\(\int x^3 (a+b \arcsin (c x))^3 \, dx\) [21]

Optimal result

Integrand size = 14, antiderivative size = 208 \[ \int x^3 (a+b \arcsin (c x))^3 \, dx=-\frac {45 b^3 x \sqrt {1-c^2 x^2}}{256 c^3}-\frac {3 b^3 x^3 \sqrt {1-c^2 x^2}}{128 c}+\frac {45 b^3 \arcsin (c x)}{256 c^4}-\frac {9 b^2 x^2 (a+b \arcsin (c x))}{32 c^2}-\frac {3}{32} b^2 x^4 (a+b \arcsin (c x))+\frac {9 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{32 c^3}+\frac {3 b x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{16 c}-\frac {3 (a+b \arcsin (c x))^3}{32 c^4}+\frac {1}{4} x^4 (a+b \arcsin (c x))^3 \] Output:

-45/256*b^3*x*(-c^2*x^2+1)^(1/2)/c^3-3/128*b^3*x^3*(-c^2*x^2+1)^(1/2)/c+45 
/256*b^3*arcsin(c*x)/c^4-9/32*b^2*x^2*(a+b*arcsin(c*x))/c^2-3/32*b^2*x^4*( 
a+b*arcsin(c*x))+9/32*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/c^3+3/16* 
b*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/c-3/32*(a+b*arcsin(c*x))^3/c^ 
4+1/4*x^4*(a+b*arcsin(c*x))^3
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.23 \[ \int x^3 (a+b \arcsin (c x))^3 \, dx=\frac {-24 a b^2 c^2 x^2 \left (3+c^2 x^2\right )+24 a^2 b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )-3 b^3 c x \sqrt {1-c^2 x^2} \left (15+2 c^2 x^2\right )+8 a^3 \left (-3+8 c^4 x^4\right )+3 b \left (16 a b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+b^2 \left (15-24 c^2 x^2-8 c^4 x^4\right )+8 a^2 \left (-3+8 c^4 x^4\right )\right ) \arcsin (c x)+24 b^2 \left (b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+a \left (-3+8 c^4 x^4\right )\right ) \arcsin (c x)^2+8 b^3 \left (-3+8 c^4 x^4\right ) \arcsin (c x)^3}{256 c^4} \] Input:

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

Output:

(-24*a*b^2*c^2*x^2*(3 + c^2*x^2) + 24*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c 
^2*x^2) - 3*b^3*c*x*Sqrt[1 - c^2*x^2]*(15 + 2*c^2*x^2) + 8*a^3*(-3 + 8*c^4 
*x^4) + 3*b*(16*a*b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + b^2*(15 - 24*c 
^2*x^2 - 8*c^4*x^4) + 8*a^2*(-3 + 8*c^4*x^4))*ArcSin[c*x] + 24*b^2*(b*c*x* 
Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + a*(-3 + 8*c^4*x^4))*ArcSin[c*x]^2 + 8* 
b^3*(-3 + 8*c^4*x^4)*ArcSin[c*x]^3)/(256*c^4)
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5138, 5210, 5138, 262, 262, 223, 5210, 5138, 262, 223, 5152}

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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

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

Input:

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

Output:

(x^4*(a + b*ArcSin[c*x])^3)/4 - (3*b*c*(-1/4*(x^3*Sqrt[1 - c^2*x^2]*(a + b 
*ArcSin[c*x])^2)/c^2 + (b*((x^4*(a + b*ArcSin[c*x]))/4 - (b*c*(-1/4*(x^3*S 
qrt[1 - c^2*x^2])/c^2 + (3*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/( 
2*c^3)))/(4*c^2)))/4))/(2*c) + (3*(-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin 
[c*x])^2)/c^2 + (a + b*ArcSin[c*x])^3/(6*b*c^3) + (b*((x^2*(a + b*ArcSin[c 
*x]))/2 - (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)) 
/c))/(4*c^2)))/4
 

Defintions of rubi rules used

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 262

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

rule 5138

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

rule 5152

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

rule 5210

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S 
imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f* 
x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; 
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.55

method	result	size
derivativedivides	\(\frac {\frac {a^{3} c^{4} x^{4}}{4}+b^{3} \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )^{3}}{4}-\frac {3 \arcsin \left (c x \right )^{2} \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{32}-\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{32}-\frac {3 c x \left (2 c^{2} x^{2}+3\right ) \sqrt {-c^{2} x^{2}+1}}{256}-\frac {27 \arcsin \left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{32}-\frac {9 c x \sqrt {-c^{2} x^{2}+1}}{64}+\frac {3 \arcsin \left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )^{2}}{4}-\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}+\frac {3 \arcsin \left (c x \right )^{2}}{32}-\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{32}-\frac {3 \arcsin \left (c x \right )}{32}\right )}{c^{4}}\)	\(323\)
default	\(\frac {\frac {a^{3} c^{4} x^{4}}{4}+b^{3} \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )^{3}}{4}-\frac {3 \arcsin \left (c x \right )^{2} \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{32}-\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{32}-\frac {3 c x \left (2 c^{2} x^{2}+3\right ) \sqrt {-c^{2} x^{2}+1}}{256}-\frac {27 \arcsin \left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{32}-\frac {9 c x \sqrt {-c^{2} x^{2}+1}}{64}+\frac {3 \arcsin \left (c x \right )^{3}}{16}\right )+3 a \,b^{2} \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )^{2}}{4}-\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}+\frac {3 \arcsin \left (c x \right )^{2}}{32}-\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{32}-\frac {3 \arcsin \left (c x \right )}{32}\right )}{c^{4}}\)	\(323\)
parts	\(\frac {a^{3} x^{4}}{4}+\frac {b^{3} \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )^{3}}{4}-\frac {3 \arcsin \left (c x \right )^{2} \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{32}-\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{32}-\frac {3 c x \left (2 c^{2} x^{2}+3\right ) \sqrt {-c^{2} x^{2}+1}}{256}-\frac {27 \arcsin \left (c x \right )}{256}-\frac {9 \left (c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{32}-\frac {9 c x \sqrt {-c^{2} x^{2}+1}}{64}+\frac {3 \arcsin \left (c x \right )^{3}}{16}\right )}{c^{4}}+\frac {3 a \,b^{2} \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )^{2}}{4}-\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}+\frac {3 \arcsin \left (c x \right )^{2}}{32}-\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}\right )}{c^{4}}+\frac {3 a^{2} b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{32}-\frac {3 \arcsin \left (c x \right )}{32}\right )}{c^{4}}\)	\(325\)
orering	\(\frac {\left (350 c^{6} x^{6}+399 c^{4} x^{4}-1800 c^{2} x^{2}+1080\right ) \left (a +b \arcsin \left (c x \right )\right )^{3}}{512 c^{6} x^{2}}-\frac {\left (110 c^{6} x^{6}+263 c^{4} x^{4}-1020 c^{2} x^{2}+630\right ) \left (3 x^{2} \left (a +b \arcsin \left (c x \right )\right )^{3}+\frac {3 x^{3} \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{512 c^{6} x^{4}}+\frac {\left (c x -1\right ) \left (c x +1\right ) \left (10 c^{4} x^{4}+48 c^{2} x^{2}-75\right ) \left (6 x \left (a +b \arcsin \left (c x \right )\right )^{3}+\frac {18 x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {6 x^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {3 x^{4} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{256 c^{6} x^{3}}-\frac {\left (2 c^{2} x^{2}+15\right ) \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (6 \left (a +b \arcsin \left (c x \right )\right )^{3}+\frac {54 x \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {54 x^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {30 x^{3} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x^{3} b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 x^{4} \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{4}}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {9 x^{5} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{5}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{512 x^{2} c^{6}}\)	\(485\)

Input:

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

Output:

1/c^4*(1/4*a^3*c^4*x^4+b^3*(1/4*c^4*x^4*arcsin(c*x)^3-3/32*arcsin(c*x)^2*( 
-2*c^3*x^3*(-c^2*x^2+1)^(1/2)-3*c*x*(-c^2*x^2+1)^(1/2)+3*arcsin(c*x))-3/32 
*c^4*x^4*arcsin(c*x)-3/256*c*x*(2*c^2*x^2+3)*(-c^2*x^2+1)^(1/2)-27/256*arc 
sin(c*x)-9/32*(c^2*x^2-1)*arcsin(c*x)-9/64*c*x*(-c^2*x^2+1)^(1/2)+3/16*arc 
sin(c*x)^3)+3*a*b^2*(1/4*c^4*x^4*arcsin(c*x)^2-1/16*arcsin(c*x)*(-2*c^3*x^ 
3*(-c^2*x^2+1)^(1/2)-3*c*x*(-c^2*x^2+1)^(1/2)+3*arcsin(c*x))+3/32*arcsin(c 
*x)^2-1/128*(2*c^2*x^2+3)^2)+3*a^2*b*(1/4*c^4*x^4*arcsin(c*x)+1/16*c^3*x^3 
*(-c^2*x^2+1)^(1/2)+3/32*c*x*(-c^2*x^2+1)^(1/2)-3/32*arcsin(c*x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int x^3 (a+b \arcsin (c x))^3 \, dx=\frac {8 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} c^{4} x^{4} - 72 \, a b^{2} c^{2} x^{2} + 8 \, {\left (8 \, b^{3} c^{4} x^{4} - 3 \, b^{3}\right )} \arcsin \left (c x\right )^{3} + 24 \, {\left (8 \, a b^{2} c^{4} x^{4} - 3 \, a b^{2}\right )} \arcsin \left (c x\right )^{2} + 3 \, {\left (8 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{4} x^{4} - 24 \, b^{3} c^{2} x^{2} - 24 \, a^{2} b + 15 \, b^{3}\right )} \arcsin \left (c x\right ) + 3 \, {\left (2 \, {\left (8 \, a^{2} b - b^{3}\right )} c^{3} x^{3} + 3 \, {\left (8 \, a^{2} b - 5 \, b^{3}\right )} c x + 8 \, {\left (2 \, b^{3} c^{3} x^{3} + 3 \, b^{3} c x\right )} \arcsin \left (c x\right )^{2} + 16 \, {\left (2 \, a b^{2} c^{3} x^{3} + 3 \, a b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{256 \, c^{4}} \] Input:

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

Output:

1/256*(8*(8*a^3 - 3*a*b^2)*c^4*x^4 - 72*a*b^2*c^2*x^2 + 8*(8*b^3*c^4*x^4 - 
 3*b^3)*arcsin(c*x)^3 + 24*(8*a*b^2*c^4*x^4 - 3*a*b^2)*arcsin(c*x)^2 + 3*( 
8*(8*a^2*b - b^3)*c^4*x^4 - 24*b^3*c^2*x^2 - 24*a^2*b + 15*b^3)*arcsin(c*x 
) + 3*(2*(8*a^2*b - b^3)*c^3*x^3 + 3*(8*a^2*b - 5*b^3)*c*x + 8*(2*b^3*c^3* 
x^3 + 3*b^3*c*x)*arcsin(c*x)^2 + 16*(2*a*b^2*c^3*x^3 + 3*a*b^2*c*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. 416 vs. \(2 (199) = 398\).

Time = 0.58 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.00 \[ \int x^3 (a+b \arcsin (c x))^3 \, dx=\begin {cases} \frac {a^{3} x^{4}}{4} + \frac {3 a^{2} b x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 a^{2} b x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {9 a^{2} b x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {9 a^{2} b \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {3 a b^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {3 a b^{2} x^{4}}{32} + \frac {3 a b^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{8 c} - \frac {9 a b^{2} x^{2}}{32 c^{2}} + \frac {9 a b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{16 c^{3}} - \frac {9 a b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{32 c^{4}} + \frac {b^{3} x^{4} \operatorname {asin}^{3}{\left (c x \right )}}{4} - \frac {3 b^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{32} + \frac {3 b^{3} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{16 c} - \frac {3 b^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{128 c} - \frac {9 b^{3} x^{2} \operatorname {asin}{\left (c x \right )}}{32 c^{2}} + \frac {9 b^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{32 c^{3}} - \frac {45 b^{3} x \sqrt {- c^{2} x^{2} + 1}}{256 c^{3}} - \frac {3 b^{3} \operatorname {asin}^{3}{\left (c x \right )}}{32 c^{4}} + \frac {45 b^{3} \operatorname {asin}{\left (c x \right )}}{256 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{3} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**3*x**4/4 + 3*a**2*b*x**4*asin(c*x)/4 + 3*a**2*b*x**3*sqrt(-c 
**2*x**2 + 1)/(16*c) + 9*a**2*b*x*sqrt(-c**2*x**2 + 1)/(32*c**3) - 9*a**2* 
b*asin(c*x)/(32*c**4) + 3*a*b**2*x**4*asin(c*x)**2/4 - 3*a*b**2*x**4/32 + 
3*a*b**2*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(8*c) - 9*a*b**2*x**2/(32*c** 
2) + 9*a*b**2*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(16*c**3) - 9*a*b**2*asin(c 
*x)**2/(32*c**4) + b**3*x**4*asin(c*x)**3/4 - 3*b**3*x**4*asin(c*x)/32 + 3 
*b**3*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)**2/(16*c) - 3*b**3*x**3*sqrt(-c* 
*2*x**2 + 1)/(128*c) - 9*b**3*x**2*asin(c*x)/(32*c**2) + 9*b**3*x*sqrt(-c* 
*2*x**2 + 1)*asin(c*x)**2/(32*c**3) - 45*b**3*x*sqrt(-c**2*x**2 + 1)/(256* 
c**3) - 3*b**3*asin(c*x)**3/(32*c**4) + 45*b**3*asin(c*x)/(256*c**4), Ne(c 
, 0)), (a**3*x**4/4, True))
 

Maxima [F]

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

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

Output:

1/4*b^3*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^3 + 1/4*a^3*x^4 + 3 
/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)*a^2*b + integrate(3/4*(sqrt(c*x + 1)*sqr 
t(-c*x + 1)*b^3*c*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 4*(a* 
b^2*c^2*x^5 - a*b^2*x^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2)/(c^ 
2*x^2 - 1), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (182) = 364\).

Time = 0.15 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.36 \[ \int x^3 (a+b \arcsin (c x))^3 \, dx=\frac {1}{4} \, a^{3} x^{4} - \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3} x \arcsin \left (c x\right )^{2}}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{3} \arcsin \left (c x\right )^{3}}{4 \, c^{4}} - \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b^{2} x \arcsin \left (c x\right )}{8 \, c^{3}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x \arcsin \left (c x\right )^{2}}{32 \, c^{3}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{3} \arcsin \left (c x\right )^{3}}{2 \, c^{4}} - \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} b x}{16 \, c^{3}} + \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3} x}{128 \, c^{3}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x \arcsin \left (c x\right )}{16 \, c^{3}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} a^{2} b \arcsin \left (c x\right )}{4 \, c^{4}} - \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{3} \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a b^{2} \arcsin \left (c x\right )^{2}}{2 \, c^{4}} + \frac {5 \, b^{3} \arcsin \left (c x\right )^{3}}{32 \, c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b x}{32 \, c^{3}} - \frac {51 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x}{256 \, c^{3}} - \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b^{2}}{32 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a^{2} b \arcsin \left (c x\right )}{2 \, c^{4}} - \frac {15 \, {\left (c^{2} x^{2} - 1\right )} b^{3} \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {15 \, a b^{2} \arcsin \left (c x\right )^{2}}{32 \, c^{4}} - \frac {15 \, {\left (c^{2} x^{2} - 1\right )} a b^{2}}{32 \, c^{4}} + \frac {15 \, a^{2} b \arcsin \left (c x\right )}{32 \, c^{4}} - \frac {51 \, b^{3} \arcsin \left (c x\right )}{256 \, c^{4}} - \frac {51 \, a b^{2}}{256 \, c^{4}} \] Input:

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

Output:

1/4*a^3*x^4 - 3/16*(-c^2*x^2 + 1)^(3/2)*b^3*x*arcsin(c*x)^2/c^3 + 1/4*(c^2 
*x^2 - 1)^2*b^3*arcsin(c*x)^3/c^4 - 3/8*(-c^2*x^2 + 1)^(3/2)*a*b^2*x*arcsi 
n(c*x)/c^3 + 15/32*sqrt(-c^2*x^2 + 1)*b^3*x*arcsin(c*x)^2/c^3 + 3/4*(c^2*x 
^2 - 1)^2*a*b^2*arcsin(c*x)^2/c^4 + 1/2*(c^2*x^2 - 1)*b^3*arcsin(c*x)^3/c^ 
4 - 3/16*(-c^2*x^2 + 1)^(3/2)*a^2*b*x/c^3 + 3/128*(-c^2*x^2 + 1)^(3/2)*b^3 
*x/c^3 + 15/16*sqrt(-c^2*x^2 + 1)*a*b^2*x*arcsin(c*x)/c^3 + 3/4*(c^2*x^2 - 
 1)^2*a^2*b*arcsin(c*x)/c^4 - 3/32*(c^2*x^2 - 1)^2*b^3*arcsin(c*x)/c^4 + 3 
/2*(c^2*x^2 - 1)*a*b^2*arcsin(c*x)^2/c^4 + 5/32*b^3*arcsin(c*x)^3/c^4 + 15 
/32*sqrt(-c^2*x^2 + 1)*a^2*b*x/c^3 - 51/256*sqrt(-c^2*x^2 + 1)*b^3*x/c^3 - 
 3/32*(c^2*x^2 - 1)^2*a*b^2/c^4 + 3/2*(c^2*x^2 - 1)*a^2*b*arcsin(c*x)/c^4 
- 15/32*(c^2*x^2 - 1)*b^3*arcsin(c*x)/c^4 + 15/32*a*b^2*arcsin(c*x)^2/c^4 
- 15/32*(c^2*x^2 - 1)*a*b^2/c^4 + 15/32*a^2*b*arcsin(c*x)/c^4 - 51/256*b^3 
*arcsin(c*x)/c^4 - 51/256*a*b^2/c^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 (a+b \arcsin (c x))^3 \, dx=\frac {24 \mathit {asin} \left (c x \right ) a^{2} b \,c^{4} x^{4}-9 \mathit {asin} \left (c x \right ) a^{2} b +6 \sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{3} x^{3}+9 \sqrt {-c^{2} x^{2}+1}\, a^{2} b c x +32 \left (\int \mathit {asin} \left (c x \right )^{3} x^{3}d x \right ) b^{3} c^{4}+96 \left (\int \mathit {asin} \left (c x \right )^{2} x^{3}d x \right ) a \,b^{2} c^{4}+8 a^{3} c^{4} x^{4}}{32 c^{4}} \] Input:

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

Output:

(24*asin(c*x)*a**2*b*c**4*x**4 - 9*asin(c*x)*a**2*b + 6*sqrt( - c**2*x**2 
+ 1)*a**2*b*c**3*x**3 + 9*sqrt( - c**2*x**2 + 1)*a**2*b*c*x + 32*int(asin( 
c*x)**3*x**3,x)*b**3*c**4 + 96*int(asin(c*x)**2*x**3,x)*a*b**2*c**4 + 8*a* 
*3*c**4*x**4)/(32*c**4)