3.2.73 \(\int x^2 (a+b \arcsin (c+d x))^n \, dx\) [173]

3.2.73.1 Optimal result

Integrand size = 16, antiderivative size = 611 \[ \int x^2 (a+b \arcsin (c+d x))^n \, dx=-\frac {i e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )}{8 d^3}-\frac {i c^2 e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d^3}+\frac {i e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )}{8 d^3}+\frac {i c^2 e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d^3}+\frac {2^{-2-n} c e^{-\frac {2 i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )}{d^3}+\frac {2^{-2-n} c e^{\frac {2 i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c+d x))}{b}\right )}{d^3}+\frac {i 3^{-1-n} e^{-\frac {3 i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c+d x))}{b}\right )}{8 d^3}-\frac {i 3^{-1-n} e^{\frac {3 i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c+d x))}{b}\right )}{8 d^3} \]

output

-1/8*I*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,-I*(a+b*arcsin(d*x+c))/b)/d^3/exp(I 
*a/b)/((-I*(a+b*arcsin(d*x+c))/b)^n)-1/2*I*c^2*(a+b*arcsin(d*x+c))^n*GAMMA 
(1+n,-I*(a+b*arcsin(d*x+c))/b)/d^3/exp(I*a/b)/((-I*(a+b*arcsin(d*x+c))/b)^ 
n)+1/8*I*exp(I*a/b)*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,I*(a+b*arcsin(d*x+c))/ 
b)/d^3/((I*(a+b*arcsin(d*x+c))/b)^n)+1/2*I*c^2*exp(I*a/b)*(a+b*arcsin(d*x+ 
c))^n*GAMMA(1+n,I*(a+b*arcsin(d*x+c))/b)/d^3/((I*(a+b*arcsin(d*x+c))/b)^n) 
+2^(-2-n)*c*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,-2*I*(a+b*arcsin(d*x+c))/b)/d^ 
3/exp(2*I*a/b)/((-I*(a+b*arcsin(d*x+c))/b)^n)+2^(-2-n)*c*exp(2*I*a/b)*(a+b 
*arcsin(d*x+c))^n*GAMMA(1+n,2*I*(a+b*arcsin(d*x+c))/b)/d^3/((I*(a+b*arcsin 
(d*x+c))/b)^n)+1/8*I*3^(-1-n)*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,-3*I*(a+b*ar 
csin(d*x+c))/b)/d^3/exp(3*I*a/b)/((-I*(a+b*arcsin(d*x+c))/b)^n)-1/8*I*3^(- 
1-n)*exp(3*I*a/b)*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,3*I*(a+b*arcsin(d*x+c))/ 
b)/d^3/((I*(a+b*arcsin(d*x+c))/b)^n)
 

3.2.73.2 Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.69 \[ \int x^2 (a+b \arcsin (c+d x))^n \, dx=\frac {2^{-3-n} 3^{-1-n} e^{-\frac {3 i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {(a+b \arcsin (c+d x))^2}{b^2}\right )^{-n} \left (-i 2^n 3^{1+n} \left (1+4 c^2\right ) e^{\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )+i 2^n 3^{1+n} \left (1+4 c^2\right ) e^{\frac {4 i a}{b}} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )+2\ 3^{1+n} c e^{\frac {i a}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+2\ 3^{1+n} c e^{\frac {5 i a}{b}} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+i 2^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c+d x))}{b}\right )-i 2^n e^{\frac {6 i a}{b}} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c+d x))}{b}\right )\right )}{d^3} \]

input

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

output

(2^(-3 - n)*3^(-1 - n)*(a + b*ArcSin[c + d*x])^n*((-I)*2^n*3^(1 + n)*(1 + 
4*c^2)*E^(((2*I)*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-I 
)*(a + b*ArcSin[c + d*x]))/b] + I*2^n*3^(1 + n)*(1 + 4*c^2)*E^(((4*I)*a)/b 
)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d 
*x]))/b] + 2*3^(1 + n)*c*E^((I*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gam 
ma[1 + n, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + 2*3^(1 + n)*c*E^(((5*I)*a) 
/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin 
[c + d*x]))/b] + I*2^n*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-3 
*I)*(a + b*ArcSin[c + d*x]))/b] - I*2^n*E^(((6*I)*a)/b)*(((-I)*(a + b*ArcS 
in[c + d*x]))/b)^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c + d*x]))/b]))/(d^3* 
E^(((3*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^n)
 

3.2.73.3 Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5304, 27, 5246, 7293, 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.

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

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int x^2 (a+b \arcsin (c+d x))^nd(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int d^2 x^2 (a+b \arcsin (c+d x))^nd(c+d x)}{d^3}\)

\(\Big \downarrow \) 5246

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{2} i c^2 e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{2} i c^2 e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {i (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} i e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i (a+b \arcsin (c+d x))}{b}\right )+c 2^{-n-2} e^{-\frac {2 i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} i 3^{-n-1} e^{-\frac {3 i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 i (a+b \arcsin (c+d x))}{b}\right )+\frac {1}{8} i e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {i (a+b \arcsin (c+d x))}{b}\right )+c 2^{-n-2} e^{\frac {2 i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\frac {1}{8} i 3^{-n-1} e^{\frac {3 i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 i (a+b \arcsin (c+d x))}{b}\right )}{d^3}\)

input

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

output

(((-1/8*I)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + 
d*x]))/b])/(E^((I*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n) - ((I/2)*c^2 
*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b]) 
/(E^((I*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/8)*E^((I*a)/b)*( 
a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/((I* 
(a + b*ArcSin[c + d*x]))/b)^n + ((I/2)*c^2*E^((I*a)/b)*(a + b*ArcSin[c + d 
*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/((I*(a + b*ArcSin[c + 
d*x]))/b)^n + (2^(-2 - n)*c*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-2*I) 
*(a + b*ArcSin[c + d*x]))/b])/(E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d* 
x]))/b)^n) + (2^(-2 - n)*c*E^(((2*I)*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma 
[1 + n, ((2*I)*(a + b*ArcSin[c + d*x]))/b])/((I*(a + b*ArcSin[c + d*x]))/b 
)^n + ((I/8)*3^(-1 - n)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-3*I)*(a 
+ b*ArcSin[c + d*x]))/b])/(E^(((3*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x])) 
/b)^n) - ((I/8)*3^(-1 - n)*E^(((3*I)*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma 
[1 + n, ((3*I)*(a + b*ArcSin[c + d*x]))/b])/((I*(a + b*ArcSin[c + d*x]))/b 
)^n)/d^3
 

3.2.73.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 5246

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

rule 5304

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.2.73.4 Maple [F]

\[\int x^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{n}d x\]

input

int(x^2*(a+b*arcsin(d*x+c))^n,x)
 

output

int(x^2*(a+b*arcsin(d*x+c))^n,x)
 

3.2.73.5 Fricas [F]

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

input

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

output

integral((b*arcsin(d*x + c) + a)^n*x^2, x)
 

3.2.73.6 Sympy [F]

\[ \int x^2 (a+b \arcsin (c+d x))^n \, dx=\int x^{2} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{n}\, dx \]

input

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

output

Integral(x**2*(a + b*asin(c + d*x))**n, x)
 

3.2.73.7 Maxima [F]

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

input

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

output

integrate((b*arcsin(d*x + c) + a)^n*x^2, x)
 

3.2.73.8 Giac [F]

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

input

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

output

integrate((b*arcsin(d*x + c) + a)^n*x^2, x)
 

3.2.73.9 Mupad [F(-1)]

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

input

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

output

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