Integrand size = 31, antiderivative size = 773 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {32 b^2 d^2 g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f x \sqrt {d-c^2 d x^2}}{1152}+\frac {16 b^2 d g \left (d-c^2 d x^2\right )^{3/2}}{735 c^2}-\frac {65 b^2 d f x \left (d-c^2 d x^2\right )^{3/2}}{1728}+\frac {12 b^2 g \left (d-c^2 d x^2\right )^{5/2}}{1225 c^2}-\frac {1}{108} b^2 f x \left (d-c^2 d x^2\right )^{5/2}+\frac {2 b^2 g \left (d-c^2 d x^2\right )^{7/2}}{343 c^2 d}+\frac {115 b^2 d^2 f \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 f \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d f x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{6} f x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {g \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d}+\frac {5 d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}} \] Output:
32/245*b^2*d^2*g*(-c^2*d*x^2+d)^(1/2)/c^2-245/1152*b^2*d^2*f*x*(-c^2*d*x^2 +d)^(1/2)+16/735*b^2*d*g*(-c^2*d*x^2+d)^(3/2)/c^2-65/1728*b^2*d*f*x*(-c^2* d*x^2+d)^(3/2)+12/1225*b^2*g*(-c^2*d*x^2+d)^(5/2)/c^2-1/108*b^2*f*x*(-c^2* d*x^2+d)^(5/2)+2/343*b^2*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+115/1152*b^2*d^2*f*( -c^2*d*x^2+d)^(1/2)*arcsin(c*x)/c/(-c^2*x^2+1)^(1/2)+2/7*b*d^2*g*x*(-c^2*d *x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)-5/16*b*c*d^2*f*x^2*(- c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-2/7*b*c*d^2*g*x^3* (-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+6/35*b*c^3*d^2*g *x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-2/49*b*c^5* d^2*g*x^7*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+5/48*b *d^2*f*(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c+1/18*b* d^2*f*(-c^2*x^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c+5/16*d^2 *f*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2+5/24*d*f*x*(-c^2*d*x^2+d)^(3 /2)*(a+b*arcsin(c*x))^2+1/6*f*x*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2-1 /7*g*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))^2/c^2/d+5/48*d^2*f*(-c^2*d*x^2 +d)^(1/2)*(a+b*arcsin(c*x))^3/b/c/(-c^2*x^2+1)^(1/2)
Time = 0.51 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.61 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (3087000 a^3 c f+88200 a^2 b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )-840 a b^2 c x \left (245 c^2 f x \left (99-39 c^2 x^2+8 c^4 x^4\right )+288 g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )\right )+b^3 \sqrt {1-c^2 x^2} \left (-8575 c^2 f x \left (897-194 c^2 x^2+32 c^4 x^4\right )-2304 g \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )\right )+105 b \left (88200 a^2 c f+1680 a b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )+b^2 c \left (-2304 g x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )-245 f \left (-299+792 c^2 x^2-312 c^4 x^4+64 c^6 x^6\right )\right )\right ) \arcsin (c x)+88200 b^2 \left (105 a c f+b \sqrt {1-c^2 x^2} \left (48 g \left (-1+c^2 x^2\right )^3+7 c^2 f x \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)^2+3087000 b^3 c f \arcsin (c x)^3\right )}{29635200 b c^2 \sqrt {1-c^2 x^2}} \] Input:
Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(d^2*Sqrt[d - c^2*d*x^2]*(3087000*a^3*c*f + 88200*a^2*b*Sqrt[1 - c^2*x^2]* (48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) - 840*a* b^2*c*x*(245*c^2*f*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) + 288*g*(-35 + 35*c^2*x ^2 - 21*c^4*x^4 + 5*c^6*x^6)) + b^3*Sqrt[1 - c^2*x^2]*(-8575*c^2*f*x*(897 - 194*c^2*x^2 + 32*c^4*x^4) - 2304*g*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6)) + 105*b*(88200*a^2*c*f + 1680*a*b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + b^2*c*(-2304*g* x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) - 245*f*(-299 + 792*c^2*x^2 - 312*c^4*x^4 + 64*c^6*x^6)))*ArcSin[c*x] + 88200*b^2*(105*a*c*f + b*Sqrt[ 1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x ^4)))*ArcSin[c*x]^2 + 3087000*b^3*c*f*ArcSin[c*x]^3))/(29635200*b*c^2*Sqrt [1 - c^2*x^2])
Time = 1.21 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.66, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x) (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x) \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (f (a+b \arcsin (c x))^2 \left (1-c^2 x^2\right )^{5/2}+g x (a+b \arcsin (c x))^2 \left (1-c^2 x^2\right )^{5/2}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {2}{49} b c^5 g x^7 (a+b \arcsin (c x))+\frac {6}{35} b c^3 g x^5 (a+b \arcsin (c x))+\frac {1}{6} f x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2+\frac {5}{24} f x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {5}{16} f x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {b f \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{18 c}+\frac {5 b f \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{48 c}-\frac {g \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2}-\frac {5}{16} b c f x^2 (a+b \arcsin (c x))+\frac {5 f (a+b \arcsin (c x))^3}{48 b c}-\frac {2}{7} b c g x^3 (a+b \arcsin (c x))+\frac {2 b g x (a+b \arcsin (c x))}{7 c}+\frac {115 b^2 f \arcsin (c x)}{1152 c}-\frac {1}{108} b^2 f x \left (1-c^2 x^2\right )^{5/2}-\frac {65 b^2 f x \left (1-c^2 x^2\right )^{3/2}}{1728}-\frac {245 b^2 f x \sqrt {1-c^2 x^2}}{1152}+\frac {2 b^2 g \left (1-c^2 x^2\right )^{7/2}}{343 c^2}+\frac {12 b^2 g \left (1-c^2 x^2\right )^{5/2}}{1225 c^2}+\frac {16 b^2 g \left (1-c^2 x^2\right )^{3/2}}{735 c^2}+\frac {32 b^2 g \sqrt {1-c^2 x^2}}{245 c^2}\right )}{\sqrt {1-c^2 x^2}}\) |
Input:
Int[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(d^2*Sqrt[d - c^2*d*x^2]*((32*b^2*g*Sqrt[1 - c^2*x^2])/(245*c^2) - (245*b^ 2*f*x*Sqrt[1 - c^2*x^2])/1152 + (16*b^2*g*(1 - c^2*x^2)^(3/2))/(735*c^2) - (65*b^2*f*x*(1 - c^2*x^2)^(3/2))/1728 + (12*b^2*g*(1 - c^2*x^2)^(5/2))/(1 225*c^2) - (b^2*f*x*(1 - c^2*x^2)^(5/2))/108 + (2*b^2*g*(1 - c^2*x^2)^(7/2 ))/(343*c^2) + (115*b^2*f*ArcSin[c*x])/(1152*c) + (2*b*g*x*(a + b*ArcSin[c *x]))/(7*c) - (5*b*c*f*x^2*(a + b*ArcSin[c*x]))/16 - (2*b*c*g*x^3*(a + b*A rcSin[c*x]))/7 + (6*b*c^3*g*x^5*(a + b*ArcSin[c*x]))/35 - (2*b*c^5*g*x^7*( a + b*ArcSin[c*x]))/49 + (5*b*f*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(48*c ) + (b*f*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/(18*c) + (5*f*x*Sqrt[1 - c^2 *x^2]*(a + b*ArcSin[c*x])^2)/16 + (5*f*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin [c*x])^2)/24 + (f*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/6 - (g*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x])^2)/(7*c^2) + (5*f*(a + b*ArcSin[c*x])^ 3)/(48*b*c)))/Sqrt[1 - c^2*x^2]
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 = 1.23 (sec) , antiderivative size = 2852, normalized size of antiderivative = 3.69
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2852\) |
| parts | \(\text {Expression too large to display}\) | \(2852\) |
Input:
int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBO SE)
Output:
1/6*a^2*f*x*(-c^2*d*x^2+d)^(5/2)+5/24*a^2*f*d*x*(-c^2*d*x^2+d)^(3/2)+5/16* a^2*f*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a^2*f*d^3/(c^2*d)^(1/2)*arctan((c^2* d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/7*a^2*g*(-c^2*d*x^2+d)^(7/2)/c^2/d+b^2* (-5/48*(-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*d^2+1/43904*(-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*(14*I* arcsin(c*x)+49*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)+1/6912*(-d*(c^2*x^2-1) )^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*x^6*c^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*(6*I*arcsin(c*x)+18*arcsin(c*x)^2-1)*d^2/c/(c^2*x^2-1 )-5/128*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*g*(arc sin(c*x)^2-2+2*I*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-5/128*(-d*(c^2*x^2-1))^( 1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g*(arcsin(c*x)^2-2-2*I*arcsin(c* x))*d^2/c^2/(c^2*x^2-1)+15/256*(-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*arcsin(c*x)^2-1-2*I *arcsin(c*x))*d^2/c/(c^2*x^2-1)+1/384*(-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)*d^2/c^2/(c^2*x^2-1)-1/137200*(-d*(c^2*x^ 2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g*(385*I*arcsin(c*x)+1...
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="f ricas")
Output:
integral((a^2*c^4*d^2*g*x^5 + a^2*c^4*d^2*f*x^4 - 2*a^2*c^2*d^2*g*x^3 - 2* a^2*c^2*d^2*f*x^2 + a^2*d^2*g*x + a^2*d^2*f + (b^2*c^4*d^2*g*x^5 + b^2*c^4 *d^2*f*x^4 - 2*b^2*c^2*d^2*g*x^3 - 2*b^2*c^2*d^2*f*x^2 + b^2*d^2*g*x + b^2 *d^2*f)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*g*x^5 + a*b*c^4*d^2*f*x^4 - 2*a*b*c ^2*d^2*g*x^3 - 2*a*b*c^2*d^2*f*x^2 + a*b*d^2*g*x + a*b*d^2*f)*arcsin(c*x)) *sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2,x)
Output:
Timed out
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="m axima")
Output:
1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt (-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a^2*f - 1/7*(-c^2*d*x^2 + d)^(7/2)*a^2*g/(c^2*d) + sqrt(d)*integrate(((b^2*c^4*d^2*g*x^5 + b^2*c^ 4*d^2*f*x^4 - 2*b^2*c^2*d^2*g*x^3 - 2*b^2*c^2*d^2*f*x^2 + b^2*d^2*g*x + b^ 2*d^2*f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*g*x ^5 + a*b*c^4*d^2*f*x^4 - 2*a*b*c^2*d^2*g*x^3 - 2*a*b*c^2*d^2*f*x^2 + a*b*d ^2*g*x + a*b*d^2*f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="g iac")
Output:
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. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int \left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int((f + g*x)*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2),x)
Output:
int((f + g*x)*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2), x)
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (105 \mathit {asin} \left (c x \right ) a^{2} c f +56 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{6} f \,x^{5}+48 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{6} g \,x^{6}-182 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} f \,x^{3}-144 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} g \,x^{4}+231 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} f x +144 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} g \,x^{2}-48 \sqrt {-c^{2} x^{2}+1}\, a^{2} g +672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{5}d x \right ) a b \,c^{6} g +672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) a b \,c^{6} f -1344 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{3}d x \right ) a b \,c^{4} g -1344 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{4} f +672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x d x \right ) a b \,c^{2} g +672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )d x \right ) a b \,c^{2} f +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{6} f -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4} g -672 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{4} f +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2} c^{2} g +336 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} c^{2} f +48 a^{2} g \right )}{336 c^{2}} \] Input:
int((g*x+f)*(-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x))^2,x)
Output:
(sqrt(d)*d**2*(105*asin(c*x)*a**2*c*f + 56*sqrt( - c**2*x**2 + 1)*a**2*c** 6*f*x**5 + 48*sqrt( - c**2*x**2 + 1)*a**2*c**6*g*x**6 - 182*sqrt( - c**2*x **2 + 1)*a**2*c**4*f*x**3 - 144*sqrt( - c**2*x**2 + 1)*a**2*c**4*g*x**4 + 231*sqrt( - c**2*x**2 + 1)*a**2*c**2*f*x + 144*sqrt( - c**2*x**2 + 1)*a**2 *c**2*g*x**2 - 48*sqrt( - c**2*x**2 + 1)*a**2*g + 672*int(sqrt( - c**2*x** 2 + 1)*asin(c*x)*x**5,x)*a*b*c**6*g + 672*int(sqrt( - c**2*x**2 + 1)*asin( c*x)*x**4,x)*a*b*c**6*f - 1344*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**3,x )*a*b*c**4*g - 1344*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**2,x)*a*b*c**4* f + 672*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x,x)*a*b*c**2*g + 672*int(sqr t( - c**2*x**2 + 1)*asin(c*x),x)*a*b*c**2*f + 336*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**5,x)*b**2*c**6*g + 336*int(sqrt( - c**2*x**2 + 1)*asin( c*x)**2*x**4,x)*b**2*c**6*f - 672*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2* x**3,x)*b**2*c**4*g - 672*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**2,x)* b**2*c**4*f + 336*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x,x)*b**2*c**2*g + 336*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2,x)*b**2*c**2*f + 48*a**2*g) )/(336*c**2)