Integrand size = 33, antiderivative size = 634 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {6 b^2 f^2 g \sqrt {d-c^2 d x^2}}{c^2 d}+\frac {14 b^2 g^3 \sqrt {d-c^2 d x^2}}{9 c^4 d}+\frac {3 b^2 f g^2 x \sqrt {d-c^2 d x^2}}{4 c^2 d}-\frac {2 b^2 g^3 \left (d-c^2 d x^2\right )^{3/2}}{27 c^4 d^2}-\frac {3 b^2 f g^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^3 \sqrt {d-c^2 d x^2}}+\frac {6 b f^2 g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d-c^2 d x^2}}+\frac {4 b g^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {3 b f g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c \sqrt {d-c^2 d x^2}}+\frac {2 b g^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}-\frac {2 g^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^4 d}-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {f^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{2 b c^3 \sqrt {d-c^2 d x^2}} \] Output:
6*b^2*f^2*g*(-c^2*d*x^2+d)^(1/2)/c^2/d+14/9*b^2*g^3*(-c^2*d*x^2+d)^(1/2)/c ^4/d+3/4*b^2*f*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2/d-2/27*b^2*g^3*(-c^2*d*x^2+d )^(3/2)/c^4/d^2-3/4*b^2*f*g^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)/c^3/(-c^2*d*x ^2+d)^(1/2)+6*b*f^2*g*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*d*x^2 +d)^(1/2)+4/3*b*g^3*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3/(-c^2*d*x^2 +d)^(1/2)+3/2*b*f*g^2*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*d*x ^2+d)^(1/2)+2/9*b*g^3*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*d*x ^2+d)^(1/2)-3*f^2*g*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^2/d-2/3*g^3 *(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^4/d-3/2*f*g^2*x*(-c^2*d*x^2+d) ^(1/2)*(a+b*arcsin(c*x))^2/c^2/d-1/3*g^3*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arc sin(c*x))^2/c^2/d+1/3*f^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^3/b/c/(-c^2 *d*x^2+d)^(1/2)+1/2*f*g^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^3/b/c^3/(-c ^2*d*x^2+d)^(1/2)
Time = 1.25 (sec) , antiderivative size = 582, normalized size of antiderivative = 0.92 \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-36 a^2 d \left (1-c^2 x^2\right )^{3/2} \left (4 g^3+c^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )-216 a b c^3 d f^3 \left (-1+c^2 x^2\right ) \arcsin (c x)^2-72 b^2 c^3 d f^3 \left (-1+c^2 x^2\right ) \arcsin (c x)^3-1296 a b c^2 d f^2 g \left (-1+c^2 x^2\right ) \left (c x-\sqrt {1-c^2 x^2} \arcsin (c x)\right )-48 a b d g^3 \left (-1+c^2 x^2\right ) \left (6 c x+c^3 x^3-3 \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)\right )+648 b^2 c^2 d f^2 g \left (1-c^2 x^2\right ) \left (2 c x \arcsin (c x)-\sqrt {1-c^2 x^2} \left (-2+\arcsin (c x)^2\right )\right )-108 a^2 c \sqrt {d} f \left (2 c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+162 a b c d f g^2 \left (-1+c^2 x^2\right ) \left (-2 \arcsin (c x)^2+\cos (2 \arcsin (c x))+2 \arcsin (c x) \sin (2 \arcsin (c x))\right )+27 b^2 c d f g^2 \left (1-c^2 x^2\right ) \left (4 \arcsin (c x)^3-6 \arcsin (c x) \cos (2 \arcsin (c x))+\left (3-6 \arcsin (c x)^2\right ) \sin (2 \arcsin (c x))\right )-2 b^2 d g^3 \left (1-c^2 x^2\right ) \left (81 \sqrt {1-c^2 x^2} \left (-2+\arcsin (c x)^2\right )-\left (-2+9 \arcsin (c x)^2\right ) \cos (3 \arcsin (c x))+6 \arcsin (c x) (-27 c x+\sin (3 \arcsin (c x)))\right )}{216 c^4 d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \] Input:
Integrate[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
(-36*a^2*d*(1 - c^2*x^2)^(3/2)*(4*g^3 + c^2*g*(18*f^2 + 9*f*g*x + 2*g^2*x^ 2)) - 216*a*b*c^3*d*f^3*(-1 + c^2*x^2)*ArcSin[c*x]^2 - 72*b^2*c^3*d*f^3*(- 1 + c^2*x^2)*ArcSin[c*x]^3 - 1296*a*b*c^2*d*f^2*g*(-1 + c^2*x^2)*(c*x - Sq rt[1 - c^2*x^2]*ArcSin[c*x]) - 48*a*b*d*g^3*(-1 + c^2*x^2)*(6*c*x + c^3*x^ 3 - 3*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]) + 648*b^2*c^2*d*f^2*g*( 1 - c^2*x^2)*(2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2)) - 108*a^2*c*Sqrt[d]*f*(2*c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d *x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 162*a*b *c*d*f*g^2*(-1 + c^2*x^2)*(-2*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 2*ArcSi n[c*x]*Sin[2*ArcSin[c*x]]) + 27*b^2*c*d*f*g^2*(1 - c^2*x^2)*(4*ArcSin[c*x] ^3 - 6*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + (3 - 6*ArcSin[c*x]^2)*Sin[2*ArcSin [c*x]]) - 2*b^2*d*g^3*(1 - c^2*x^2)*(81*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x ]^2) - (-2 + 9*ArcSin[c*x]^2)*Cos[3*ArcSin[c*x]] + 6*ArcSin[c*x]*(-27*c*x + Sin[3*ArcSin[c*x]])))/(216*c^4*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])
Time = 0.92 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {5276, 5272, 3042, 3798, 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.
| \(\displaystyle \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5272 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int (c f+c g x)^3 (a+b \arcsin (c x))^2d\arcsin (c x)}{c^4 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))^2 (c f+g \sin (\arcsin (c x)))^3d\arcsin (c x)}{c^4 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \left (f^3 (a+b \arcsin (c x))^2 c^3+g^3 x^3 (a+b \arcsin (c x))^2 c^3+3 f g^2 x^2 (a+b \arcsin (c x))^2 c^3+3 f^2 g x (a+b \arcsin (c x))^2 c^3\right )d\arcsin (c x)}{c^4 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (\frac {c^3 f^3 (a+b \arcsin (c x))^3}{3 b}+6 b c^3 f^2 g x (a+b \arcsin (c x))+\frac {3}{2} b c^3 f g^2 x^2 (a+b \arcsin (c x))+\frac {2}{9} b c^3 g^3 x^3 (a+b \arcsin (c x))-3 c^2 f^2 g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {3}{2} c^2 f g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {1}{3} c^2 g^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {2}{3} g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {c f g^2 (a+b \arcsin (c x))^3}{2 b}+\frac {4}{3} b c g^3 x (a+b \arcsin (c x))-\frac {3}{4} b^2 c f g^2 \arcsin (c x)+6 b^2 c^2 f^2 g \sqrt {1-c^2 x^2}+\frac {3}{4} b^2 c^2 f g^2 x \sqrt {1-c^2 x^2}-\frac {2}{27} b^2 g^3 \left (1-c^2 x^2\right )^{3/2}+\frac {14}{9} b^2 g^3 \sqrt {1-c^2 x^2}\right )}{c^4 \sqrt {d-c^2 d x^2}}\) |
Input:
Int[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[1 - c^2*x^2]*(6*b^2*c^2*f^2*g*Sqrt[1 - c^2*x^2] + (14*b^2*g^3*Sqrt[1 - c^2*x^2])/9 + (3*b^2*c^2*f*g^2*x*Sqrt[1 - c^2*x^2])/4 - (2*b^2*g^3*(1 - c^2*x^2)^(3/2))/27 - (3*b^2*c*f*g^2*ArcSin[c*x])/4 + 6*b*c^3*f^2*g*x*(a + b*ArcSin[c*x]) + (4*b*c*g^3*x*(a + b*ArcSin[c*x]))/3 + (3*b*c^3*f*g^2*x^2 *(a + b*ArcSin[c*x]))/2 + (2*b*c^3*g^3*x^3*(a + b*ArcSin[c*x]))/9 - 3*c^2* f^2*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - (2*g^3*Sqrt[1 - c^2*x^2]*( a + b*ArcSin[c*x])^2)/3 - (3*c^2*f*g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c *x])^2)/2 - (c^2*g^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/3 + (c^3 *f^3*(a + b*ArcSin[c*x])^3)/(3*b) + (c*f*g^2*(a + b*ArcSin[c*x])^3)/(2*b)) )/(c^4*Sqrt[d - c^2*d*x^2])
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d]) Subst[In t[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c , d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (G tQ[m, 0] || IGtQ[n, 0])
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 = 0.99 (sec) , antiderivative size = 1636, normalized size of antiderivative = 2.58
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1636\) |
| parts | \(\text {Expression too large to display}\) | \(1636\) |
Input:
int((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVER BOSE)
Output:
a^2*(f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+g^3*(- 1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(1/2))+3*f*g^2 *(-1/2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^( 1/2)*x/(-c^2*d*x^2+d)^(1/2)))-3*f^2*g/c^2/d*(-c^2*d*x^2+d)^(1/2))+b^2*(-1/ 6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)^ 3*f*(2*c^2*f^2+3*g^2)+1/432*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2 )*c*x+2*c^2*x^2-1)*g^3*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^4/d/(c^2*x^2- 1)-3/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*g*(8*I* arcsin(c*x)*c^2*f^2+4*arcsin(c*x)^2*c^2*f^2+2*I*arcsin(c*x)*g^2+arcsin(c*x )^2*g^2-8*c^2*f^2-2*g^2)/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)*(I*( -c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g*(4*arcsin(c*x)^2*c^2*f^2-8*I*arcsin(c*x )*c^2*f^2+arcsin(c*x)^2*g^2-2*I*arcsin(c*x)*g^2-8*c^2*f^2-2*g^2)/c^4/d/(c^ 2*x^2-1)+1/432*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^ 2-1)*g^3*(-6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^4/d/(c^2*x^2-1)+3/8*(-d*(c ^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2*arcsin(c*x)+3/ 16*(-d*(c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)*f*g^2*(2*arcsin(c*x)^2-1)*x-1/ 216*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*g^3*(9*arcsin(c*x)^2-2)*cos(4 *arcsin(c*x))+1/36*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*arcsin(c*x)*g^ 3*sin(4*arcsin(c*x))+3/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2*ar csin(c*x)*cos(3*arcsin(c*x))+3/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2...
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm= "fricas")
Output:
integral(-(a^2*g^3*x^3 + 3*a^2*f*g^2*x^2 + 3*a^2*f^2*g*x + a^2*f^3 + (b^2* g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arcsin(c*x)^2 + 2*(a* b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x + a*b*f^3)*arcsin(c*x))*sqrt(- c^2*d*x^2 + d)/(c^2*d*x^2 - d), x)
Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((g*x+f)**3*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)
Output:
Exception raised: TypeError >> Invalid comparison of non-real zoo
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm= "maxima")
Output:
-1/3*a^2*g^3*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^2*d*x^2 + d)/(c ^4*d)) - 3/2*a^2*f*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3* sqrt(d))) + a*b*f^3*arcsin(c*x)^2/(c*sqrt(d)) + 6*a*b*f^2*g*x/(c*sqrt(d)) + a^2*f^3*arcsin(c*x)/(c*sqrt(d)) - 6*sqrt(-c^2*d*x^2 + d)*a*b*f^2*g*arcsi n(c*x)/(c^2*d) - 3*sqrt(-c^2*d*x^2 + d)*a^2*f^2*g/(c^2*d) - sqrt(d)*integr ate(((b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arctan2(c*x , sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2)*arct an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^2* d*x^2 - d), x)
Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm= "giac")
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. \[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int(((f + g*x)^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)
Output:
int(((f + g*x)^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {(f+g x)^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 \mathit {asin} \left (c x \right )^{3} b^{2} c^{3} f^{3}-18 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} b^{2} c^{2} f^{2} g +6 \mathit {asin} \left (c x \right )^{2} a b \,c^{3} f^{3}-36 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a b \,c^{2} f^{2} g +6 \mathit {asin} \left (c x \right ) a^{2} c^{3} f^{3}+9 \mathit {asin} \left (c x \right ) a^{2} c f \,g^{2}+36 \mathit {asin} \left (c x \right ) b^{2} c^{3} f^{2} g x -18 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} f^{2} g -9 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} f \,g^{2} x -2 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} g^{3} x^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a^{2} g^{3}+36 \sqrt {-c^{2} x^{2}+1}\, b^{2} c^{2} f^{2} g +12 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{3}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{4} g^{3}+36 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{4} f \,g^{2}+6 \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{3}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{4} g^{3}+18 \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{4} f \,g^{2}+36 a b \,c^{3} f^{2} g x}{6 \sqrt {d}\, c^{4}} \] Input:
int((g*x+f)^3*(a+b*asin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)
Output:
(2*asin(c*x)**3*b**2*c**3*f**3 - 18*sqrt( - c**2*x**2 + 1)*asin(c*x)**2*b* *2*c**2*f**2*g + 6*asin(c*x)**2*a*b*c**3*f**3 - 36*sqrt( - c**2*x**2 + 1)* asin(c*x)*a*b*c**2*f**2*g + 6*asin(c*x)*a**2*c**3*f**3 + 9*asin(c*x)*a**2* c*f*g**2 + 36*asin(c*x)*b**2*c**3*f**2*g*x - 18*sqrt( - c**2*x**2 + 1)*a** 2*c**2*f**2*g - 9*sqrt( - c**2*x**2 + 1)*a**2*c**2*f*g**2*x - 2*sqrt( - c* *2*x**2 + 1)*a**2*c**2*g**3*x**2 - 4*sqrt( - c**2*x**2 + 1)*a**2*g**3 + 36 *sqrt( - c**2*x**2 + 1)*b**2*c**2*f**2*g + 12*int((asin(c*x)*x**3)/sqrt( - c**2*x**2 + 1),x)*a*b*c**4*g**3 + 36*int((asin(c*x)*x**2)/sqrt( - c**2*x* *2 + 1),x)*a*b*c**4*f*g**2 + 6*int((asin(c*x)**2*x**3)/sqrt( - c**2*x**2 + 1),x)*b**2*c**4*g**3 + 18*int((asin(c*x)**2*x**2)/sqrt( - c**2*x**2 + 1), x)*b**2*c**4*f*g**2 + 36*a*b*c**3*f**2*g*x)/(6*sqrt(d)*c**4)