x\right )^{2}}{2 \, c \sqrt {d}} + \frac {b g x}{c \sqrt {d}} + \frac {a f \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} + d} b g \arcsin \left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a g}{c^{2} d} \]
1/2*b*f*arcsin(c*x)^2/(c*sqrt(d)) + b*g*x/(c*sqrt(d)) + a*f*arcsin(c*x)/(c *sqrt(d)) - sqrt(-c^2*d*x^2 + d)*b*g*arcsin(c*x)/(c^2*d) - sqrt(-c^2*d*x^2 + d)*a*g/(c^2*d)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]