Integrand size = 31, antiderivative size = 396 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {4 b^2 g \sqrt {d-c^2 d x^2}}{9 c^2}-\frac {1}{4} b^2 f x \sqrt {d-c^2 d x^2}+\frac {2 b^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {b^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {2 b g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}-\frac {2 b c g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}} \]
4/9*b^2*g*(-c^2*d*x^2+d)^(1/2)/c^2-1/4*b^2*f*x*(-c^2*d*x^2+d)^(1/2)+2/27*b ^2*g*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+1/2*f*x*(a+b*arcsin(c*x))^2*(-c ^2*d*x^2+d)^(1/2)-1/3*g*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1 /2)/c^2+1/4*b^2*f*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+2/ 3*b*g*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/2*b* c*f*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/9*b*c* g*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/6*f*(a+b *arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)
Time = 0.18 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.57 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (54 f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {36 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c^2}+\frac {18 f (a+b \arcsin (c x))^3}{b c}-\frac {8 b g \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )+3 b c x \left (-3+c^2 x^2\right ) \arcsin (c x)\right )}{c^2}-\frac {27 b f \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )}{c}\right )}{108 \sqrt {1-c^2 x^2}} \]
(Sqrt[d - c^2*d*x^2]*(54*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - (36 *g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/c^2 + (18*f*(a + b*ArcSin[c* x])^3)/(b*c) - (8*b*g*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2) + 3*b*c*x*(-3 + c^2*x^2)*ArcSin[c*x]))/c^2 - (27*b*f*(c*x*(2*a*c* x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/c))/(108*Sqrt[ 1 - c^2*x^2])
Time = 0.70 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.65, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, 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.
\(\displaystyle \int \sqrt {d-c^2 d x^2} (f+g x) (a+b \arcsin (c x))^2 \, dx\) |
\(\Big \downarrow \) 5276 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (f+g x) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5262 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )dx}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {1}{2} f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}-\frac {1}{2} b c f x^2 (a+b \arcsin (c x))+\frac {f (a+b \arcsin (c x))^3}{6 b c}-\frac {2}{9} b c g x^3 (a+b \arcsin (c x))+\frac {2 b g x (a+b \arcsin (c x))}{3 c}+\frac {b^2 f \arcsin (c x)}{4 c}-\frac {1}{4} b^2 f x \sqrt {1-c^2 x^2}+\frac {2 b^2 g \left (1-c^2 x^2\right )^{3/2}}{27 c^2}+\frac {4 b^2 g \sqrt {1-c^2 x^2}}{9 c^2}\right )}{\sqrt {1-c^2 x^2}}\) |
(Sqrt[d - c^2*d*x^2]*((4*b^2*g*Sqrt[1 - c^2*x^2])/(9*c^2) - (b^2*f*x*Sqrt[ 1 - c^2*x^2])/4 + (2*b^2*g*(1 - c^2*x^2)^(3/2))/(27*c^2) + (b^2*f*ArcSin[c *x])/(4*c) + (2*b*g*x*(a + b*ArcSin[c*x]))/(3*c) - (b*c*f*x^2*(a + b*ArcSi n[c*x]))/2 - (2*b*c*g*x^3*(a + b*ArcSin[c*x]))/9 + (f*x*Sqrt[1 - c^2*x^2]* (a + b*ArcSin[c*x])^2)/2 - (g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/( 3*c^2) + (f*(a + b*ArcSin[c*x])^3)/(6*b*c)))/Sqrt[1 - c^2*x^2]
3.1.60.3.1 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 = 0.65 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.12
method | result | size |
default | \(\text {Expression too large to display}\) | \(1236\) |
parts | \(\text {Expression too large to display}\) | \(1236\) |
1/2*a^2*f*x*(-c^2*d*x^2+d)^(1/2)+1/2*a^2*f*d/(c^2*d)^(1/2)*arctan((c^2*d)^ (1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/3*a^2*g*(-c^2*d*x^2+d)^(3/2)/c^2/d+b^2*(-1 /6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^3*f +1/216*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1 )^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*g*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2 )/c^2/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2 *c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(2*I*arcsin(c*x)+2*arcsin(c*x )^2-1)/c/(c^2*x^2-1)-1/8*(-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)^2-2+2*I*arcsin(c*x))/c^2/(c^2*x^2-1)-1/8*(-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)^2-2-2* I*arcsin(c*x))/c^2/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+ 1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(-2*I*arcsin(c*x) +2*arcsin(c*x)^2-1)/c/(c^2*x^2-1)+1/216*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^ 3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*g*( -6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^2/(c^2*x^2-1))+2*a*b*(-1/4*(-d*(c^2* x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*f+1/72*(-d*(c ^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*( -c^2*x^2+1)^(1/2)*x*c+1)*g*(I+3*arcsin(c*x))/c^2/(c^2*x^2-1)+1/16*(-d*(c^2 *x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^( 1/2)-2*c*x)*f*(I+2*arcsin(c*x))/c/(c^2*x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2...
\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(a^2*g*x + a^2*f + (b^2*g*x + b^2*f)*arcsin( c*x)^2 + 2*(a*b*g*x + a*b*f)*arcsin(c*x)), x)
\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )\, dx \]
\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a^2*f - 1/3*(-c^2*d*x ^2 + d)^(3/2)*a^2*g/(c^2*d) + sqrt(d)*integrate(((b^2*g*x + b^2*f)*arctan2 (c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g*x + a*b*f)*arctan2(c*x, s qrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
Exception generated. \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int \left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]