\(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \arcsin (c x))^n \, dx\) [406]

Optimal result

Integrand size = 29, antiderivative size = 684 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx=\frac {d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^{1+n}}{16 b c^3 (1+n) \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-n} d e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} d e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-2 n} d e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 i (a+b \arcsin (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-2 n} d e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 i (a+b \arcsin (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} 3^{-1-n} d e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 i (a+b \arcsin (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-n} 3^{-1-n} d e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 i (a+b \arcsin (c x))}{b}\right )}{c^3 \sqrt {1-c^2 x^2}} \] Output:

1/16*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^(1+n)/b/c^3/(1+n)/(-c^2*x^2+ 
1)^(1/2)-I*2^(-7-n)*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n*GAMMA(1+n,- 
2*I*(a+b*arcsin(c*x))/b)/c^3/exp(2*I*a/b)/(-c^2*x^2+1)^(1/2)/((-I*(a+b*arc 
sin(c*x))/b)^n)+I*2^(-7-n)*d*exp(2*I*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin 
(c*x))^n*GAMMA(1+n,2*I*(a+b*arcsin(c*x))/b)/c^3/(-c^2*x^2+1)^(1/2)/((I*(a+ 
b*arcsin(c*x))/b)^n)+I*2^(-7-2*n)*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x)) 
^n*GAMMA(1+n,-4*I*(a+b*arcsin(c*x))/b)/c^3/exp(4*I*a/b)/(-c^2*x^2+1)^(1/2) 
/((-I*(a+b*arcsin(c*x))/b)^n)-I*2^(-7-2*n)*d*exp(4*I*a/b)*(-c^2*d*x^2+d)^( 
1/2)*(a+b*arcsin(c*x))^n*GAMMA(1+n,4*I*(a+b*arcsin(c*x))/b)/c^3/(-c^2*x^2+ 
1)^(1/2)/((I*(a+b*arcsin(c*x))/b)^n)+I*2^(-7-n)*3^(-1-n)*d*(-c^2*d*x^2+d)^ 
(1/2)*(a+b*arcsin(c*x))^n*GAMMA(1+n,-6*I*(a+b*arcsin(c*x))/b)/c^3/exp(6*I* 
a/b)/(-c^2*x^2+1)^(1/2)/((-I*(a+b*arcsin(c*x))/b)^n)-I*2^(-7-n)*3^(-1-n)*d 
*exp(6*I*a/b)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n*GAMMA(1+n,6*I*(a+b* 
arcsin(c*x))/b)/c^3/(-c^2*x^2+1)^(1/2)/((I*(a+b*arcsin(c*x))/b)^n)
 

Mathematica [A] (verified)

Time = 2.52 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.74 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx=\frac {d^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {24 a}{b+b n}+\frac {24 \arcsin (c x)}{1+n}-3 i 2^{-n} e^{-\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c x))}{b}\right )+3 i 2^{-n} e^{\frac {2 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c x))}{b}\right )+3 i 4^{-n} e^{-\frac {4 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {4 i (a+b \arcsin (c x))}{b}\right )-3 i 4^{-n} e^{\frac {4 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {4 i (a+b \arcsin (c x))}{b}\right )+i 6^{-n} e^{-\frac {6 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {6 i (a+b \arcsin (c x))}{b}\right )-i 6^{-n} e^{\frac {6 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {6 i (a+b \arcsin (c x))}{b}\right )\right )}{384 c^3 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n,x]
 

Output:

(d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((24*a)/(b + b*n) + (24*ArcSi 
n[c*x])/(1 + n) - ((3*I)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-2*I 
)*(a + b*ArcSin[c*x]))/b])/(2^n*E^(((2*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2 
)^n) + ((3*I)*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n 
, ((2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*((a + b*ArcSin[c*x])^2/b^2)^n) + (( 
3*I)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x] 
))/b])/(4^n*E^(((4*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n) - ((3*I)*E^(((4 
*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcS 
in[c*x]))/b])/(4^n*((a + b*ArcSin[c*x])^2/b^2)^n) + (I*((I*(a + b*ArcSin[c 
*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b])/(6^n*E^(((6*I)*a) 
/b)*((a + b*ArcSin[c*x])^2/b^2)^n) - (I*E^(((6*I)*a)/b)*(((-I)*(a + b*ArcS 
in[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b])/(6^n*((a + b*A 
rcSin[c*x])^2/b^2)^n)))/(384*c^3*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 493, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5224, 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.

\(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx\)

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 4906

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (-\frac {1}{32} \cos \left (\frac {6 a}{b}-\frac {6 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n-\frac {1}{16} \cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {1}{32} \cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {1}{16} (a+b \arcsin (c x))^n\right )d(a+b \arcsin (c x))}{b c^3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {(a+b \arcsin (c x))^{n+1}}{16 (n+1)}-i b 2^{-n-7} e^{-\frac {2 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i (a+b \arcsin (c x))}{b}\right )+i b 2^{-2 n-7} e^{-\frac {4 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 i (a+b \arcsin (c x))}{b}\right )+i b 2^{-n-7} 3^{-n-1} e^{-\frac {6 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 i (a+b \arcsin (c x))}{b}\right )+i b 2^{-n-7} e^{\frac {2 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i (a+b \arcsin (c x))}{b}\right )-i b 2^{-2 n-7} e^{\frac {4 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 i (a+b \arcsin (c x))}{b}\right )-i b 2^{-n-7} 3^{-n-1} e^{\frac {6 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 i (a+b \arcsin (c x))}{b}\right )\right )}{b c^3 \sqrt {1-c^2 x^2}}\)

Input:

Int[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n,x]
 

Output:

(d*Sqrt[d - c^2*d*x^2]*((a + b*ArcSin[c*x])^(1 + n)/(16*(1 + n)) - (I*2^(- 
7 - n)*b*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b 
])/(E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*2^(-7 - n)*b*E^ 
(((2*I)*a)/b)*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x] 
))/b])/((I*(a + b*ArcSin[c*x]))/b)^n + (I*2^(-7 - 2*n)*b*(a + b*ArcSin[c*x 
])^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b])/(E^(((4*I)*a)/b)*(((-I) 
*(a + b*ArcSin[c*x]))/b)^n) - (I*2^(-7 - 2*n)*b*E^(((4*I)*a)/b)*(a + b*Arc 
Sin[c*x])^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin 
[c*x]))/b)^n + (I*2^(-7 - n)*3^(-1 - n)*b*(a + b*ArcSin[c*x])^n*Gamma[1 + 
n, ((-6*I)*(a + b*ArcSin[c*x]))/b])/(E^(((6*I)*a)/b)*(((-I)*(a + b*ArcSin[ 
c*x]))/b)^n) - (I*2^(-7 - n)*3^(-1 - n)*b*E^(((6*I)*a)/b)*(a + b*ArcSin[c* 
x])^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c*x]) 
)/b)^n))/(b*c^3*Sqrt[1 - c^2*x^2])
 

Defintions of rubi rules used

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]

\[\int x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}d x\]

Input:

int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x)
 

Output:

int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x)
 

Fricas [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x, algorithm="frica 
s")
 

Output:

integral(-(c^2*d*x^4 - d*x^2)*sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n, 
x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx=\text {Timed out} \] Input:

integrate(x**2*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**n,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x, algorithm="maxim 
a")
 

Output:

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^n*x^2, x)
 

Giac [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x^{2} \,d x } \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x, algorithm="giac" 
)
 

Output:

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^n*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:

int(x^2*(a + b*asin(c*x))^n*(d - c^2*d*x^2)^(3/2),x)
 

Output:

int(x^2*(a + b*asin(c*x))^n*(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

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

int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*asin(c*x))^n,x)
 

Output:

sqrt(d)*d*( - int((asin(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**4,x)*c**2 
 + int((asin(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**2,x))