\(\int x (d-c^2 d x^2)^3 (a+b \arcsin (c x)) \, dx\) [22]

Optimal result

Integrand size = 23, antiderivative size = 150 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {35 b d^3 x \sqrt {1-c^2 x^2}}{1024 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}+\frac {35 b d^3 \arcsin (c x)}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))}{8 c^2} \] Output:

35/1024*b*d^3*x*(-c^2*x^2+1)^(1/2)/c+35/1536*b*d^3*x*(-c^2*x^2+1)^(3/2)/c+ 
7/384*b*d^3*x*(-c^2*x^2+1)^(5/2)/c+1/64*b*d^3*x*(-c^2*x^2+1)^(7/2)/c+35/10 
24*b*d^3*arcsin(c*x)/c^2-1/8*d^3*(-c^2*x^2+1)^4*(a+b*arcsin(c*x))/c^2
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.73 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {d^3 \left (384 a \left (-1+c^2 x^2\right )^4+b c x \sqrt {1-c^2 x^2} \left (-279+326 c^2 x^2-200 c^4 x^4+48 c^6 x^6\right )+3 b \left (93-512 c^2 x^2+768 c^4 x^4-512 c^6 x^6+128 c^8 x^8\right ) \arcsin (c x)\right )}{3072 c^2} \] Input:

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

Output:

-1/3072*(d^3*(384*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[1 - c^2*x^2]*(-279 + 326 
*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6) + 3*b*(93 - 512*c^2*x^2 + 768*c^4*x^4 
 - 512*c^6*x^6 + 128*c^8*x^8)*ArcSin[c*x]))/c^2
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5182, 211, 211, 211, 211, 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.

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

\(\Big \downarrow \) 5182

\(\displaystyle \frac {b d^3 \int \left (1-c^2 x^2\right )^{7/2}dx}{8 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))}{8 c^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{8} \int \left (1-c^2 x^2\right )^{5/2}dx+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )}{8 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))}{8 c^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )}{8 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))}{8 c^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )}{8 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))}{8 c^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )}{8 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))}{8 c^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )}{8 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arcsin (c x))}{8 c^2}\)

Input:

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

Output:

-1/8*(d^3*(1 - c^2*x^2)^4*(a + b*ArcSin[c*x]))/c^2 + (b*d^3*((x*(1 - c^2*x 
^2)^(7/2))/8 + (7*((x*(1 - c^2*x^2)^(5/2))/6 + (5*((x*(1 - c^2*x^2)^(3/2)) 
/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/6))/8))/(8*c)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 5182

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {-\frac {d^{3} a \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\)	\(160\)
default	\(\frac {-\frac {d^{3} a \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\)	\(160\)
parts	\(-\frac {d^{3} a \left (c^{2} x^{2}-1\right )^{4}}{8 c^{2}}-\frac {d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\)	\(162\)
orering	\(\frac {\left (720 c^{8} x^{8}-2984 c^{6} x^{6}+4786 c^{4} x^{4}-3815 c^{2} x^{2}+558\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )}{3072 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (48 c^{6} x^{6}-200 c^{4} x^{4}+326 c^{2} x^{2}-279\right ) \left (\left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )-6 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x \left (-c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3072 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\)	\(214\)

Input:

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

Output:

1/c^2*(-1/8*d^3*a*(c^2*x^2-1)^4-d^3*b*(1/8*arcsin(c*x)*c^8*x^8-1/2*arcsin( 
c*x)*c^6*x^6+3/4*c^4*x^4*arcsin(c*x)-1/2*c^2*x^2*arcsin(c*x)+93/1024*arcsi 
n(c*x)+1/64*c^7*x^7*(-c^2*x^2+1)^(1/2)-25/384*c^5*x^5*(-c^2*x^2+1)^(1/2)+1 
63/1536*c^3*x^3*(-c^2*x^2+1)^(1/2)-93/1024*c*x*(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.15 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {384 \, a c^{8} d^{3} x^{8} - 1536 \, a c^{6} d^{3} x^{6} + 2304 \, a c^{4} d^{3} x^{4} - 1536 \, a c^{2} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} d^{3} x^{8} - 512 \, b c^{6} d^{3} x^{6} + 768 \, b c^{4} d^{3} x^{4} - 512 \, b c^{2} d^{3} x^{2} + 93 \, b d^{3}\right )} \arcsin \left (c x\right ) + {\left (48 \, b c^{7} d^{3} x^{7} - 200 \, b c^{5} d^{3} x^{5} + 326 \, b c^{3} d^{3} x^{3} - 279 \, b c d^{3} x\right )} \sqrt {-c^{2} x^{2} + 1}}{3072 \, c^{2}} \] Input:

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

Output:

-1/3072*(384*a*c^8*d^3*x^8 - 1536*a*c^6*d^3*x^6 + 2304*a*c^4*d^3*x^4 - 153 
6*a*c^2*d^3*x^2 + 3*(128*b*c^8*d^3*x^8 - 512*b*c^6*d^3*x^6 + 768*b*c^4*d^3 
*x^4 - 512*b*c^2*d^3*x^2 + 93*b*d^3)*arcsin(c*x) + (48*b*c^7*d^3*x^7 - 200 
*b*c^5*d^3*x^5 + 326*b*c^3*d^3*x^3 - 279*b*c*d^3*x)*sqrt(-c^2*x^2 + 1))/c^ 
2
 

Sympy [A] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.69 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\begin {cases} - \frac {a c^{6} d^{3} x^{8}}{8} + \frac {a c^{4} d^{3} x^{6}}{2} - \frac {3 a c^{2} d^{3} x^{4}}{4} + \frac {a d^{3} x^{2}}{2} - \frac {b c^{6} d^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{8} - \frac {b c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64} + \frac {b c^{4} d^{3} x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {25 b c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{384} - \frac {3 b c^{2} d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{4} - \frac {163 b c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1536} + \frac {b d^{3} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {93 b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{1024 c} - \frac {93 b d^{3} \operatorname {asin}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a*c**6*d**3*x**8/8 + a*c**4*d**3*x**6/2 - 3*a*c**2*d**3*x**4/4 
 + a*d**3*x**2/2 - b*c**6*d**3*x**8*asin(c*x)/8 - b*c**5*d**3*x**7*sqrt(-c 
**2*x**2 + 1)/64 + b*c**4*d**3*x**6*asin(c*x)/2 + 25*b*c**3*d**3*x**5*sqrt 
(-c**2*x**2 + 1)/384 - 3*b*c**2*d**3*x**4*asin(c*x)/4 - 163*b*c*d**3*x**3* 
sqrt(-c**2*x**2 + 1)/1536 + b*d**3*x**2*asin(c*x)/2 + 93*b*d**3*x*sqrt(-c* 
*2*x**2 + 1)/(1024*c) - 93*b*d**3*asin(c*x)/(1024*c**2), Ne(c, 0)), (a*d** 
3*x**2/2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (129) = 258\).

Time = 0.12 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.39 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {1}{8} \, a c^{6} d^{3} x^{8} + \frac {1}{2} \, a c^{4} d^{3} x^{6} - \frac {1}{3072} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b c^{6} d^{3} - \frac {3}{4} \, a c^{2} d^{3} x^{4} + \frac {1}{96} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b c^{4} d^{3} - \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} \] Input:

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

Output:

-1/8*a*c^6*d^3*x^8 + 1/2*a*c^4*d^3*x^6 - 1/3072*(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*c^6*d^3 - 3/4*a*c^2*d^3*x^4 + 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*c^4*d^3 - 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 
)*b*c^2*d^3 + 1/2*a*d^3*x^2 + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 
1)*x/c^2 - arcsin(c*x)/c^3))*b*d^3
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.35 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {1}{8} \, a c^{6} d^{3} x^{8} + \frac {1}{2} \, a c^{4} d^{3} x^{6} - \frac {3}{4} \, a c^{2} d^{3} x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{64 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b d^{3} \arcsin \left (c x\right )}{8 \, c^{2}} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{384 \, c} + \frac {35 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3} x}{1536 \, c} + \frac {35 \, \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{1024 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{3}}{2 \, c^{2}} + \frac {35 \, b d^{3} \arcsin \left (c x\right )}{1024 \, c^{2}} \] Input:

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

Output:

-1/8*a*c^6*d^3*x^8 + 1/2*a*c^4*d^3*x^6 - 3/4*a*c^2*d^3*x^4 - 1/64*(c^2*x^2 
 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d^3*x/c - 1/8*(c^2*x^2 - 1)^4*b*d^3*arcsin(c* 
x)/c^2 + 7/384*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^3*x/c + 35/1536*(-c^ 
2*x^2 + 1)^(3/2)*b*d^3*x/c + 35/1024*sqrt(-c^2*x^2 + 1)*b*d^3*x/c + 1/2*(c 
^2*x^2 - 1)*a*d^3/c^2 + 35/1024*b*d^3*arcsin(c*x)/c^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {d^{3} \left (-384 \mathit {asin} \left (c x \right ) b \,c^{8} x^{8}+1536 \mathit {asin} \left (c x \right ) b \,c^{6} x^{6}-2304 \mathit {asin} \left (c x \right ) b \,c^{4} x^{4}+1536 \mathit {asin} \left (c x \right ) b \,c^{2} x^{2}-279 \mathit {asin} \left (c x \right ) b -48 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} x^{7}+200 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} x^{5}-326 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} x^{3}+279 \sqrt {-c^{2} x^{2}+1}\, b c x -384 a \,c^{8} x^{8}+1536 a \,c^{6} x^{6}-2304 a \,c^{4} x^{4}+1536 a \,c^{2} x^{2}\right )}{3072 c^{2}} \] Input:

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

Output:

(d**3*( - 384*asin(c*x)*b*c**8*x**8 + 1536*asin(c*x)*b*c**6*x**6 - 2304*as 
in(c*x)*b*c**4*x**4 + 1536*asin(c*x)*b*c**2*x**2 - 279*asin(c*x)*b - 48*sq 
rt( - c**2*x**2 + 1)*b*c**7*x**7 + 200*sqrt( - c**2*x**2 + 1)*b*c**5*x**5 
- 326*sqrt( - c**2*x**2 + 1)*b*c**3*x**3 + 279*sqrt( - c**2*x**2 + 1)*b*c* 
x - 384*a*c**8*x**8 + 1536*a*c**6*x**6 - 2304*a*c**4*x**4 + 1536*a*c**2*x* 
*2))/(3072*c**2)