Integrand size = 33, antiderivative size = 737 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {8 b^2 f g \sqrt {d-c^2 d x^2}}{9 c^2}-\frac {1}{4} b^2 f^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 g^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {4 b^2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {b^2 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b^2 g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {b g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {4 b c f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}} \]
8/9*b^2*f*g*(-c^2*d*x^2+d)^(1/2)/c^2-1/4*b^2*f^2*x*(-c^2*d*x^2+d)^(1/2)+1/ 64*b^2*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2-1/32*b^2*g^2*x^3*(-c^2*d*x^2+d)^(1/2 )+4/27*b^2*f*g*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+1/2*f^2*x*(a+b*arcsin (c*x))^2*(-c^2*d*x^2+d)^(1/2)-1/8*g^2*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d) ^(1/2)/c^2+1/4*g^2*x^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)-2/3*f*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^2*arcsin (c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/64*b^2*g^2*arcsin(c*x)*( -c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+4/3*b*f*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^2*x^2*(a+b*arcsin(c*x)) *(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/8*b*g^2*x^2*(a+b*arcsin(c*x))*( -c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-4/9*b*c*f*g*x^3*(a+b*arcsin(c*x)) *(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/8*b*c*g^2*x^4*(a+b*arcsin(c*x)) *(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/6*f^2*(a+b*arcsin(c*x))^3*(-c^2 *d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)+1/24*g^2*(a+b*arcsin(c*x))^3*(-c^2* d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)
Time = 0.59 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.60 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\frac {1}{2} f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {1}{4} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {f^2 (a+b \arcsin (c x))^3}{6 b c}-\frac {4 b f g \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )+3 b c x \left (-3+c^2 x^2\right ) \arcsin (c x)\right )}{27 c^2}-\frac {b f^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )}{4 c}-\frac {b g^2 \left (8 a c^4 x^4+b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+b \left (-3+8 c^4 x^4\right ) \arcsin (c x)\right )}{64 c^3}-\frac {g^2 \left (6 b c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 (a+b \arcsin (c x))^3-3 b^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )\right )}{48 b c^3}\right )}{\sqrt {1-c^2 x^2}} \]
(Sqrt[d - c^2*d*x^2]*((f^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/4 - (2*f*g*(1 - c^2*x^2) ^(3/2)*(a + b*ArcSin[c*x])^2)/(3*c^2) + (f^2*(a + b*ArcSin[c*x])^3)/(6*b*c ) - (4*b*f*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]))/(27*c^2) - (b*f^2*(c*x*(2*a*c*x + b *Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(4*c) - (b*g^2*(8*a *c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + b*(-3 + 8*c^4*x^4)*Ar cSin[c*x]))/(64*c^3) - (g^2*(6*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) ^2 - 2*(a + b*ArcSin[c*x])^3 - 3*b^2*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x])))/(48*b*c^3)))/Sqrt[1 - c^2*x^2]
Time = 1.14 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.65, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \sqrt {d-c^2 d x^2} (f+g x)^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (f+g x)^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+2 f g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {g^2 (a+b \arcsin (c x))^3}{24 b c^3}+\frac {1}{2} f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {2 f g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}-\frac {g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{2} b c f^2 x^2 (a+b \arcsin (c x))+\frac {f^2 (a+b \arcsin (c x))^3}{6 b c}-\frac {4}{9} b c f g x^3 (a+b \arcsin (c x))+\frac {4 b f g x (a+b \arcsin (c x))}{3 c}-\frac {1}{8} b c g^2 x^4 (a+b \arcsin (c x))+\frac {b g^2 x^2 (a+b \arcsin (c x))}{8 c}-\frac {b^2 g^2 \arcsin (c x)}{64 c^3}+\frac {b^2 f^2 \arcsin (c x)}{4 c}-\frac {1}{4} b^2 f^2 x \sqrt {1-c^2 x^2}+\frac {4 b^2 f g \left (1-c^2 x^2\right )^{3/2}}{27 c^2}+\frac {8 b^2 f g \sqrt {1-c^2 x^2}}{9 c^2}+\frac {b^2 g^2 x \sqrt {1-c^2 x^2}}{64 c^2}-\frac {1}{32} b^2 g^2 x^3 \sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}\) |
(Sqrt[d - c^2*d*x^2]*((8*b^2*f*g*Sqrt[1 - c^2*x^2])/(9*c^2) - (b^2*f^2*x*S qrt[1 - c^2*x^2])/4 + (b^2*g^2*x*Sqrt[1 - c^2*x^2])/(64*c^2) - (b^2*g^2*x^ 3*Sqrt[1 - c^2*x^2])/32 + (4*b^2*f*g*(1 - c^2*x^2)^(3/2))/(27*c^2) + (b^2* f^2*ArcSin[c*x])/(4*c) - (b^2*g^2*ArcSin[c*x])/(64*c^3) + (4*b*f*g*x*(a + b*ArcSin[c*x]))/(3*c) - (b*c*f^2*x^2*(a + b*ArcSin[c*x]))/2 + (b*g^2*x^2*( a + b*ArcSin[c*x]))/(8*c) - (4*b*c*f*g*x^3*(a + b*ArcSin[c*x]))/9 - (b*c*g ^2*x^4*(a + b*ArcSin[c*x]))/8 + (f^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x ])^2)/2 - (g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) + (g^2*x ^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/4 - (2*f*g*(1 - c^2*x^2)^(3/2) *(a + b*ArcSin[c*x])^2)/(3*c^2) + (f^2*(a + b*ArcSin[c*x])^3)/(6*b*c) + (g ^2*(a + b*ArcSin[c*x])^3)/(24*b*c^3)))/Sqrt[1 - c^2*x^2]
3.1.59.3.1 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 = 0.73 (sec) , antiderivative size = 1852, normalized size of antiderivative = 2.51
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1852\) |
| parts | \(\text {Expression too large to display}\) | \(1852\) |
a^2*(f^2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1 /2)*x/(-c^2*d*x^2+d)^(1/2)))+g^2*(-1/4*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/4/c^ 2*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/( -c^2*d*x^2+d)^(1/2))))-2/3*f*g*(-c^2*d*x^2+d)^(3/2)/c^2/d)+b^2*(-1/24*(-d* (c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^3*(4*c^2 *f^2+g^2)+1/512*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8* c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c *x)*g^2*(4*I*arcsin(c*x)+8*arcsin(c*x)^2-1)/c^3/(c^2*x^2-1)+1/108*(-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)*f*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*(2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)/c/(c ^2*x^2-1)-1/4*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)* f*g*(arcsin(c*x)^2-2+2*I*arcsin(c*x))/c^2/(c^2*x^2-1)-1/4*(-d*(c^2*x^2-1)) ^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arcsin(c*x)^2-2-2*I*arcsi n(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*(-2*I*arcsin(c*x)+2*ar csin(c*x)^2-1)/c/(c^2*x^2-1)+1/108*(-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)*f*g*(-6* I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^2/(c^2*x^2-1)+1/512*(-d*(c^2*x^2-1))...
\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
integral((a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arcsin(c*x)^2 + 2*(a*b*g^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arcsin (c*x))*sqrt(-c^2*d*x^2 + d), x)
\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}\, dx \]
\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a^2*f^2 + 1/8*a^2*g^2 *(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2*d) + sqrt(d )*arcsin(c*x)/c^3) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a^2*f*g/(c^2*d) + sqrt(d)* integrate(((b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arctan2(c*x, sqrt(c*x + 1 )*sqrt(-c*x + 1))^2 + 2*(a*b*g^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
Exception generated. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]