\(\int x (a+b \arcsin (c+d x))^n \, dx\) [53]

Optimal result

Integrand size = 14, antiderivative size = 301 \[ \int x (a+b \arcsin (c+d x))^n \, dx=\frac {i c e^{-\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d^2}-\frac {i c e^{\frac {i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )}{2 d^2}-\frac {2^{-3-n} e^{-\frac {2 i a}{b}} (a+b \arcsin (c+d x))^n \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )}{d^2}-\frac {2^{-3-n} e^{\frac {2 i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c+d x))}{b}\right )}{d^2} \] Output:

1/2*I*c*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,-I*(a+b*arcsin(d*x+c))/b)/d^2/exp( 
I*a/b)/((-I*(a+b*arcsin(d*x+c))/b)^n)-1/2*I*c*exp(I*a/b)*(a+b*arcsin(d*x+c 
))^n*GAMMA(1+n,I*(a+b*arcsin(d*x+c))/b)/d^2/((I*(a+b*arcsin(d*x+c))/b)^n)- 
2^(-3-n)*(a+b*arcsin(d*x+c))^n*GAMMA(1+n,-2*I*(a+b*arcsin(d*x+c))/b)/d^2/e 
xp(2*I*a/b)/((-I*(a+b*arcsin(d*x+c))/b)^n)-2^(-3-n)*exp(2*I*a/b)*(a+b*arcs 
in(d*x+c))^n*GAMMA(1+n,2*I*(a+b*arcsin(d*x+c))/b)/d^2/((I*(a+b*arcsin(d*x+ 
c))/b)^n)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.89 \[ \int x (a+b \arcsin (c+d x))^n \, dx=-\frac {i 2^{-3-n} e^{-\frac {2 i a}{b}} (a+b \arcsin (c+d x))^n \left (\frac {(a+b \arcsin (c+d x))^2}{b^2}\right )^{-n} \left (-2^{2+n} c e^{\frac {i a}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,-\frac {i (a+b \arcsin (c+d x))}{b}\right )+2^{2+n} c e^{\frac {3 i a}{b}} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,\frac {i (a+b \arcsin (c+d x))}{b}\right )-i \left (\left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {4 i a}{b}} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^n \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5304, 25, 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.

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

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5246

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((((-1/2*I)*c*(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*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^(-3 - n)*(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^(-3 - n)*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)/d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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] 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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