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

Optimal result

Integrand size = 14, antiderivative size = 102 \[ \int x^2 (a+b \arcsin (c x))^2 \, dx=-\frac {4 b^2 x}{9 c^2}-\frac {2 b^2 x^3}{27}+\frac {4 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3}+\frac {2 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^2 \] Output:

-4/9*b^2*x/c^2-2/27*b^2*x^3+4/9*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3 
+2/9*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c+1/3*x^3*(a+b*arcsin(c*x) 
)^2
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93 \[ \int x^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{3} \left (x^3 (a+b \arcsin (c x))^2-\frac {2 b \left (6 b c x+b c^3 x^3-6 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-3 c^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )}{9 c^3}\right ) \] Input:

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

Output:

(x^3*(a + b*ArcSin[c*x])^2 - (2*b*(6*b*c*x + b*c^3*x^3 - 6*Sqrt[1 - c^2*x^ 
2]*(a + b*ArcSin[c*x]) - 3*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))) 
/(9*c^3))/3
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5138, 5210, 15, 5182, 24}

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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \left (\frac {2 \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {2 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}+\frac {b x^3}{9 c}\right )\)

Input:

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

Output:

(x^3*(a + b*ArcSin[c*x])^2)/3 - (2*b*c*((b*x^3)/(9*c) - (x^2*Sqrt[1 - c^2* 
x^2]*(a + b*ArcSin[c*x]))/(3*c^2) + (2*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + 
b*ArcSin[c*x]))/c^2))/(3*c^2)))/3
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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]
 

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 = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.23

method	result	size
parts	\(\frac {x^{3} a^{2}}{3}+\frac {b^{2} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{2}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(125\)
derivativedivides	\(\frac {\frac {c^{3} x^{3} a^{2}}{3}+b^{2} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{2}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+2 a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(126\)
default	\(\frac {\frac {c^{3} x^{3} a^{2}}{3}+b^{2} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{2}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+2 a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(126\)
orering	\(\frac {\left (19 c^{4} x^{4}+24 c^{2} x^{2}-48\right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{27 c^{4} x}-\frac {\left (6 c^{4} x^{4}+17 c^{2} x^{2}-30\right ) \left (2 x \left (a +b \arcsin \left (c x \right )\right )^{2}+\frac {2 x^{2} \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{27 c^{4} x^{2}}+\frac {\left (c^{2} x^{2}+6\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 \left (a +b \arcsin \left (c x \right )\right )^{2}+\frac {8 x \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {2 x^{3} \left (a +b \arcsin \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 x \,c^{4}}\)	\(222\)

Input:

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

Output:

1/3*x^3*a^2+b^2/c^3*(1/3*c^3*x^3*arcsin(c*x)^2+2/9*arcsin(c*x)*(c^2*x^2+2) 
*(-c^2*x^2+1)^(1/2)-2/27*c^3*x^3-4/9*c*x)+2*a*b/c^3*(1/3*c^3*x^3*arcsin(c* 
x)+1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)+2/9*(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09 \[ \int x^2 (a+b \arcsin (c x))^2 \, dx=\frac {9 \, b^{2} c^{3} x^{3} \arcsin \left (c x\right )^{2} + 18 \, a b c^{3} x^{3} \arcsin \left (c x\right ) + {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x + 6 \, {\left (a b c^{2} x^{2} + 2 \, a b + {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \] Input:

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

Output:

1/27*(9*b^2*c^3*x^3*arcsin(c*x)^2 + 18*a*b*c^3*x^3*arcsin(c*x) + (9*a^2 - 
2*b^2)*c^3*x^3 - 12*b^2*c*x + 6*(a*b*c^2*x^2 + 2*a*b + (b^2*c^2*x^2 + 2*b^ 
2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^3
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.67 \[ \int x^2 (a+b \arcsin (c x))^2 \, dx=\begin {cases} \frac {a^{2} x^{3}}{3} + \frac {2 a b x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {4 a b \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} x^{3}}{27} + \frac {2 b^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} - \frac {4 b^{2} x}{9 c^{2}} + \frac {4 b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*x**3/3 + 2*a*b*x**3*asin(c*x)/3 + 2*a*b*x**2*sqrt(-c**2*x* 
*2 + 1)/(9*c) + 4*a*b*sqrt(-c**2*x**2 + 1)/(9*c**3) + b**2*x**3*asin(c*x)* 
*2/3 - 2*b**2*x**3/27 + 2*b**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) - 
 4*b**2*x/(9*c**2) + 4*b**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c**3), Ne(c, 
 0)), (a**2*x**3/3, True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.39 \[ \int x^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{3} \, b^{2} x^{3} \arcsin \left (c x\right )^{2} + \frac {1}{3} \, a^{2} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b + \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} \] Input:

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

Output:

1/3*b^2*x^3*arcsin(c*x)^2 + 1/3*a^2*x^3 + 2/9*(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 + 2/27*(3*c*(sqrt( 
-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 
 6*x)/c^2)*b^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (88) = 176\).

Time = 0.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.90 \[ \int x^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{3} \, a^{2} x^{3} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b^{2} x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} x}{27 \, c^{2}} + \frac {2 \, a b x \arcsin \left (c x\right )}{3 \, c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} \arcsin \left (c x\right )}{9 \, c^{3}} - \frac {14 \, b^{2} x}{27 \, c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b}{9 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b}{3 \, c^{3}} \] Input:

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

Output:

1/3*a^2*x^3 + 1/3*(c^2*x^2 - 1)*b^2*x*arcsin(c*x)^2/c^2 + 2/3*(c^2*x^2 - 1 
)*a*b*x*arcsin(c*x)/c^2 + 1/3*b^2*x*arcsin(c*x)^2/c^2 - 2/27*(c^2*x^2 - 1) 
*b^2*x/c^2 + 2/3*a*b*x*arcsin(c*x)/c^2 - 2/9*(-c^2*x^2 + 1)^(3/2)*b^2*arcs 
in(c*x)/c^3 - 14/27*b^2*x/c^2 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b/c^3 + 2/3*sqr 
t(-c^2*x^2 + 1)*b^2*arcsin(c*x)/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*a*b/c^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(6*asin(c*x)*a*b*c**3*x**3 + 2*sqrt( - c**2*x**2 + 1)*a*b*c**2*x**2 + 4*sq 
rt( - c**2*x**2 + 1)*a*b + 9*int(asin(c*x)**2*x**2,x)*b**2*c**3 + 3*a**2*c 
**3*x**3)/(9*c**3)