\(\int (f+g x)^3 (d-c^2 d x^2)^{3/2} (a+b \arcsin (c x))^2 \, dx\) [135]

Optimal result

Integrand size = 33, antiderivative size = 1538 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Output:

-1/35*d*g^3*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^2+3/8*d*f*g^2*x 
^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2-43/576*b^2*d*f*g^2*x^3*(-c^2*d 
*x^2+d)^(1/2)+16/25*b^2*d*f^2*g*(-c^2*d*x^2+d)^(1/2)/c^2-3/5*f^2*g*(-c^2*d 
*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/c^2/d+6/125*b^2*f^2*g*(-c^2*d*x^2+d)^(5/ 
2)/c^2/d-7/384*b^2*d*f*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+1/36*b^2*c^2*d*f*g^2 
*x^5*(-c^2*d*x^2+d)^(1/2)+1/8*b*d*f^3*(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1 
/2)*(a+b*arcsin(c*x))/c+8/75*b^2*f^2*g*(-c^2*d*x^2+d)^(3/2)/c^2+38/6125*b^ 
2*g^3*(-c^2*d*x^2+d)^(5/2)/c^4/d-2/343*b^2*g^3*(-c^2*d*x^2+d)^(7/2)/c^4/d^ 
2+1/2*f*g^2*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2+2/105*b*d*g^3*x^3 
*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)-16/175*b*c*d* 
g^3*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 
^3*d*g^3*x^7*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+1/1 
6*d*f*g^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^3/b/c^3/(-c^2*x^2+1)^(1/2 
)+6/5*b*d*f^2*g*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1 
/2)+3/16*b*d*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+ 
1)^(1/2)-4/5*b*c*d*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2* 
x^2+1)^(1/2)-7/16*b*c*d*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/( 
-c^2*x^2+1)^(1/2)+6/25*b*c^3*d*f^2*g*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin( 
c*x))/(-c^2*x^2+1)^(1/2)+1/6*b*c^3*d*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)*(a+b*a 
rcsin(c*x))/(-c^2*x^2+1)^(1/2)-3/16*d*f*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b...
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 872, normalized size of antiderivative = 0.57 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (3087000 a^3 c f \left (2 c^2 f^2+g^2\right )-88200 a^2 b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )+840 a b^2 c x \left (6720 g^3+35 c^2 g \left (2016 f^2+315 f g x+32 g^2 x^2\right )-21 c^4 x \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+2 c^6 x^3 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )\right )+b^3 \sqrt {1-c^2 x^2} \left (4785152 g^3+c^2 g \left (39250176 f^2-900375 f g x-429824 g^2 x^2\right )+4 c^6 x^3 \left (385875 f^3+592704 f^2 g x+343000 f g^2 x^2+72000 g^3 x^3\right )-2 c^4 x \left (6559875 f^3+5005056 f^2 g x+1843625 f g^2 x^2+278784 g^3 x^3\right )\right )+105 b \left (88200 a^2 c f \left (2 c^2 f^2+g^2\right )-1680 a b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )+b^2 c \left (35 g^2 (245 f+1536 g x)+70 c^2 \left (1785 f^3+8064 f^2 g x+1260 f g^2 x^2+128 g^3 x^3\right )-168 c^4 x^2 \left (1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right )+16 c^6 x^4 \left (3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right )\right )\right ) \arcsin (c x)-88200 b^2 \left (-105 a c f \left (2 c^2 f^2+g^2\right )+b \sqrt {1-c^2 x^2} \left (32 g^3+c^2 g \left (336 f^2+105 f g x+16 g^2 x^2\right )+4 c^6 x^3 \left (35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right )-2 c^4 x \left (175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right )\right )\right ) \arcsin (c x)^2+3087000 b^3 c f \left (2 c^2 f^2+g^2\right ) \arcsin (c x)^3\right )}{49392000 b c^4 \sqrt {1-c^2 x^2}} \] Input:

Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]
 

Output:

(d*Sqrt[d - c^2*d*x^2]*(3087000*a^3*c*f*(2*c^2*f^2 + g^2) - 88200*a^2*b*Sq 
rt[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)) + 840*a*b^2*c*x*(6720*g^3 + 3 
5*c^2*g*(2016*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)) + b^3*Sqrt[1 - c^2*x^2]*(4785152*g^3 + 
c^2*g*(39250176*f^2 - 900375*f*g*x - 429824*g^2*x^2) + 4*c^6*x^3*(385875*f 
^3 + 592704*f^2*g*x + 343000*f*g^2*x^2 + 72000*g^3*x^3) - 2*c^4*x*(6559875 
*f^3 + 5005056*f^2*g*x + 1843625*f*g^2*x^2 + 278784*g^3*x^3)) + 105*b*(882 
00*a^2*c*f*(2*c^2*f^2 + g^2) - 1680*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*(35*g^2*(245*f + 1536*g*x) + 70*c^2*(1785*f^3 + 8064* 
f^2*g*x + 1260*f*g^2*x^2 + 128*g^3*x^3) - 168*c^4*x^2*(1750*f^3 + 2240*f^2 
*g*x + 1225*f*g^2*x^2 + 256*g^3*x^3) + 16*c^6*x^4*(3675*f^3 + 7056*f^2*g*x 
 + 4900*f*g^2*x^2 + 1200*g^3*x^3)))*ArcSin[c*x] - 88200*b^2*(-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*(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)))*Arc...
 

Rubi [A] (verified)

Time = 2.48 (sec) , antiderivative size = 1066, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x)^3 (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5276

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x)^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5262

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 f^3+3 g x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 f^2+3 g^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 f+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {2}{49} b c^3 g^3 (a+b \arcsin (c x)) x^7+\frac {1}{6} b c^3 f g^2 (a+b \arcsin (c x)) x^6-\frac {16}{175} b c g^3 (a+b \arcsin (c x)) x^5+\frac {6}{25} b c^3 f^2 g (a+b \arcsin (c x)) x^5+\frac {1}{36} b^2 c^2 f g^2 \sqrt {1-c^2 x^2} x^5+\frac {1}{7} g^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x^4+\frac {3}{35} g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x^4-\frac {7}{16} b c f g^2 (a+b \arcsin (c x)) x^4+\frac {1}{2} f g^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x^3+\frac {3}{8} f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x^3+\frac {2 b g^3 (a+b \arcsin (c x)) x^3}{105 c}-\frac {4}{5} b c f^2 g (a+b \arcsin (c x)) x^3-\frac {43}{576} b^2 f g^2 \sqrt {1-c^2 x^2} x^3-\frac {g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x^2}{35 c^2}-\frac {3}{8} b c f^3 (a+b \arcsin (c x)) x^2+\frac {3 b f g^2 (a+b \arcsin (c x)) x^2}{16 c}+\frac {4 a b g^3 x}{35 c^3}+\frac {1}{4} f^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x+\frac {3}{8} f^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x-\frac {3 f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x}{16 c^2}-\frac {1}{32} b^2 f^3 \left (1-c^2 x^2\right )^{3/2} x+\frac {4 b^2 g^3 \arcsin (c x) x}{35 c^3}+\frac {6 b f^2 g (a+b \arcsin (c x)) x}{5 c}-\frac {15}{64} b^2 f^3 \sqrt {1-c^2 x^2} x-\frac {7 b^2 f g^2 \sqrt {1-c^2 x^2} x}{384 c^2}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^{7/2}}{343 c^4}+\frac {f^3 (a+b \arcsin (c x))^3}{8 b c}+\frac {f g^2 (a+b \arcsin (c x))^3}{16 b c^3}+\frac {38 b^2 g^3 \left (1-c^2 x^2\right )^{5/2}}{6125 c^4}+\frac {6 b^2 f^2 g \left (1-c^2 x^2\right )^{5/2}}{125 c^2}-\frac {3 f^2 g \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2}-\frac {2 g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{35 c^4}+\frac {152 b^2 g^3 \left (1-c^2 x^2\right )^{3/2}}{11025 c^4}+\frac {8 b^2 f^2 g \left (1-c^2 x^2\right )^{3/2}}{75 c^2}+\frac {9 b^2 f^3 \arcsin (c x)}{64 c}+\frac {7 b^2 f g^2 \arcsin (c x)}{384 c^3}+\frac {b f^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{8 c}+\frac {304 b^2 g^3 \sqrt {1-c^2 x^2}}{3675 c^4}+\frac {16 b^2 f^2 g \sqrt {1-c^2 x^2}}{25 c^2}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

Int[(f + g*x)^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]
 

Output:

(d*Sqrt[d - c^2*d*x^2]*((4*a*b*g^3*x)/(35*c^3) + (16*b^2*f^2*g*Sqrt[1 - c^ 
2*x^2])/(25*c^2) + (304*b^2*g^3*Sqrt[1 - c^2*x^2])/(3675*c^4) - (15*b^2*f^ 
3*x*Sqrt[1 - c^2*x^2])/64 - (7*b^2*f*g^2*x*Sqrt[1 - c^2*x^2])/(384*c^2) - 
(43*b^2*f*g^2*x^3*Sqrt[1 - c^2*x^2])/576 + (b^2*c^2*f*g^2*x^5*Sqrt[1 - c^2 
*x^2])/36 + (8*b^2*f^2*g*(1 - c^2*x^2)^(3/2))/(75*c^2) + (152*b^2*g^3*(1 - 
 c^2*x^2)^(3/2))/(11025*c^4) - (b^2*f^3*x*(1 - c^2*x^2)^(3/2))/32 + (6*b^2 
*f^2*g*(1 - c^2*x^2)^(5/2))/(125*c^2) + (38*b^2*g^3*(1 - c^2*x^2)^(5/2))/( 
6125*c^4) - (2*b^2*g^3*(1 - c^2*x^2)^(7/2))/(343*c^4) + (9*b^2*f^3*ArcSin[ 
c*x])/(64*c) + (7*b^2*f*g^2*ArcSin[c*x])/(384*c^3) + (4*b^2*g^3*x*ArcSin[c 
*x])/(35*c^3) + (6*b*f^2*g*x*(a + b*ArcSin[c*x]))/(5*c) - (3*b*c*f^3*x^2*( 
a + b*ArcSin[c*x]))/8 + (3*b*f*g^2*x^2*(a + b*ArcSin[c*x]))/(16*c) - (4*b* 
c*f^2*g*x^3*(a + b*ArcSin[c*x]))/5 + (2*b*g^3*x^3*(a + b*ArcSin[c*x]))/(10 
5*c) - (7*b*c*f*g^2*x^4*(a + b*ArcSin[c*x]))/16 + (6*b*c^3*f^2*g*x^5*(a + 
b*ArcSin[c*x]))/25 - (16*b*c*g^3*x^5*(a + b*ArcSin[c*x]))/175 + (b*c^3*f*g 
^2*x^6*(a + b*ArcSin[c*x]))/6 + (2*b*c^3*g^3*x^7*(a + b*ArcSin[c*x]))/49 + 
 (b*f^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(8*c) - (2*g^3*Sqrt[1 - c^2*x 
^2]*(a + b*ArcSin[c*x])^2)/(35*c^4) + (3*f^3*x*Sqrt[1 - c^2*x^2]*(a + b*Ar 
cSin[c*x])^2)/8 - (3*f*g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(16* 
c^2) - (g^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^2) + (3*f*g 
^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/8 + (3*g^3*x^4*Sqrt[1 -...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5262

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))
 

rule 5276

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.51 (sec) , antiderivative size = 4176, normalized size of antiderivative = 2.72

method	result	size
default	\(\text {Expression too large to display}\)	\(4176\)
parts	\(\text {Expression too large to display}\)	\(4176\)

Input:

int((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVER 
BOSE)
 

Output:

a^2*(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*x^2+d)^(5/2)/c^2/d)+b^2*(-1/16*(- 
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)^3*f*(2 
*c^2*f^2+g^2)*d-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 
^3*(14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d/c^4/(c^2*x^2-1)-1/2304*(-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*g^2*(6*I*arcsin(c*x)+18*arcsin(c*x)^2-1)*d/c^3 
/(c^2*x^2-1)-1/16000*(-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*(120*I*arcsin(c*x)*c^2*f^2+300*arcsin(c*x)^2*c^2 
*f^2-10*I*arcsin(c*x)*g^2-25*arcsin(c*x)^2*g^2-24*c^2*f^2+2*g^2)*d/c^4/(c^ 
2*x^2-1)-1/1024*(-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)+...
 

Fricas [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm= 
"fricas")
 

Output:

integral(-(a^2*c^2*d*g^3*x^5 + 3*a^2*c^2*d*f*g^2*x^4 - 3*a^2*d*f^2*g*x - a 
^2*d*f^3 + (3*a^2*c^2*d*f^2*g - a^2*d*g^3)*x^3 + (a^2*c^2*d*f^3 - 3*a^2*d* 
f*g^2)*x^2 + (b^2*c^2*d*g^3*x^5 + 3*b^2*c^2*d*f*g^2*x^4 - 3*b^2*d*f^2*g*x 
- b^2*d*f^3 + (3*b^2*c^2*d*f^2*g - b^2*d*g^3)*x^3 + (b^2*c^2*d*f^3 - 3*b^2 
*d*f*g^2)*x^2)*arcsin(c*x)^2 + 2*(a*b*c^2*d*g^3*x^5 + 3*a*b*c^2*d*f*g^2*x^ 
4 - 3*a*b*d*f^2*g*x - a*b*d*f^3 + (3*a*b*c^2*d*f^2*g - a*b*d*g^3)*x^3 + (a 
*b*c^2*d*f^3 - 3*a*b*d*f*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
 

Sympy [F(-1)]

Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \] Input:

integrate((g*x+f)**3*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm= 
"maxima")
 

Output:

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^2*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^2*g^3 + 1/16*a^2*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^2*f^2* 
g/(c^2*d) + sqrt(d)*integrate(-((b^2*c^2*d*g^3*x^5 + 3*b^2*c^2*d*f*g^2*x^4 
 - 3*b^2*d*f^2*g*x - b^2*d*f^3 + (3*b^2*c^2*d*f^2*g - b^2*d*g^3)*x^3 + (b^ 
2*c^2*d*f^3 - 3*b^2*d*f*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1 
))^2 + 2*(a*b*c^2*d*g^3*x^5 + 3*a*b*c^2*d*f*g^2*x^4 - 3*a*b*d*f^2*g*x - a* 
b*d*f^3 + (3*a*b*c^2*d*f^2*g - a*b*d*g^3)*x^3 + (a*b*c^2*d*f^3 - 3*a*b*d*f 
*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt( 
-c*x + 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^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
 

Mupad [F(-1)]

Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:

int((f + g*x)^3*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2),x)
 

Output:

int((f + g*x)^3*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Input:

int((g*x+f)^3*(-c^2*d*x^2+d)^(3/2)*(a+b*asin(c*x))^2,x)
 

Output:

(sqrt(d)*d*(210*asin(c*x)*a**2*c**3*f**3 + 105*asin(c*x)*a**2*c*f*g**2 - 1 
40*sqrt( - c**2*x**2 + 1)*a**2*c**6*f**3*x**3 - 336*sqrt( - c**2*x**2 + 1) 
*a**2*c**6*f**2*g*x**4 - 280*sqrt( - c**2*x**2 + 1)*a**2*c**6*f*g**2*x**5 
- 80*sqrt( - c**2*x**2 + 1)*a**2*c**6*g**3*x**6 + 350*sqrt( - c**2*x**2 + 
1)*a**2*c**4*f**3*x + 672*sqrt( - c**2*x**2 + 1)*a**2*c**4*f**2*g*x**2 + 4 
90*sqrt( - c**2*x**2 + 1)*a**2*c**4*f*g**2*x**3 + 128*sqrt( - c**2*x**2 + 
1)*a**2*c**4*g**3*x**4 - 336*sqrt( - c**2*x**2 + 1)*a**2*c**2*f**2*g - 105 
*sqrt( - c**2*x**2 + 1)*a**2*c**2*f*g**2*x - 16*sqrt( - c**2*x**2 + 1)*a** 
2*c**2*g**3*x**2 - 32*sqrt( - c**2*x**2 + 1)*a**2*g**3 - 1120*int(sqrt( - 
c**2*x**2 + 1)*asin(c*x)*x**5,x)*a*b*c**6*g**3 - 3360*int(sqrt( - c**2*x** 
2 + 1)*asin(c*x)*x**4,x)*a*b*c**6*f*g**2 - 3360*int(sqrt( - c**2*x**2 + 1) 
*asin(c*x)*x**3,x)*a*b*c**6*f**2*g + 1120*int(sqrt( - c**2*x**2 + 1)*asin( 
c*x)*x**3,x)*a*b*c**4*g**3 - 1120*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x** 
2,x)*a*b*c**6*f**3 + 3360*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**2,x)*a*b 
*c**4*f*g**2 + 3360*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x,x)*a*b*c**4*f** 
2*g + 1120*int(sqrt( - c**2*x**2 + 1)*asin(c*x),x)*a*b*c**4*f**3 - 560*int 
(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**5,x)*b**2*c**6*g**3 - 1680*int(sqr 
t( - c**2*x**2 + 1)*asin(c*x)**2*x**4,x)*b**2*c**6*f*g**2 - 1680*int(sqrt( 
 - c**2*x**2 + 1)*asin(c*x)**2*x**3,x)*b**2*c**6*f**2*g + 560*int(sqrt( - 
c**2*x**2 + 1)*asin(c*x)**2*x**3,x)*b**2*c**4*g**3 - 560*int(sqrt( - c*...