\(\int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx\) [402]

Optimal result

Integrand size = 27, antiderivative size = 391 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=-\frac {e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}}-\frac {e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-1-n} e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-1-n} e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c x))}{b}\right )}{8 c^2 \sqrt {1-c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.70 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\frac {d e^{-\frac {3 i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (3 e^{\frac {2 i a}{b}} \left (-\left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )-e^{\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )\right )-3^{-n} \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \left (\left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{24 c^2 \sqrt {d \left (1-c^2 x^2\right )}} \] Input:

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

Output:

(d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(3*E^(((2*I)*a)/b)*(-(Gamma[1 + 
 n, ((-I)*(a + b*ArcSin[c*x]))/b]/(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (E^( 
((2*I)*a)/b)*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c* 
x]))/b)^n) - (((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-3*I)*(a + b*Ar 
cSin[c*x]))/b] + E^(((6*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 
+ n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(3^n*((a + b*ArcSin[c*x])^2/b^2)^n))) 
/(24*c^2*E^(((3*I)*a)/b)*Sqrt[d*(1 - c^2*x^2)])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5224, 25, 4906, 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.

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 2009

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

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5224

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

Maple [F]

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

Input:

int(x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x)
 

Output:

int(x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x)
 

Fricas [F]

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

integrate(x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x, algorithm="fricas" 
)
 

Output:

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

Sympy [F]

\[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\int x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}\, dx \] Input:

integrate(x*(-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))**n,x)
 

Output:

Integral(x*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**n, x)
 

Maxima [F]

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

integrate(x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x, algorithm="maxima" 
)
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

int(x*(a + b*asin(c*x))^n*(d - c^2*d*x^2)^(1/2),x)
 

Output:

int(x*(a + b*asin(c*x))^n*(d - c^2*d*x^2)^(1/2), x)
 

Reduce [F]

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

int(x*(-c^2*d*x^2+d)^(1/2)*(a+b*asin(c*x))^n,x)
 

Output:

sqrt(d)*int((asin(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x,x)