Integrand size = 31, antiderivative size = 959 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {2 b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}-\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}-\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {1-c^2 x^2}}+\frac {b d g^3 x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}+\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {8 b c d g^3 x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c^4}+\frac {3 d f^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {3 d f g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{32 b c^3 \sqrt {1-c^2 x^2}} \]
3/8*d*f^3*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-3/16*d*f*g^2*x*(a+b*arc sin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3/8*d*f*g^2*x^3*(a+b*arcsin(c*x))*(-c^2 *d*x^2+d)^(1/2)+1/4*d*f^3*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^ (1/2)+1/2*d*f*g^2*x^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)- 3/5*d*f^2*g*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2-1/5* d*g^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4+1/7*d*g^3* (-c^2*x^2+1)^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4+3/5*b*d*f^2*g*x* (-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+2/35*b*d*g^3*x*(-c^2*d*x^2+d)^(1 /2)/c^3/(-c^2*x^2+1)^(1/2)-5/16*b*c*d*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x ^2+1)^(1/2)+3/32*b*d*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2 /5*b*c*d*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/105*b*d*g^3*x ^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+1/16*b*c^3*d*f^3*x^4*(-c^2*d* x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-7/32*b*c*d*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/ (-c^2*x^2+1)^(1/2)+3/25*b*c^3*d*f^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1 )^(1/2)-8/175*b*c*d*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/12*b *c^3*d*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/49*b*c^3*d*g^3* x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+3/16*d*f^3*(a+b*arcsin(c*x))^2 *(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)+3/32*d*f*g^2*(a+b*arcsin(c*x) )^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)
Time = 0.83 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.48 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (11025 a^2 c f \left (2 c^2 f^2+g^2\right )-210 a b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )+b^2 c x \left (6720 g^3+35 c^2 g \left (2016 f^2+315 f g x+32 g^2 x^2\right )-21 c^4 x \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+2 c^6 x^3 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )\right )-210 b \left (-105 a c f \left (2 c^2 f^2+g^2\right )+b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )\right ) \arcsin (c x)+11025 b^2 c f \left (2 c^2 f^2+g^2\right ) \arcsin (c x)^2\right )}{117600 b c^4 \sqrt {1-c^2 x^2}} \]
(d*Sqrt[d - c^2*d*x^2]*(11025*a^2*c*f*(2*c^2*f^2 + g^2) - 210*a*b*Sqrt[1 - c^2*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*( 35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336* f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)) + b^2*c*x*(6720*g^3 + 35*c^2*g*(201 6*f^2 + 315*f*g*x + 32*g^2*x^2) - 21*c^4*x*(1750*f^3 + 2240*f^2*g*x + 1225 *f*g^2*x^2 + 256*g^3*x^3) + 2*c^6*x^3*(3675*f^3 + 7056*f^2*g*x + 4900*f*g^ 2*x^2 + 1200*g^3*x^3)) - 210*b*(-105*a*c*f*(2*c^2*f^2 + g^2) + b*Sqrt[1 - c^2*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(3 5*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f ^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)))*ArcSin[c*x] + 11025*b^2*c*f*(2*c^2* f^2 + g^2)*ArcSin[c*x]^2))/(117600*b*c^4*Sqrt[1 - c^2*x^2])
Time = 1.18 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x)^3 (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x)^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) f^3+3 g x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) f^2+3 g^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) f+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {3 f g^2 (a+b \arcsin (c x))^2}{32 b c^3}+\frac {1}{4} f^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{8} f^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {3 f^2 g \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^2}-\frac {3 f g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^2}+\frac {1}{2} f g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{8} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {g^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {g^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {3 f^3 (a+b \arcsin (c x))^2}{16 b c}+\frac {1}{16} b c^3 f^3 x^4+\frac {3}{25} b c^3 f^2 g x^5+\frac {1}{12} b c^3 f g^2 x^6+\frac {1}{49} b c^3 g^3 x^7+\frac {2 b g^3 x}{35 c^3}-\frac {5}{16} b c f^3 x^2-\frac {2}{5} b c f^2 g x^3+\frac {3 b f^2 g x}{5 c}-\frac {7}{32} b c f g^2 x^4+\frac {3 b f g^2 x^2}{32 c}-\frac {8}{175} b c g^3 x^5+\frac {b g^3 x^3}{105 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d*Sqrt[d - c^2*d*x^2]*((3*b*f^2*g*x)/(5*c) + (2*b*g^3*x)/(35*c^3) - (5*b* c*f^3*x^2)/16 + (3*b*f*g^2*x^2)/(32*c) - (2*b*c*f^2*g*x^3)/5 + (b*g^3*x^3) /(105*c) + (b*c^3*f^3*x^4)/16 - (7*b*c*f*g^2*x^4)/32 + (3*b*c^3*f^2*g*x^5) /25 - (8*b*c*g^3*x^5)/175 + (b*c^3*f*g^2*x^6)/12 + (b*c^3*g^3*x^7)/49 + (3 *f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/8 - (3*f*g^2*x*Sqrt[1 - c^2* x^2]*(a + b*ArcSin[c*x]))/(16*c^2) + (3*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b *ArcSin[c*x]))/8 + (f^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/4 + (f* g^2*x^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/2 - (3*f^2*g*(1 - c^2*x^2 )^(5/2)*(a + b*ArcSin[c*x]))/(5*c^2) - (g^3*(1 - c^2*x^2)^(5/2)*(a + b*Arc Sin[c*x]))/(5*c^4) + (g^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^4) + (3*f^3*(a + b*ArcSin[c*x])^2)/(16*b*c) + (3*f*g^2*(a + b*ArcSin[c*x])^2 )/(32*b*c^3)))/Sqrt[1 - c^2*x^2]
3.1.36.3.1 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 = 0.72 (sec) , antiderivative size = 2074, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2074\) |
| parts | \(\text {Expression too large to display}\) | \(2074\) |
a*(f^3*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(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^3*(-1/7*x^ 2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+3*f*g^2*(-1/ 6*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/6/c^2*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*( 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)))))-3/5*f^2*g/c^2/d*(-c^2*d*x^2+d)^(5/2))+b*(-3/32*(-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)^2*f*(2*c^2 *f^2+g^2)*d-1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7 *x^7*(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^ 2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*g^3*(I +7*arcsin(c*x))*d/c^4/(c^2*x^2-1)-1/768*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^ 2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^5*x ^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c*x)* f*g^2*(I+6*arcsin(c*x))*d/c^3/(c^2*x^2-1)-1/3200*(-d*(c^2*x^2-1))^(1/2)*(1 6*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*(12*I*f^2*c^2+60*arc sin(c*x)*c^2*f^2-I*g^2-5*arcsin(c*x)*g^2)*d/c^4/(c^2*x^2-1)-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)*f*(2*I*c^2*f^2+8*arc sin(c*x)*c^2*f^2-3*I*g^2-12*arcsin(c*x)*g^2)*d/c^3/(c^2*x^2-1)-3/128*(-...
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
integral(-(a*c^2*d*g^3*x^5 + 3*a*c^2*d*f*g^2*x^4 - 3*a*d*f^2*g*x - a*d*f^3 + (3*a*c^2*d*f^2*g - a*d*g^3)*x^3 + (a*c^2*d*f^3 - 3*a*d*f*g^2)*x^2 + (b* c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2*g*x - b*d*f^3 + (3*b*c^2*d *f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g^2)*x^2)*arcsin(c*x))*sqrt (-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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^3 - 1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2 *d*x^2 + d)^(5/2)/(c^4*d))*a*g^3 + 1/16*a*f*g^2*(2*(-c^2*d*x^2 + d)^(3/2)* x/c^2 - 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^ 2 + 3*d^(3/2)*arcsin(c*x)/c^3) - 3/5*(-c^2*d*x^2 + d)^(5/2)*a*f^2*g/(c^2*d ) + sqrt(d)*integrate(-(b*c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2* g*x - b*d*f^3 + (3*b*c^2*d*f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g ^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x)
Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]