\(\int (d+e x)^3 (a+b \arcsin (c x))^2 \, dx\) [9]

Optimal result

Integrand size = 18, antiderivative size = 374 \[ \int (d+e x)^3 (a+b \arcsin (c x))^2 \, dx=-2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}-\frac {2}{9} b^2 d e^2 x^3-\frac {1}{32} b^2 e^3 x^4+\frac {2 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c}+\frac {4 b d e^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^3}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c}+\frac {3 b e^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{16 c^3}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c}-\frac {d^4 (a+b \arcsin (c x))^2}{4 e}-\frac {3 d^2 e (a+b \arcsin (c x))^2}{4 c^2}-\frac {3 e^3 (a+b \arcsin (c x))^2}{32 c^4}+\frac {(d+e x)^4 (a+b \arcsin (c x))^2}{4 e} \] Output:

-2*b^2*d^3*x-4/3*b^2*d*e^2*x/c^2-3/4*b^2*d^2*e*x^2-3/32*b^2*e^3*x^2/c^2-2/ 
9*b^2*d*e^2*x^3-1/32*b^2*e^3*x^4+2*b*d^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c* 
x))/c+4/3*b*d*e^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3+3/2*b*d^2*e*x*( 
-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+3/16*b*e^3*x*(-c^2*x^2+1)^(1/2)*(a+b 
*arcsin(c*x))/c^3+2/3*b*d*e^2*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+1 
/8*b*e^3*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c-1/4*d^4*(a+b*arcsin(c* 
x))^2/e-3/4*d^2*e*(a+b*arcsin(c*x))^2/c^2-3/32*e^3*(a+b*arcsin(c*x))^2/c^4 
+1/4*(e*x+d)^4*(a+b*arcsin(c*x))^2/e
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.95 \[ \int (d+e x)^3 (a+b \arcsin (c x))^2 \, dx=\frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a b \sqrt {1-c^2 x^2} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )-b^2 c x \left (3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )+6 b \left (3 a \left (-24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+b c \sqrt {1-c^2 x^2} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \arcsin (c x)+9 b^2 \left (-24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \arcsin (c x)^2}{288 c^4} \] Input:

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

Output:

(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*b*Sqrt[ 
1 - c^2*x^2]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 
 + 6*e^3*x^3)) - b^2*c*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2*e 
*x + 64*d*e^2*x^2 + 9*e^3*x^3))) + 6*b*(3*a*(-24*c^2*d^2*e - 3*e^3 + 8*c^4 
*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)) + b*c*Sqrt[1 - c^2*x^2]*(e 
^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3))) 
*ArcSin[c*x] + 9*b^2*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 
 4*d*e^2*x^2 + e^3*x^3))*ArcSin[c*x]^2)/(288*c^4)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5242, 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.

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

\(\Big \downarrow \) 5242

\(\displaystyle \frac {(d+e x)^4 (a+b \arcsin (c x))^2}{4 e}-\frac {b c \int \frac {(d+e x)^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{2 e}\)

\(\Big \downarrow \) 5262

\(\displaystyle \frac {(d+e x)^4 (a+b \arcsin (c x))^2}{4 e}-\frac {b c \int \left (\frac {(a+b \arcsin (c x)) d^4}{\sqrt {1-c^2 x^2}}+\frac {4 e x (a+b \arcsin (c x)) d^3}{\sqrt {1-c^2 x^2}}+\frac {6 e^2 x^2 (a+b \arcsin (c x)) d^2}{\sqrt {1-c^2 x^2}}+\frac {4 e^3 x^3 (a+b \arcsin (c x)) d}{\sqrt {1-c^2 x^2}}+\frac {e^4 x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}\right )dx}{2 e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^4 (a+b \arcsin (c x))^2}{4 e}-\frac {b c \left (\frac {3 e^4 (a+b \arcsin (c x))^2}{16 b c^5}+\frac {3 d^2 e^2 (a+b \arcsin (c x))^2}{2 b c^3}-\frac {4 d^3 e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}-\frac {3 d^2 e^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}-\frac {4 d e^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}-\frac {e^4 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}-\frac {8 d e^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^4}-\frac {3 e^4 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c^4}+\frac {d^4 (a+b \arcsin (c x))^2}{2 b c}+\frac {8 b d e^3 x}{3 c^3}+\frac {3 b e^4 x^2}{16 c^3}+\frac {4 b d^3 e x}{c}+\frac {3 b d^2 e^2 x^2}{2 c}+\frac {4 b d e^3 x^3}{9 c}+\frac {b e^4 x^4}{16 c}\right )}{2 e}\)

Input:

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

Output:

((d + e*x)^4*(a + b*ArcSin[c*x])^2)/(4*e) - (b*c*((4*b*d^3*e*x)/c + (8*b*d 
*e^3*x)/(3*c^3) + (3*b*d^2*e^2*x^2)/(2*c) + (3*b*e^4*x^2)/(16*c^3) + (4*b* 
d*e^3*x^3)/(9*c) + (b*e^4*x^4)/(16*c) - (4*d^3*e*Sqrt[1 - c^2*x^2]*(a + b* 
ArcSin[c*x]))/c^2 - (8*d*e^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^4 
) - (3*d^2*e^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2 - (3*e^4*x*Sqr 
t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c^4) - (4*d*e^3*x^2*Sqrt[1 - c^2*x^ 
2]*(a + b*ArcSin[c*x]))/(3*c^2) - (e^4*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin 
[c*x]))/(4*c^2) + (d^4*(a + b*ArcSin[c*x])^2)/(2*b*c) + (3*d^2*e^2*(a + b* 
ArcSin[c*x])^2)/(2*b*c^3) + (3*e^4*(a + b*ArcSin[c*x])^2)/(16*b*c^5)))/(2* 
e)
 

Defintions of rubi rules used

rule 2009

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

rule 5242

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - 
Simp[b*c*(n/(e*(m + 1)))   Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 
1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] 
 && NeQ[m, -1]
 

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))
 

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (c x e +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (c^{3} d^{3} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {3 e \,c^{2} d^{2} \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {c d \,e^{2} \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {e^{3} \left (32 \arcsin \left (c x \right )^{2} x^{4} c^{4}+16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-4 c^{4} x^{4}+24 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -12 \arcsin \left (c x \right )^{2}-12 c^{2} x^{2}-9\right )}{128}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{4} d^{4}}{4 e}+\arcsin \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arcsin \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arcsin \left (c x \right ) c^{4} d \,x^{3}+\frac {\arcsin \left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \arcsin \left (c x \right )+e^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d c \,e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+6 c^{2} d^{2} e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-4 c^{3} d^{3} e \sqrt {-c^{2} x^{2}+1}}{4 e}\right )}{c^{3}}}{c}\)	\(541\)
default	\(\frac {\frac {a^{2} \left (c x e +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (c^{3} d^{3} \left (\arcsin \left (c x \right )^{2} c x -2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {3 e \,c^{2} d^{2} \left (2 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}+\frac {c d \,e^{2} \left (9 \arcsin \left (c x \right )^{2} c^{3} x^{3}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 c^{3} x^{3}+12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {e^{3} \left (32 \arcsin \left (c x \right )^{2} x^{4} c^{4}+16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{3} c^{3}-4 c^{4} x^{4}+24 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -12 \arcsin \left (c x \right )^{2}-12 c^{2} x^{2}-9\right )}{128}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{4} d^{4}}{4 e}+\arcsin \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arcsin \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arcsin \left (c x \right ) c^{4} d \,x^{3}+\frac {\arcsin \left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \arcsin \left (c x \right )+e^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d c \,e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+6 c^{2} d^{2} e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-4 c^{3} d^{3} e \sqrt {-c^{2} x^{2}+1}}{4 e}\right )}{c^{3}}}{c}\)	\(541\)
orering	\(\frac {\left (111 c^{4} e^{5} x^{6}+699 e^{4} c^{4} x^{5} d +1928 x^{4} e^{3} d^{2} c^{4}+3480 e^{2} c^{4} x^{3} d^{3}+672 c^{4} d^{4} e \,x^{2}+63 c^{2} e^{5} x^{4}+192 c^{4} d^{5} x +1079 c^{2} d \,e^{4} x^{3}-1632 x^{2} e^{3} d^{2} c^{2}-3600 e^{2} c^{2} d^{3} x -720 c^{2} d^{4} e -180 e^{5} x^{2}-2010 d \,e^{4} x -402 d^{2} e^{3}\right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{192 \left (e x +d \right )^{2} c^{4}}-\frac {\left (81 e^{4} c^{4} x^{6}+539 e^{3} c^{4} x^{5} d +1640 x^{4} e^{2} d^{2} c^{4}+3672 c^{4} d^{3} e \,x^{3}+99 e^{4} c^{2} x^{4}+1719 c^{2} d \,e^{3} x^{3}-1920 x^{2} e^{2} d^{2} c^{2}-4464 c^{2} d^{3} e x -576 c^{2} d^{4}-216 e^{4} x^{2}-2742 d \,e^{3} x -384 d^{2} e^{2}\right ) \left (3 \left (e x +d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} e +\frac {2 \left (e x +d \right )^{3} \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{576 \left (e x +d \right )^{4} c^{4}}+\frac {x \left (9 e^{3} c^{2} x^{3}+64 x^{2} e^{2} d \,c^{2}+216 c^{2} d^{2} e x +576 d^{3} c^{2}+27 x \,e^{3}+384 d \,e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (6 \left (e x +d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2} e^{2}+\frac {12 \left (e x +d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) e b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 \left (e x +d \right )^{3} b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {2 \left (e x +d \right )^{3} \left (a +b \arcsin \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{576 c^{4} \left (e x +d \right )^{3}}\)	\(561\)
parts	\(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (288 \arcsin \left (c x \right )^{2} c^{4} x^{4} e^{3}+1152 \arcsin \left (c x \right )^{2} c^{4} x^{3} d \,e^{2}+1728 \arcsin \left (c x \right )^{2} c^{4} x^{2} d^{2} e +1152 \arcsin \left (c x \right )^{2} c^{4} x \,d^{3}+144 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} e^{3} x^{3}+768 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} d \,e^{2} x^{2}+1728 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} d^{2} e x +2304 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} d^{3}-864 \arcsin \left (c x \right )^{2} c^{2} d^{2} e -36 c^{4} e^{3} x^{4}-256 c^{4} d \,e^{2} x^{3}-864 x^{2} e \,d^{2} c^{4}-2304 c^{4} d^{3} x +216 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x \,e^{3}+1536 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c d \,e^{2}-108 \arcsin \left (c x \right )^{2} e^{3}-108 c^{2} e^{3} x^{2}-1536 c^{2} d \,e^{2} x -81 e^{3}\right )}{1152 c^{4}}+\frac {2 a b \left (\frac {c \,e^{3} \arcsin \left (c x \right ) x^{4}}{4}+c \,e^{2} \arcsin \left (c x \right ) x^{3} d +\frac {3 c e \arcsin \left (c x \right ) x^{2} d^{2}}{2}+\arcsin \left (c x \right ) d^{3} c x +\frac {c \arcsin \left (c x \right ) d^{4}}{4 e}-\frac {c^{4} d^{4} \arcsin \left (c x \right )+e^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d c \,e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+6 c^{2} d^{2} e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-4 c^{3} d^{3} e \sqrt {-c^{2} x^{2}+1}}{4 c^{3} e}\right )}{c}\)	\(576\)

Input:

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

Output:

1/c*(1/4*a^2/c^3*(c*e*x+c*d)^4/e+b^2/c^3*(c^3*d^3*(arcsin(c*x)^2*c*x-2*c*x 
+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+3/4*e*c^2*d^2*(2*arcsin(c*x)^2*x^2*c^2+ 
2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x-arcsin(c*x)^2-c^2*x^2)+1/9*c*d*e^2*(9 
*arcsin(c*x)^2*c^3*x^3+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-2*c^3*x^3+ 
12*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-12*c*x)+1/128*e^3*(32*arcsin(c*x)^2*x^4* 
c^4+16*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^3*c^3-4*c^4*x^4+24*arcsin(c*x)*(-c 
^2*x^2+1)^(1/2)*c*x-12*arcsin(c*x)^2-12*c^2*x^2-9))+2*a*b/c^3*(1/4/e*arcsi 
n(c*x)*c^4*d^4+arcsin(c*x)*c^4*d^3*x+3/2*e*arcsin(c*x)*c^4*d^2*x^2+e^2*arc 
sin(c*x)*c^4*d*x^3+1/4*arcsin(c*x)*e^3*c^4*x^4-1/4/e*(c^4*d^4*arcsin(c*x)+ 
e^4*(-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))+4*d*c*e^3*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+ 
6*c^2*d^2*e^2*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))-4*c^3*d^3*e*(- 
c^2*x^2+1)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.19 \[ \int (d+e x)^3 (a+b \arcsin (c x))^2 \, dx=\frac {9 \, {\left (8 \, a^{2} - b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} d^{2} e - b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x - 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \arcsin \left (c x\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{4} d^{3} - 4 \, b^{2} c^{2} d e^{2}\right )} x + 18 \, {\left (8 \, a b c^{4} e^{3} x^{4} + 32 \, a b c^{4} d e^{2} x^{3} + 48 \, a b c^{4} d^{2} e x^{2} + 32 \, a b c^{4} d^{3} x - 24 \, a b c^{2} d^{2} e - 3 \, a b e^{3}\right )} \arcsin \left (c x\right ) + 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} + 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e + a b c e^{3}\right )} x + {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} + 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{4}} \] Input:

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

Output:

1/288*(9*(8*a^2 - b^2)*c^4*e^3*x^4 + 32*(9*a^2 - 2*b^2)*c^4*d*e^2*x^3 + 27 
*(8*(2*a^2 - b^2)*c^4*d^2*e - b^2*c^2*e^3)*x^2 + 9*(8*b^2*c^4*e^3*x^4 + 32 
*b^2*c^4*d*e^2*x^3 + 48*b^2*c^4*d^2*e*x^2 + 32*b^2*c^4*d^3*x - 24*b^2*c^2* 
d^2*e - 3*b^2*e^3)*arcsin(c*x)^2 + 96*(3*(a^2 - 2*b^2)*c^4*d^3 - 4*b^2*c^2 
*d*e^2)*x + 18*(8*a*b*c^4*e^3*x^4 + 32*a*b*c^4*d*e^2*x^3 + 48*a*b*c^4*d^2* 
e*x^2 + 32*a*b*c^4*d^3*x - 24*a*b*c^2*d^2*e - 3*a*b*e^3)*arcsin(c*x) + 6*( 
6*a*b*c^3*e^3*x^3 + 32*a*b*c^3*d*e^2*x^2 + 96*a*b*c^3*d^3 + 64*a*b*c*d*e^2 
 + 9*(8*a*b*c^3*d^2*e + a*b*c*e^3)*x + (6*b^2*c^3*e^3*x^3 + 32*b^2*c^3*d*e 
^2*x^2 + 96*b^2*c^3*d^3 + 64*b^2*c*d*e^2 + 9*(8*b^2*c^3*d^2*e + b^2*c*e^3) 
*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (364) = 728\).

Time = 0.47 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.99 \[ \int (d+e x)^3 (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((a**2*d**3*x + 3*a**2*d**2*e*x**2/2 + a**2*d*e**2*x**3 + a**2*e* 
*3*x**4/4 + 2*a*b*d**3*x*asin(c*x) + 3*a*b*d**2*e*x**2*asin(c*x) + 2*a*b*d 
*e**2*x**3*asin(c*x) + a*b*e**3*x**4*asin(c*x)/2 + 2*a*b*d**3*sqrt(-c**2*x 
**2 + 1)/c + 3*a*b*d**2*e*x*sqrt(-c**2*x**2 + 1)/(2*c) + 2*a*b*d*e**2*x**2 
*sqrt(-c**2*x**2 + 1)/(3*c) + a*b*e**3*x**3*sqrt(-c**2*x**2 + 1)/(8*c) - 3 
*a*b*d**2*e*asin(c*x)/(2*c**2) + 4*a*b*d*e**2*sqrt(-c**2*x**2 + 1)/(3*c**3 
) + 3*a*b*e**3*x*sqrt(-c**2*x**2 + 1)/(16*c**3) - 3*a*b*e**3*asin(c*x)/(16 
*c**4) + b**2*d**3*x*asin(c*x)**2 - 2*b**2*d**3*x + 3*b**2*d**2*e*x**2*asi 
n(c*x)**2/2 - 3*b**2*d**2*e*x**2/4 + b**2*d*e**2*x**3*asin(c*x)**2 - 2*b** 
2*d*e**2*x**3/9 + b**2*e**3*x**4*asin(c*x)**2/4 - b**2*e**3*x**4/32 + 2*b* 
*2*d**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/c + 3*b**2*d**2*e*x*sqrt(-c**2*x**2 
 + 1)*asin(c*x)/(2*c) + 2*b**2*d*e**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/ 
(3*c) + b**2*e**3*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(8*c) - 3*b**2*d**2* 
e*asin(c*x)**2/(4*c**2) - 4*b**2*d*e**2*x/(3*c**2) - 3*b**2*e**3*x**2/(32* 
c**2) + 4*b**2*d*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c**3) + 3*b**2*e** 
3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(16*c**3) - 3*b**2*e**3*asin(c*x)**2/(3 
2*c**4), Ne(c, 0)), (a**2*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x 
**4/4), True))
 

Maxima [F]

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

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

Output:

1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + b^2*d^3*x*arcsin(c*x)^2 + 3/2*a^2*d^2*e* 
x^2 + 3/2*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c 
^3))*a*b*d^2*e + 2/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 
2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*d*e^2 + 1/16*(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)* 
a*b*e^3 - 2*b^2*d^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d^3*x + 2 
*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^3/c + 1/4*(b^2*e^3*x^4 + 4*b 
^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) 
^2 + integrate(1/2*(b^2*c*e^3*x^4 + 4*b^2*c*d*e^2*x^3 + 6*b^2*c*d^2*e*x^2) 
*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/( 
c^2*x^2 - 1), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (334) = 668\).

Time = 0.15 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.18 \[ \int (d+e x)^3 (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + b^2*d^3*x*arcsin(c*x)^2 + 2*a*b*d^3*x*ar 
csin(c*x) + (c^2*x^2 - 1)*b^2*d*e^2*x*arcsin(c*x)^2/c^2 + 3/2*sqrt(-c^2*x^ 
2 + 1)*b^2*d^2*e*x*arcsin(c*x)/c + a^2*d^3*x - 2*b^2*d^3*x + 2*(c^2*x^2 - 
1)*a*b*d*e^2*x*arcsin(c*x)/c^2 + 3/2*(c^2*x^2 - 1)*b^2*d^2*e*arcsin(c*x)^2 
/c^2 + b^2*d*e^2*x*arcsin(c*x)^2/c^2 + 3/2*sqrt(-c^2*x^2 + 1)*a*b*d^2*e*x/ 
c + 2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c - 1/8*(-c^2*x^2 + 1)^(3/2)* 
b^2*e^3*x*arcsin(c*x)/c^3 - 2/9*(c^2*x^2 - 1)*b^2*d*e^2*x/c^2 + 3*(c^2*x^2 
 - 1)*a*b*d^2*e*arcsin(c*x)/c^2 + 2*a*b*d*e^2*x*arcsin(c*x)/c^2 + 3/4*b^2* 
d^2*e*arcsin(c*x)^2/c^2 + 1/4*(c^2*x^2 - 1)^2*b^2*e^3*arcsin(c*x)^2/c^4 + 
2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c - 1/8*(-c^2*x^2 + 1)^(3/2)*a*b*e^3*x/c^3 - 
2/3*(-c^2*x^2 + 1)^(3/2)*b^2*d*e^2*arcsin(c*x)/c^3 + 5/16*sqrt(-c^2*x^2 + 
1)*b^2*e^3*x*arcsin(c*x)/c^3 + 3/2*(c^2*x^2 - 1)*a^2*d^2*e/c^2 - 3/4*(c^2* 
x^2 - 1)*b^2*d^2*e/c^2 - 14/9*b^2*d*e^2*x/c^2 + 3/2*a*b*d^2*e*arcsin(c*x)/ 
c^2 + 1/2*(c^2*x^2 - 1)^2*a*b*e^3*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)*b^2* 
e^3*arcsin(c*x)^2/c^4 - 2/3*(-c^2*x^2 + 1)^(3/2)*a*b*d*e^2/c^3 + 5/16*sqrt 
(-c^2*x^2 + 1)*a*b*e^3*x/c^3 + 2*sqrt(-c^2*x^2 + 1)*b^2*d*e^2*arcsin(c*x)/ 
c^3 - 3/8*b^2*d^2*e/c^2 - 1/32*(c^2*x^2 - 1)^2*b^2*e^3/c^4 + (c^2*x^2 - 1) 
*a*b*e^3*arcsin(c*x)/c^4 + 5/32*b^2*e^3*arcsin(c*x)^2/c^4 + 2*sqrt(-c^2*x^ 
2 + 1)*a*b*d*e^2/c^3 - 5/32*(c^2*x^2 - 1)*b^2*e^3/c^4 + 5/16*a*b*e^3*arcsi 
n(c*x)/c^4 - 17/256*b^2*e^3/c^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x)^3 (a+b \arcsin (c x))^2 \, dx=\frac {48 \left (\int \mathit {asin} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4} e^{3}+72 a^{2} c^{4} d^{2} e \,x^{2}+48 a^{2} c^{4} d \,e^{2} x^{3}-36 b^{2} c^{4} d^{2} e \,x^{2}+72 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2} c^{3} d^{2} e x +144 \mathit {asin} \left (c x \right ) a b \,c^{4} d^{2} e \,x^{2}+96 \mathit {asin} \left (c x \right ) a b \,c^{4} d \,e^{2} x^{3}+72 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} d^{2} e x +32 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} d \,e^{2} x^{2}-9 \mathit {asin} \left (c x \right ) a b \,e^{3}+48 a^{2} c^{4} d^{3} x +12 a^{2} c^{4} e^{3} x^{4}-96 b^{2} c^{4} d^{3} x +48 \mathit {asin} \left (c x \right )^{2} b^{2} c^{4} d^{3} x -36 \mathit {asin} \left (c x \right )^{2} b^{2} c^{2} d^{2} e +96 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2} c^{3} d^{3}+96 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} d^{3}+144 \left (\int \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{4} d \,e^{2}+72 \mathit {asin} \left (c x \right )^{2} b^{2} c^{4} d^{2} e \,x^{2}+96 \mathit {asin} \left (c x \right ) a b \,c^{4} d^{3} x +24 \mathit {asin} \left (c x \right ) a b \,c^{4} e^{3} x^{4}-72 \mathit {asin} \left (c x \right ) a b \,c^{2} d^{2} e +6 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} e^{3} x^{3}+64 \sqrt {-c^{2} x^{2}+1}\, a b c d \,e^{2}+9 \sqrt {-c^{2} x^{2}+1}\, a b c \,e^{3} x}{48 c^{4}} \] Input:

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

Output:

(48*asin(c*x)**2*b**2*c**4*d**3*x + 72*asin(c*x)**2*b**2*c**4*d**2*e*x**2 
- 36*asin(c*x)**2*b**2*c**2*d**2*e + 96*sqrt( - c**2*x**2 + 1)*asin(c*x)*b 
**2*c**3*d**3 + 72*sqrt( - c**2*x**2 + 1)*asin(c*x)*b**2*c**3*d**2*e*x + 9 
6*asin(c*x)*a*b*c**4*d**3*x + 144*asin(c*x)*a*b*c**4*d**2*e*x**2 + 96*asin 
(c*x)*a*b*c**4*d*e**2*x**3 + 24*asin(c*x)*a*b*c**4*e**3*x**4 - 72*asin(c*x 
)*a*b*c**2*d**2*e - 9*asin(c*x)*a*b*e**3 + 96*sqrt( - c**2*x**2 + 1)*a*b*c 
**3*d**3 + 72*sqrt( - c**2*x**2 + 1)*a*b*c**3*d**2*e*x + 32*sqrt( - c**2*x 
**2 + 1)*a*b*c**3*d*e**2*x**2 + 6*sqrt( - c**2*x**2 + 1)*a*b*c**3*e**3*x** 
3 + 64*sqrt( - c**2*x**2 + 1)*a*b*c*d*e**2 + 9*sqrt( - c**2*x**2 + 1)*a*b* 
c*e**3*x + 48*int(asin(c*x)**2*x**3,x)*b**2*c**4*e**3 + 144*int(asin(c*x)* 
*2*x**2,x)*b**2*c**4*d*e**2 + 48*a**2*c**4*d**3*x + 72*a**2*c**4*d**2*e*x* 
*2 + 48*a**2*c**4*d*e**2*x**3 + 12*a**2*c**4*e**3*x**4 - 96*b**2*c**4*d**3 
*x - 36*b**2*c**4*d**2*e*x**2)/(48*c**4)