\(\int x (d-c^2 d x^2)^2 (a+b \arcsin (c x)) \, dx\) [13]

Optimal result

Integrand size = 23, antiderivative size = 124 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {5 b d^2 x \sqrt {1-c^2 x^2}}{96 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2}}{144 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2}}{36 c}+\frac {5 b d^2 \arcsin (c x)}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2} \] Output:

5/96*b*d^2*x*(-c^2*x^2+1)^(1/2)/c+5/144*b*d^2*x*(-c^2*x^2+1)^(3/2)/c+1/36* 
b*d^2*x*(-c^2*x^2+1)^(5/2)/c+5/96*b*d^2*arcsin(c*x)/c^2-1/6*d^2*(-c^2*x^2+ 
1)^3*(a+b*arcsin(c*x))/c^2
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {d^2 \left (48 a \left (-1+c^2 x^2\right )^3+b c x \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )+3 b \left (-11+48 c^2 x^2-48 c^4 x^4+16 c^6 x^6\right ) \arcsin (c x)\right )}{288 c^2} \] Input:

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

Output:

(d^2*(48*a*(-1 + c^2*x^2)^3 + b*c*x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8 
*c^4*x^4) + 3*b*(-11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x])) 
/(288*c^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5182, 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.

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

Input:

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

Output:

-1/6*(d^2*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/c^2 + (b*d^2*((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))/(6*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 = 127, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {\frac {a \,d^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\)	\(127\)
default	\(\frac {\frac {a \,d^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\)	\(127\)
parts	\(\frac {a \,d^{2} \left (c^{2} x^{2}-1\right )^{3}}{6 c^{2}}+\frac {d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{2}+\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}-\frac {11 \arcsin \left (c x \right )}{96}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}+\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}\right )}{c^{2}}\)	\(129\)
orering	\(\frac {\left (88 c^{6} x^{6}-282 c^{4} x^{4}+335 c^{2} x^{2}-66\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )}{288 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (8 c^{4} x^{4}-26 c^{2} x^{2}+33\right ) \left (\left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )-4 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x \left (-c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{288 c^{2} \left (c x -1\right ) \left (c x +1\right )}\)	\(196\)

Input:

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

Output:

1/c^2*(1/6*a*d^2*(c^2*x^2-1)^3+d^2*b*(1/6*arcsin(c*x)*c^6*x^6-1/2*c^4*x^4* 
arcsin(c*x)+1/2*c^2*x^2*arcsin(c*x)-11/96*arcsin(c*x)+1/36*c^5*x^5*(-c^2*x 
^2+1)^(1/2)-13/144*c^3*x^3*(-c^2*x^2+1)^(1/2)+11/96*c*x*(-c^2*x^2+1)^(1/2) 
))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {48 \, a c^{6} d^{2} x^{6} - 144 \, a c^{4} d^{2} x^{4} + 144 \, a c^{2} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} d^{2} x^{6} - 48 \, b c^{4} d^{2} x^{4} + 48 \, b c^{2} d^{2} x^{2} - 11 \, b d^{2}\right )} \arcsin \left (c x\right ) + {\left (8 \, b c^{5} d^{2} x^{5} - 26 \, b c^{3} d^{2} x^{3} + 33 \, b c d^{2} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{2}} \] Input:

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

Output:

1/288*(48*a*c^6*d^2*x^6 - 144*a*c^4*d^2*x^4 + 144*a*c^2*d^2*x^2 + 3*(16*b* 
c^6*d^2*x^6 - 48*b*c^4*d^2*x^4 + 48*b*c^2*d^2*x^2 - 11*b*d^2)*arcsin(c*x) 
+ (8*b*c^5*d^2*x^5 - 26*b*c^3*d^2*x^3 + 33*b*c*d^2*x)*sqrt(-c^2*x^2 + 1))/ 
c^2
 

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.53 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{6}}{6} - \frac {a c^{2} d^{2} x^{4}}{2} + \frac {a d^{2} x^{2}}{2} + \frac {b c^{4} d^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{36} - \frac {b c^{2} d^{2} x^{4} \operatorname {asin}{\left (c x \right )}}{2} - \frac {13 b c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{144} + \frac {b d^{2} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {11 b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{96 c} - \frac {11 b d^{2} \operatorname {asin}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*c**4*d**2*x**6/6 - a*c**2*d**2*x**4/2 + a*d**2*x**2/2 + b*c** 
4*d**2*x**6*asin(c*x)/6 + b*c**3*d**2*x**5*sqrt(-c**2*x**2 + 1)/36 - b*c** 
2*d**2*x**4*asin(c*x)/2 - 13*b*c*d**2*x**3*sqrt(-c**2*x**2 + 1)/144 + b*d* 
*2*x**2*asin(c*x)/2 + 11*b*d**2*x*sqrt(-c**2*x**2 + 1)/(96*c) - 11*b*d**2* 
asin(c*x)/(96*c**2), Ne(c, 0)), (a*d**2*x**2/2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (107) = 214\).

Time = 0.12 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.91 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{6} \, a c^{4} d^{2} x^{6} - \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b c^{4} d^{2} - \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} \] Input:

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

Output:

1/6*a*c^4*d^2*x^6 - 1/2*a*c^2*d^2*x^4 + 1/288*(48*x^6*arcsin(c*x) + (8*sqr 
t(-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^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)*b*c^2*d^2 + 1/2*a*d^2*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^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.27 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{6} \, a c^{4} d^{2} x^{6} - \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{36 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} \arcsin \left (c x\right )}{6 \, c^{2}} + \frac {5 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} x}{144 \, c} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{96 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{2}}{2 \, c^{2}} + \frac {5 \, b d^{2} \arcsin \left (c x\right )}{96 \, c^{2}} \] Input:

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

Output:

1/6*a*c^4*d^2*x^6 - 1/2*a*c^2*d^2*x^4 + 1/36*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 
 + 1)*b*d^2*x/c + 1/6*(c^2*x^2 - 1)^3*b*d^2*arcsin(c*x)/c^2 + 5/144*(-c^2* 
x^2 + 1)^(3/2)*b*d^2*x/c + 5/96*sqrt(-c^2*x^2 + 1)*b*d^2*x/c + 1/2*(c^2*x^ 
2 - 1)*a*d^2/c^2 + 5/96*b*d^2*arcsin(c*x)/c^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.11 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {d^{2} \left (48 \mathit {asin} \left (c x \right ) b \,c^{6} x^{6}-144 \mathit {asin} \left (c x \right ) b \,c^{4} x^{4}+144 \mathit {asin} \left (c x \right ) b \,c^{2} x^{2}-33 \mathit {asin} \left (c x \right ) b +8 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} x^{5}-26 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} x^{3}+33 \sqrt {-c^{2} x^{2}+1}\, b c x +48 a \,c^{6} x^{6}-144 a \,c^{4} x^{4}+144 a \,c^{2} x^{2}\right )}{288 c^{2}} \] Input:

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

Output:

(d**2*(48*asin(c*x)*b*c**6*x**6 - 144*asin(c*x)*b*c**4*x**4 + 144*asin(c*x 
)*b*c**2*x**2 - 33*asin(c*x)*b + 8*sqrt( - c**2*x**2 + 1)*b*c**5*x**5 - 26 
*sqrt( - c**2*x**2 + 1)*b*c**3*x**3 + 33*sqrt( - c**2*x**2 + 1)*b*c*x + 48 
*a*c**6*x**6 - 144*a*c**4*x**4 + 144*a*c**2*x**2))/(288*c**2)