Integrand size = 29, antiderivative size = 370 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}-\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {3 d f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c \sqrt {1-c^2 x^2}} \]
3/8*d*f*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+1/4*d*f*x*(-c^2*x^2+1)*(a +b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-1/5*d*g*(-c^2*x^2+1)^2*(a+b*arcsin(c* x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/5*b*d*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+ 1)^(1/2)-5/16*b*c*d*f*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/15*b*c *d*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/16*b*c^3*d*f*x^4*(-c^2* d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/25*b*c^3*d*g*x^5*(-c^2*d*x^2+d)^(1/2)/ (-c^2*x^2+1)^(1/2)+3/16*d*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 = 0.22 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.58 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (225 a^2 c f-30 a b \sqrt {1-c^2 x^2} \left (8 g \left (-1+c^2 x^2\right )^2+5 c^2 f x \left (-5+2 c^2 x^2\right )\right )+b^2 c x \left (75 c^2 f x \left (-5+c^2 x^2\right )+16 g \left (15-10 c^2 x^2+3 c^4 x^4\right )\right )+30 b \left (15 a c f+b \sqrt {1-c^2 x^2} \left (5 c^2 f x \left (5-2 c^2 x^2\right )-8 g \left (-1+c^2 x^2\right )^2\right )\right ) \arcsin (c x)+225 b^2 c f \arcsin (c x)^2\right )}{1200 b c^2 \sqrt {1-c^2 x^2}} \]
(d*Sqrt[d - c^2*d*x^2]*(225*a^2*c*f - 30*a*b*Sqrt[1 - c^2*x^2]*(8*g*(-1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*c^2*x^2)) + b^2*c*x*(75*c^2*f*x*(-5 + c^2*x ^2) + 16*g*(15 - 10*c^2*x^2 + 3*c^4*x^4)) + 30*b*(15*a*c*f + b*Sqrt[1 - c^ 2*x^2]*(5*c^2*f*x*(5 - 2*c^2*x^2) - 8*g*(-1 + c^2*x^2)^2))*ArcSin[c*x] + 2 25*b^2*c*f*ArcSin[c*x]^2))/(1200*b*c^2*Sqrt[1 - c^2*x^2])
Time = 0.55 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.52, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x) (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x) \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (f (a+b \arcsin (c x)) \left (1-c^2 x^2\right )^{3/2}+g x (a+b \arcsin (c x)) \left (1-c^2 x^2\right )^{3/2}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {1}{4} f x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{8} f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {g \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}+\frac {3 f (a+b \arcsin (c x))^2}{16 b c}+\frac {1}{16} b c^3 f x^4+\frac {1}{25} b c^3 g x^5-\frac {5}{16} b c f x^2-\frac {2}{15} b c g x^3+\frac {b g x}{5 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d*Sqrt[d - c^2*d*x^2]*((b*g*x)/(5*c) - (5*b*c*f*x^2)/16 - (2*b*c*g*x^3)/1 5 + (b*c^3*f*x^4)/16 + (b*c^3*g*x^5)/25 + (3*f*x*Sqrt[1 - c^2*x^2]*(a + b* ArcSin[c*x]))/8 + (f*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/4 - (g*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^2) + (3*f*(a + b*ArcSin[c*x])^2 )/(16*b*c)))/Sqrt[1 - c^2*x^2]
3.1.38.3.1 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 = 0.56 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.74
| method | result | size |
| default | \(\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}-16 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+13 c^{2} x^{2}+20 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-5 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (i+5 \arcsin \left (c x \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) f \left (4 \arcsin \left (c x \right )+i\right ) d}{256 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 ) d}{16 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 ) d}{16 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 ) d}{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 ) d}{96 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 (11 i+45 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right ) d}{1200 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) g \left (7 i+15 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right ) d}{600 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) f \left (17 i+28 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}+\frac {3 \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 ) f \left (5 i+12 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\) | \(1014\) |
| parts | \(\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}-16 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+13 c^{2} x^{2}+20 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-5 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (i+5 \arcsin \left (c x \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) f \left (4 \arcsin \left (c x \right )+i\right ) d}{256 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 ) d}{16 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 ) d}{16 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 ) d}{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 ) d}{96 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 (11 i+45 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right ) d}{1200 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) g \left (7 i+15 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right ) d}{600 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) f \left (17 i+28 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}+\frac {3 \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 ) f \left (5 i+12 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\) | \(1014\) |
1/4*a*f*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*f*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f*d^ 2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/5*a*g/c^2/d *(-c^2*d*x^2+d)^(5/2)+b*(-3/16*(-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*d-1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28 *c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2 )*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(I+5*arcsin(c*x))*d/c^2/(c^2*x^2 -1)-1/256*(-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)*f* (4*arcsin(c*x)+I)*d/c/(c^2*x^2-1)-1/16*(-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)*d/c^2/(c^2*x^2-1)-1/16*(-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)*d/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))*d/c/(c^2*x^2-1) +1/96*(-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))*d/c^2/(c^2*x^2-1 )-1/1200*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(11 *I+45*arcsin(c*x))*cos(4*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/600*(-d*(c^2*x^2 -1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*g*(7*I+15*arcsin(c*x))*sin (4*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/256*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2- c*x*(-c^2*x^2+1)^(1/2)-I)*f*(17*I+28*arcsin(c*x))*cos(3*arcsin(c*x))*d/...
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
integral(-(a*c^2*d*g*x^3 + a*c^2*d*f*x^2 - a*d*g*x - a*d*f + (b*c^2*d*g*x^ 3 + b*c^2*d*f*x^2 - b*d*g*x - b*d*f)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a rcsin(c*x)/c)*a*f - 1/5*(-c^2*d*x^2 + d)^(5/2)*a*g/(c^2*d) + sqrt(d)*integ rate(-(b*c^2*d*g*x^3 + b*c^2*d*f*x^2 - b*d*g*x - b*d*f)*sqrt(c*x + 1)*sqrt (-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x)
Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]