Integrand size = 29, antiderivative size = 238 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}} \]
1/2*f*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-1/3*g*(-c^2*x^2+1)*(a+b*arc sin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/3*b*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2* x^2+1)^(1/2)-1/4*b*c*f*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/9*b*c *g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/4*f*(a+b*arcsin(c*x))^2*( -c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)
Time = 0.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.55 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-9 b c f x^2-\frac {4 b g x \left (-3+c^2 x^2\right )}{c}+18 f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {12 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{c^2}+\frac {9 f (a+b \arcsin (c x))^2}{b c}\right )}{36 \sqrt {1-c^2 x^2}} \]
(Sqrt[d - c^2*d*x^2]*(-9*b*c*f*x^2 - (4*b*g*x*(-3 + c^2*x^2))/c + 18*f*x*S qrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (12*g*(1 - c^2*x^2)^(3/2)*(a + b*Ar cSin[c*x]))/c^2 + (9*f*(a + b*ArcSin[c*x])^2)/(b*c)))/(36*Sqrt[1 - c^2*x^2 ])
Time = 0.48 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.59, 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.
| \(\displaystyle \int \sqrt {d-c^2 d x^2} (f+g x) (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (f+g x) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {1}{2} f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}+\frac {f (a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c f x^2-\frac {1}{9} b c g x^3+\frac {b g x}{3 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(Sqrt[d - c^2*d*x^2]*((b*g*x)/(3*c) - (b*c*f*x^2)/4 - (b*c*g*x^3)/9 + (f*x *Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 - (g*(1 - c^2*x^2)^(3/2)*(a + b* ArcSin[c*x]))/(3*c^2) + (f*(a + b*ArcSin[c*x])^2)/(4*b*c)))/Sqrt[1 - c^2*x ^2]
3.1.33.3.1 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 = 0.50 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.64
| method | result | size |
| default | \(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) g \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) g \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(628\) |
| parts | \(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) g \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arcsin \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) g \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(628\) |
1/2*a*f*x*(-c^2*d*x^2+d)^(1/2)+1/2*a*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*g*(-c^2*d*x^2+d)^(3/2)/c^2/d+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)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(arcsin(c*x)+I)/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)-I)/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)+1/72*(-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*(-I+3*arcsin(c*x))/c^2/(c^2 *x^2-1))
\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
\[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f + sqrt(d)*integra te((b*g*x + b*f)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*s qrt(-c*x + 1)), x) - 1/3*(-c^2*d*x^2 + d)^(3/2)*a*g/(c^2*d)
Exception generated. \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]