\(\int x^3 (d+e x^2)^3 (a+b \arcsin (c x)) \, dx\) [442]

Optimal result

Integrand size = 21, antiderivative size = 335 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {b \left (480 c^6 d^3+800 c^4 d^2 e+525 c^2 d e^2+126 e^3\right ) x \sqrt {1-c^2 x^2}}{5120 c^9}+\frac {b \left (480 c^6 d^3+800 c^4 d^2 e+525 c^2 d e^2+126 e^3\right ) x^3 \sqrt {1-c^2 x^2}}{7680 c^7}+\frac {b e \left (800 c^4 d^2+525 c^2 d e+126 e^2\right ) x^5 \sqrt {1-c^2 x^2}}{9600 c^5}+\frac {3 b e^2 \left (25 c^2 d+6 e\right ) x^7 \sqrt {1-c^2 x^2}}{1600 c^3}+\frac {b e^3 x^9 \sqrt {1-c^2 x^2}}{100 c}+\frac {b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \arcsin (c x)}{5120 c^{10} e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2} \] Output:

1/5120*b*(480*c^6*d^3+800*c^4*d^2*e+525*c^2*d*e^2+126*e^3)*x*(-c^2*x^2+1)^ 
(1/2)/c^9+1/7680*b*(480*c^6*d^3+800*c^4*d^2*e+525*c^2*d*e^2+126*e^3)*x^3*( 
-c^2*x^2+1)^(1/2)/c^7+1/9600*b*e*(800*c^4*d^2+525*c^2*d*e+126*e^2)*x^5*(-c 
^2*x^2+1)^(1/2)/c^5+3/1600*b*e^2*(25*c^2*d+6*e)*x^7*(-c^2*x^2+1)^(1/2)/c^3 
+1/100*b*e^3*x^9*(-c^2*x^2+1)^(1/2)/c+1/5120*b*(128*c^10*d^5-480*c^6*d^3*e 
^2-800*c^4*d^2*e^3-525*c^2*d*e^4-126*e^5)*arcsin(c*x)/c^10/e^2-1/8*d*(e*x^ 
2+d)^4*(a+b*arcsin(c*x))/e^2+1/10*(e*x^2+d)^5*(a+b*arcsin(c*x))/e^2
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.82 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {c x \left (1920 a c^9 x^3 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )+b \sqrt {1-c^2 x^2} \left (1890 e^3+315 c^2 e^2 \left (25 d+4 e x^2\right )+6 c^4 e \left (2000 d^2+875 d e x^2+168 e^2 x^4\right )+8 c^6 \left (900 d^3+1000 d^2 e x^2+525 d e^2 x^4+108 e^3 x^6\right )+16 c^8 \left (300 d^3 x^2+400 d^2 e x^4+225 d e^2 x^6+48 e^3 x^8\right )\right )\right )+15 b \left (-480 c^6 d^3-800 c^4 d^2 e-525 c^2 d e^2-126 e^3+128 c^{10} x^4 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )\right ) \arcsin (c x)}{76800 c^{10}} \] Input:

Integrate[x^3*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]
 

Output:

(c*x*(1920*a*c^9*x^3*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) + 
b*Sqrt[1 - c^2*x^2]*(1890*e^3 + 315*c^2*e^2*(25*d + 4*e*x^2) + 6*c^4*e*(20 
00*d^2 + 875*d*e*x^2 + 168*e^2*x^4) + 8*c^6*(900*d^3 + 1000*d^2*e*x^2 + 52 
5*d*e^2*x^4 + 108*e^3*x^6) + 16*c^8*(300*d^3*x^2 + 400*d^2*e*x^4 + 225*d*e 
^2*x^6 + 48*e^3*x^8))) + 15*b*(-480*c^6*d^3 - 800*c^4*d^2*e - 525*c^2*d*e^ 
2 - 126*e^3 + 128*c^10*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x 
^6))*ArcSin[c*x])/(76800*c^10)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5230, 27, 403, 27, 403, 25, 403, 25, 403, 25, 299, 223}

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.

\(\displaystyle \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5230

\(\displaystyle -b c \int -\frac {\left (d-4 e x^2\right ) \left (e x^2+d\right )^4}{40 e^2 \sqrt {1-c^2 x^2}}dx+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \int \frac {\left (d-4 e x^2\right ) \left (e x^2+d\right )^4}{\sqrt {1-c^2 x^2}}dx}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}-\frac {\int -\frac {2 \left (e x^2+d\right )^3 \left (d \left (5 c^2 d-2 e\right )-e \left (11 d c^2+18 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{10 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \left (\frac {\int \frac {\left (e x^2+d\right )^3 \left (d \left (5 c^2 d-2 e\right )-e \left (11 d c^2+18 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}-\frac {\int -\frac {\left (e x^2+d\right )^2 \left (d \left (40 d^2 c^4-27 d e c^2-18 e^2\right )-e \left (26 d^2 c^4+201 d e c^2+126 e^2\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c \left (\frac {\frac {\int \frac {\left (e x^2+d\right )^2 \left (d \left (40 d^2 c^4-27 d e c^2-18 e^2\right )-e \left (26 d^2 c^4+201 d e c^2+126 e^2\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {\frac {\frac {e x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}-\frac {\int -\frac {\left (e x^2+d\right ) \left (e \left (136 d^3 c^6-1096 d^2 e c^4-1617 d e^2 c^2-630 e^3\right ) x^2+d \left (240 d^3 c^6-188 d^2 e c^4-309 d e^2 c^2-126 e^3\right )\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}}{8 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c \left (\frac {\frac {\frac {\int \frac {\left (e x^2+d\right ) \left (e \left (136 d^3 c^6-1096 d^2 e c^4-1617 d e^2 c^2-630 e^3\right ) x^2+d \left (240 d^3 c^6-188 d^2 e c^4-309 d e^2 c^2-126 e^3\right )\right )}{\sqrt {1-c^2 x^2}}dx}{6 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {\frac {\frac {-\frac {\int -\frac {e \left (1232 d^4 c^8-2536 d^3 e c^6-7758 d^2 e^2 c^4-6615 d e^3 c^2-1890 e^4\right ) x^2+d \left (960 d^4 c^8-616 d^3 e c^6-2332 d^2 e^2 c^4-2121 d e^3 c^2-630 e^4\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c \left (\frac {\frac {\frac {\frac {\int \frac {e \left (1232 d^4 c^8-2536 d^3 e c^6-7758 d^2 e^2 c^4-6615 d e^3 c^2-1890 e^4\right ) x^2+d \left (960 d^4 c^8-616 d^3 e c^6-2332 d^2 e^2 c^4-2121 d e^3 c^2-630 e^4\right )}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b c \left (\frac {\frac {\frac {\frac {\frac {15 \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )}{2 c^2}}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\left (d+e x^2\right )^5 (a+b \arcsin (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arcsin (c x))}{8 e^2}+\frac {b c \left (\frac {\frac {\frac {\frac {\frac {15 \arcsin (c x) \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right )}{2 c^3}-\frac {e x \sqrt {1-c^2 x^2} \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )}{2 c^2}}{4 c^2}-\frac {e x \sqrt {1-c^2 x^2} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}+\frac {2 e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2}\)

Input:

Int[x^3*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]
 

Output:

-1/8*(d*(d + e*x^2)^4*(a + b*ArcSin[c*x]))/e^2 + ((d + e*x^2)^5*(a + b*Arc 
Sin[c*x]))/(10*e^2) + (b*c*((2*e*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^4)/(5*c^2 
) + ((e*(11*c^2*d + 18*e)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^3)/(8*c^2) + ((e 
*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^2)/( 
6*c^2) + (-1/4*(e*(136*c^6*d^3 - 1096*c^4*d^2*e - 1617*c^2*d*e^2 - 630*e^3 
)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/c^2 + (-1/2*(e*(1232*c^8*d^4 - 2536*c^6 
*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^4)*x*Sqrt[1 - c^2*x^2] 
)/c^2 + (15*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 800*c^4*d^2*e^3 - 525*c^2*d* 
e^4 - 126*e^5)*ArcSin[c*x])/(2*c^3))/(4*c^2))/(6*c^2))/(8*c^2))/(5*c^2)))/ 
(40*e^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 5230

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcSin[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - 
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 
0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.30

method	result	size
parts	\(a \left (\frac {1}{10} e^{3} x^{10}+\frac {3}{8} d \,e^{2} x^{8}+\frac {1}{2} d^{2} e \,x^{6}+\frac {1}{4} x^{4} d^{3}\right )+\frac {b \left (\frac {c^{4} \arcsin \left (c x \right ) e^{3} x^{10}}{10}+\frac {3 c^{4} \arcsin \left (c x \right ) d \,e^{2} x^{8}}{8}+\frac {c^{4} \arcsin \left (c x \right ) d^{2} e \,x^{6}}{2}+\frac {\arcsin \left (c x \right ) c^{4} x^{4} d^{3}}{4}-\frac {4 e^{3} \left (-\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{10}-\frac {9 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{80}-\frac {21 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{160}-\frac {21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{128}-\frac {63 c x \sqrt {-c^{2} x^{2}+1}}{256}+\frac {63 \arcsin \left (c x \right )}{256}\right )+10 d^{3} c^{6} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+15 d \,c^{2} e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )+20 d^{2} c^{4} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{40 c^{6}}\right )}{c^{4}}\)	\(436\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} c^{10} d^{3} x^{4}+\frac {1}{2} c^{10} d^{2} e \,x^{6}+\frac {3}{8} c^{10} d \,e^{2} x^{8}+\frac {1}{10} e^{3} x^{10} c^{10}\right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{3} c^{10} x^{4}}{4}+\frac {\arcsin \left (c x \right ) d^{2} c^{10} e \,x^{6}}{2}+\frac {3 \arcsin \left (c x \right ) d \,c^{10} e^{2} x^{8}}{8}+\frac {\arcsin \left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {e^{3} \left (-\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{10}-\frac {9 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{80}-\frac {21 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{160}-\frac {21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{128}-\frac {63 c x \sqrt {-c^{2} x^{2}+1}}{256}+\frac {63 \arcsin \left (c x \right )}{256}\right )}{10}-\frac {d^{3} c^{6} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}-\frac {d^{2} c^{4} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{2}\right )}{c^{6}}}{c^{4}}\)	\(449\)
default	\(\frac {\frac {a \left (\frac {1}{4} c^{10} d^{3} x^{4}+\frac {1}{2} c^{10} d^{2} e \,x^{6}+\frac {3}{8} c^{10} d \,e^{2} x^{8}+\frac {1}{10} e^{3} x^{10} c^{10}\right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{3} c^{10} x^{4}}{4}+\frac {\arcsin \left (c x \right ) d^{2} c^{10} e \,x^{6}}{2}+\frac {3 \arcsin \left (c x \right ) d \,c^{10} e^{2} x^{8}}{8}+\frac {\arcsin \left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {e^{3} \left (-\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{10}-\frac {9 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{80}-\frac {21 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{160}-\frac {21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{128}-\frac {63 c x \sqrt {-c^{2} x^{2}+1}}{256}+\frac {63 \arcsin \left (c x \right )}{256}\right )}{10}-\frac {d^{3} c^{6} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}-\frac {d^{2} c^{4} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{2}\right )}{c^{6}}}{c^{4}}\)	\(449\)
orering	\(\frac {\left (4864 c^{10} e^{4} x^{12}+23728 c^{10} d \,e^{3} x^{10}+45200 c^{10} d^{2} e^{2} x^{8}+288 c^{8} e^{4} x^{10}+40000 c^{10} d^{3} e \,x^{6}+1896 c^{8} d \,e^{3} x^{8}+11200 c^{10} d^{4} x^{4}+5400 c^{8} d^{2} e^{2} x^{6}+432 c^{6} e^{4} x^{8}+8800 c^{8} d^{3} e \,x^{4}+3294 c^{6} d \,e^{3} x^{6}+2400 c^{8} d^{4} x^{2}+13050 c^{6} d^{2} e^{2} x^{4}+756 c^{4} e^{4} x^{6}-20000 c^{6} d^{3} e \,x^{2}+8127 c^{4} d \,e^{3} x^{4}-9600 c^{6} d^{4}-37375 c^{4} d^{2} e^{2} x^{2}+1890 c^{2} e^{4} x^{4}-16000 c^{4} d^{3} e -25620 c^{2} d \,e^{3} x^{2}-10500 c^{2} d^{2} e^{2}-6300 e^{4} x^{2}-2520 d \,e^{3}\right ) \left (a +b \arcsin \left (c x \right )\right )}{25600 \left (e \,x^{2}+d \right ) c^{10}}-\frac {\left (768 c^{8} e^{3} x^{8}+3600 c^{8} d \,e^{2} x^{6}+6400 c^{8} d^{2} e \,x^{4}+864 e^{3} x^{6} c^{6}+4800 c^{8} d^{3} x^{2}+4200 c^{6} d \,e^{2} x^{4}+8000 c^{6} d^{2} e \,x^{2}+1008 c^{4} e^{3} x^{4}+7200 c^{6} d^{3}+5250 c^{4} d \,e^{2} x^{2}+12000 c^{4} d^{2} e +1260 c^{2} e^{3} x^{2}+7875 c^{2} d \,e^{2}+1890 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (3 x^{2} \left (e \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )+6 x^{4} \left (e \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) e +\frac {x^{3} \left (e \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{76800 x^{2} c^{10} \left (e \,x^{2}+d \right )^{3}}\)	\(543\)

Input:

int(x^3*(e*x^2+d)^3*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

Output:

a*(1/10*e^3*x^10+3/8*d*e^2*x^8+1/2*d^2*e*x^6+1/4*x^4*d^3)+b/c^4*(1/10*c^4* 
arcsin(c*x)*e^3*x^10+3/8*c^4*arcsin(c*x)*d*e^2*x^8+1/2*c^4*arcsin(c*x)*d^2 
*e*x^6+1/4*arcsin(c*x)*c^4*x^4*d^3-1/40/c^6*(4*e^3*(-1/10*c^9*x^9*(-c^2*x^ 
2+1)^(1/2)-9/80*c^7*x^7*(-c^2*x^2+1)^(1/2)-21/160*c^5*x^5*(-c^2*x^2+1)^(1/ 
2)-21/128*c^3*x^3*(-c^2*x^2+1)^(1/2)-63/256*c*x*(-c^2*x^2+1)^(1/2)+63/256* 
arcsin(c*x))+10*d^3*c^6*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2 
+1)^(1/2)+3/8*arcsin(c*x))+15*d*c^2*e^2*(-1/8*c^7*x^7*(-c^2*x^2+1)^(1/2)-7 
/48*c^5*x^5*(-c^2*x^2+1)^(1/2)-35/192*c^3*x^3*(-c^2*x^2+1)^(1/2)-35/128*c* 
x*(-c^2*x^2+1)^(1/2)+35/128*arcsin(c*x))+20*d^2*c^4*e*(-1/6*c^5*x^5*(-c^2* 
x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2*x^2+1)^(1/2)-5/16*c*x*(-c^2*x^2+1)^(1/2)+5 
/16*arcsin(c*x))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.95 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {7680 \, a c^{10} e^{3} x^{10} + 28800 \, a c^{10} d e^{2} x^{8} + 38400 \, a c^{10} d^{2} e x^{6} + 19200 \, a c^{10} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} e^{3} x^{10} + 1920 \, b c^{10} d e^{2} x^{8} + 2560 \, b c^{10} d^{2} e x^{6} + 1280 \, b c^{10} d^{3} x^{4} - 480 \, b c^{6} d^{3} - 800 \, b c^{4} d^{2} e - 525 \, b c^{2} d e^{2} - 126 \, b e^{3}\right )} \arcsin \left (c x\right ) + {\left (768 \, b c^{9} e^{3} x^{9} + 144 \, {\left (25 \, b c^{9} d e^{2} + 6 \, b c^{7} e^{3}\right )} x^{7} + 8 \, {\left (800 \, b c^{9} d^{2} e + 525 \, b c^{7} d e^{2} + 126 \, b c^{5} e^{3}\right )} x^{5} + 10 \, {\left (480 \, b c^{9} d^{3} + 800 \, b c^{7} d^{2} e + 525 \, b c^{5} d e^{2} + 126 \, b c^{3} e^{3}\right )} x^{3} + 15 \, {\left (480 \, b c^{7} d^{3} + 800 \, b c^{5} d^{2} e + 525 \, b c^{3} d e^{2} + 126 \, b c e^{3}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{76800 \, c^{10}} \] Input:

integrate(x^3*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="fricas")
 

Output:

1/76800*(7680*a*c^10*e^3*x^10 + 28800*a*c^10*d*e^2*x^8 + 38400*a*c^10*d^2* 
e*x^6 + 19200*a*c^10*d^3*x^4 + 15*(512*b*c^10*e^3*x^10 + 1920*b*c^10*d*e^2 
*x^8 + 2560*b*c^10*d^2*e*x^6 + 1280*b*c^10*d^3*x^4 - 480*b*c^6*d^3 - 800*b 
*c^4*d^2*e - 525*b*c^2*d*e^2 - 126*b*e^3)*arcsin(c*x) + (768*b*c^9*e^3*x^9 
 + 144*(25*b*c^9*d*e^2 + 6*b*c^7*e^3)*x^7 + 8*(800*b*c^9*d^2*e + 525*b*c^7 
*d*e^2 + 126*b*c^5*e^3)*x^5 + 10*(480*b*c^9*d^3 + 800*b*c^7*d^2*e + 525*b* 
c^5*d*e^2 + 126*b*c^3*e^3)*x^3 + 15*(480*b*c^7*d^3 + 800*b*c^5*d^2*e + 525 
*b*c^3*d*e^2 + 126*b*c*e^3)*x)*sqrt(-c^2*x^2 + 1))/c^10
 

Sympy [A] (verification not implemented)

Time = 1.58 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.78 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{4}}{4} + \frac {a d^{2} e x^{6}}{2} + \frac {3 a d e^{2} x^{8}}{8} + \frac {a e^{3} x^{10}}{10} + \frac {b d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b d^{2} e x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {3 b d e^{2} x^{8} \operatorname {asin}{\left (c x \right )}}{8} + \frac {b e^{3} x^{10} \operatorname {asin}{\left (c x \right )}}{10} + \frac {b d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b d^{2} e x^{5} \sqrt {- c^{2} x^{2} + 1}}{12 c} + \frac {3 b d e^{2} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64 c} + \frac {b e^{3} x^{9} \sqrt {- c^{2} x^{2} + 1}}{100 c} + \frac {3 b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {5 b d^{2} e x^{3} \sqrt {- c^{2} x^{2} + 1}}{48 c^{3}} + \frac {7 b d e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{128 c^{3}} + \frac {9 b e^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{800 c^{3}} - \frac {3 b d^{3} \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {5 b d^{2} e x \sqrt {- c^{2} x^{2} + 1}}{32 c^{5}} + \frac {35 b d e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{512 c^{5}} + \frac {21 b e^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{1600 c^{5}} - \frac {5 b d^{2} e \operatorname {asin}{\left (c x \right )}}{32 c^{6}} + \frac {105 b d e^{2} x \sqrt {- c^{2} x^{2} + 1}}{1024 c^{7}} + \frac {21 b e^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1280 c^{7}} - \frac {105 b d e^{2} \operatorname {asin}{\left (c x \right )}}{1024 c^{8}} + \frac {63 b e^{3} x \sqrt {- c^{2} x^{2} + 1}}{2560 c^{9}} - \frac {63 b e^{3} \operatorname {asin}{\left (c x \right )}}{2560 c^{10}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{4}}{4} + \frac {d^{2} e x^{6}}{2} + \frac {3 d e^{2} x^{8}}{8} + \frac {e^{3} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(x**3*(e*x**2+d)**3*(a+b*asin(c*x)),x)
 

Output:

Piecewise((a*d**3*x**4/4 + a*d**2*e*x**6/2 + 3*a*d*e**2*x**8/8 + a*e**3*x* 
*10/10 + b*d**3*x**4*asin(c*x)/4 + b*d**2*e*x**6*asin(c*x)/2 + 3*b*d*e**2* 
x**8*asin(c*x)/8 + b*e**3*x**10*asin(c*x)/10 + b*d**3*x**3*sqrt(-c**2*x**2 
 + 1)/(16*c) + b*d**2*e*x**5*sqrt(-c**2*x**2 + 1)/(12*c) + 3*b*d*e**2*x**7 
*sqrt(-c**2*x**2 + 1)/(64*c) + b*e**3*x**9*sqrt(-c**2*x**2 + 1)/(100*c) + 
3*b*d**3*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 5*b*d**2*e*x**3*sqrt(-c**2*x** 
2 + 1)/(48*c**3) + 7*b*d*e**2*x**5*sqrt(-c**2*x**2 + 1)/(128*c**3) + 9*b*e 
**3*x**7*sqrt(-c**2*x**2 + 1)/(800*c**3) - 3*b*d**3*asin(c*x)/(32*c**4) + 
5*b*d**2*e*x*sqrt(-c**2*x**2 + 1)/(32*c**5) + 35*b*d*e**2*x**3*sqrt(-c**2* 
x**2 + 1)/(512*c**5) + 21*b*e**3*x**5*sqrt(-c**2*x**2 + 1)/(1600*c**5) - 5 
*b*d**2*e*asin(c*x)/(32*c**6) + 105*b*d*e**2*x*sqrt(-c**2*x**2 + 1)/(1024* 
c**7) + 21*b*e**3*x**3*sqrt(-c**2*x**2 + 1)/(1280*c**7) - 105*b*d*e**2*asi 
n(c*x)/(1024*c**8) + 63*b*e**3*x*sqrt(-c**2*x**2 + 1)/(2560*c**9) - 63*b*e 
**3*asin(c*x)/(2560*c**10), Ne(c, 0)), (a*(d**3*x**4/4 + d**2*e*x**6/2 + 3 
*d*e**2*x**8/8 + e**3*x**10/10), True))
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.27 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {1}{10} \, a e^{3} x^{10} + \frac {3}{8} \, a d e^{2} x^{8} + \frac {1}{2} \, a d^{2} e x^{6} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{3} + \frac {1}{96} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b d^{2} e + \frac {1}{1024} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b d e^{2} + \frac {1}{12800} \, {\left (1280 \, x^{10} \arcsin \left (c x\right ) + {\left (\frac {128 \, \sqrt {-c^{2} x^{2} + 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \arcsin \left (c x\right )}{c^{11}}\right )} c\right )} b e^{3} \] Input:

integrate(x^3*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/10*a*e^3*x^10 + 3/8*a*d*e^2*x^8 + 1/2*a*d^2*e*x^6 + 1/4*a*d^3*x^4 + 1/32 
*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1) 
*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d^3 + 1/96*(48*x^6*arcsin(c*x) + (8*sqrt( 
-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 
 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*b*d^2*e + 1/1024*(384*x^8*arcsin(c*x) + 
 (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt( 
-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9 
)*c)*b*d*e^2 + 1/12800*(1280*x^10*arcsin(c*x) + (128*sqrt(-c^2*x^2 + 1)*x^ 
9/c^2 + 144*sqrt(-c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(-c^2*x^2 + 1)*x^5/c^6 + 
210*sqrt(-c^2*x^2 + 1)*x^3/c^8 + 315*sqrt(-c^2*x^2 + 1)*x/c^10 - 315*arcsi 
n(c*x)/c^11)*c)*b*e^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (309) = 618\).

Time = 0.14 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.41 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx =\text {Too large to display} \] Input:

integrate(x^3*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="giac")
 

Output:

1/10*a*e^3*x^10 + 3/8*a*d*e^2*x^8 + 1/2*a*d^2*e*x^6 + 1/4*a*d^3*x^4 - 1/16 
*(-c^2*x^2 + 1)^(3/2)*b*d^3*x/c^3 + 1/4*(c^2*x^2 - 1)^2*b*d^3*arcsin(c*x)/ 
c^4 + 5/32*sqrt(-c^2*x^2 + 1)*b*d^3*x/c^3 + 1/12*(c^2*x^2 - 1)^2*sqrt(-c^2 
*x^2 + 1)*b*d^2*e*x/c^5 + 1/2*(c^2*x^2 - 1)*b*d^3*arcsin(c*x)/c^4 + 1/2*(c 
^2*x^2 - 1)^3*b*d^2*e*arcsin(c*x)/c^6 - 13/48*(-c^2*x^2 + 1)^(3/2)*b*d^2*e 
*x/c^5 + 3/64*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d*e^2*x/c^7 + 5/32*b*d^ 
3*arcsin(c*x)/c^4 + 3/2*(c^2*x^2 - 1)^2*b*d^2*e*arcsin(c*x)/c^6 + 3/8*(c^2 
*x^2 - 1)^4*b*d*e^2*arcsin(c*x)/c^8 + 11/32*sqrt(-c^2*x^2 + 1)*b*d^2*e*x/c 
^5 + 25/128*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e^2*x/c^7 + 1/100*(c^2* 
x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*e^3*x/c^9 + 3/2*(c^2*x^2 - 1)*b*d^2*e*arcs 
in(c*x)/c^6 + 3/2*(c^2*x^2 - 1)^3*b*d*e^2*arcsin(c*x)/c^8 + 1/10*(c^2*x^2 
- 1)^5*b*e^3*arcsin(c*x)/c^10 - 163/512*(-c^2*x^2 + 1)^(3/2)*b*d*e^2*x/c^7 
 + 41/800*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^3*x/c^9 + 11/32*b*d^2*e*a 
rcsin(c*x)/c^6 + 9/4*(c^2*x^2 - 1)^2*b*d*e^2*arcsin(c*x)/c^8 + 1/2*(c^2*x^ 
2 - 1)^4*b*e^3*arcsin(c*x)/c^10 + 279/1024*sqrt(-c^2*x^2 + 1)*b*d*e^2*x/c^ 
7 + 171/1600*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^3*x/c^9 + 3/2*(c^2*x^2 
 - 1)*b*d*e^2*arcsin(c*x)/c^8 + (c^2*x^2 - 1)^3*b*e^3*arcsin(c*x)/c^10 - 1 
49/1280*(-c^2*x^2 + 1)^(3/2)*b*e^3*x/c^9 + 279/1024*b*d*e^2*arcsin(c*x)/c^ 
8 + (c^2*x^2 - 1)^2*b*e^3*arcsin(c*x)/c^10 + 193/2560*sqrt(-c^2*x^2 + 1)*b 
*e^3*x/c^9 + 1/2*(c^2*x^2 - 1)*b*e^3*arcsin(c*x)/c^10 + 193/2560*b*e^3*...
 

Mupad [F(-1)]

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

int(x^3*(a + b*asin(c*x))*(d + e*x^2)^3,x)
 

Output:

int(x^3*(a + b*asin(c*x))*(d + e*x^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.47 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {-1890 \mathit {asin} \left (c x \right ) b \,e^{3}-7200 \mathit {asin} \left (c x \right ) b \,c^{6} d^{3}+19200 a \,c^{10} d^{3} x^{4}+7680 a \,c^{10} e^{3} x^{10}+19200 \mathit {asin} \left (c x \right ) b \,c^{10} d^{3} x^{4}+7680 \mathit {asin} \left (c x \right ) b \,c^{10} e^{3} x^{10}-12000 \mathit {asin} \left (c x \right ) b \,c^{4} d^{2} e -7875 \mathit {asin} \left (c x \right ) b \,c^{2} d \,e^{2}+4800 \sqrt {-c^{2} x^{2}+1}\, b \,c^{9} d^{3} x^{3}+768 \sqrt {-c^{2} x^{2}+1}\, b \,c^{9} e^{3} x^{9}+7200 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d^{3} x +864 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} e^{3} x^{7}+1008 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} e^{3} x^{5}+1260 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{3} x^{3}+1890 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{3} x +38400 a \,c^{10} d^{2} e \,x^{6}+28800 a \,c^{10} d \,e^{2} x^{8}+38400 \mathit {asin} \left (c x \right ) b \,c^{10} d^{2} e \,x^{6}+28800 \mathit {asin} \left (c x \right ) b \,c^{10} d \,e^{2} x^{8}+6400 \sqrt {-c^{2} x^{2}+1}\, b \,c^{9} d^{2} e \,x^{5}+3600 \sqrt {-c^{2} x^{2}+1}\, b \,c^{9} d \,e^{2} x^{7}+8000 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d^{2} e \,x^{3}+4200 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d \,e^{2} x^{5}+12000 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d^{2} e x +5250 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d \,e^{2} x^{3}+7875 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d \,e^{2} x}{76800 c^{10}} \] Input:

int(x^3*(e*x^2+d)^3*(a+b*asin(c*x)),x)
 

Output:

(19200*asin(c*x)*b*c**10*d**3*x**4 + 38400*asin(c*x)*b*c**10*d**2*e*x**6 + 
 28800*asin(c*x)*b*c**10*d*e**2*x**8 + 7680*asin(c*x)*b*c**10*e**3*x**10 - 
 7200*asin(c*x)*b*c**6*d**3 - 12000*asin(c*x)*b*c**4*d**2*e - 7875*asin(c* 
x)*b*c**2*d*e**2 - 1890*asin(c*x)*b*e**3 + 4800*sqrt( - c**2*x**2 + 1)*b*c 
**9*d**3*x**3 + 6400*sqrt( - c**2*x**2 + 1)*b*c**9*d**2*e*x**5 + 3600*sqrt 
( - c**2*x**2 + 1)*b*c**9*d*e**2*x**7 + 768*sqrt( - c**2*x**2 + 1)*b*c**9* 
e**3*x**9 + 7200*sqrt( - c**2*x**2 + 1)*b*c**7*d**3*x + 8000*sqrt( - c**2* 
x**2 + 1)*b*c**7*d**2*e*x**3 + 4200*sqrt( - c**2*x**2 + 1)*b*c**7*d*e**2*x 
**5 + 864*sqrt( - c**2*x**2 + 1)*b*c**7*e**3*x**7 + 12000*sqrt( - c**2*x** 
2 + 1)*b*c**5*d**2*e*x + 5250*sqrt( - c**2*x**2 + 1)*b*c**5*d*e**2*x**3 + 
1008*sqrt( - c**2*x**2 + 1)*b*c**5*e**3*x**5 + 7875*sqrt( - c**2*x**2 + 1) 
*b*c**3*d*e**2*x + 1260*sqrt( - c**2*x**2 + 1)*b*c**3*e**3*x**3 + 1890*sqr 
t( - c**2*x**2 + 1)*b*c*e**3*x + 19200*a*c**10*d**3*x**4 + 38400*a*c**10*d 
**2*e*x**6 + 28800*a*c**10*d*e**2*x**8 + 7680*a*c**10*e**3*x**10)/(76800*c 
**10)