\(\int (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))^n \, dx\) [413]

Optimal result

Integrand size = 26, antiderivative size = 698 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^{1+n}}{16 b c (1+n) \sqrt {1-c^2 x^2}}-\frac {15 i 2^{-7-n} d^2 e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {15 i 2^{-7-n} d^2 e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {3 i 2^{-7-2 n} d^2 e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {3 i 2^{-7-2 n} d^2 e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 i (a+b \arcsin (c x))}{b}\right )}{c \sqrt {1-c^2 x^2}} \] Output:

5/16*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^(1+n)/b/c/(1+n)/(-c^2*x^2+ 
1)^(1/2)-15*I*2^(-7-n)*d^2*(-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/exp(2*I*a/b)/(-c^2*x^2+1)^(1/2)/((-I*(a+b* 
arcsin(c*x))/b)^n)+15*I*2^(-7-n)*d^2*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/(-c^2*x^2+1)^(1/2)/( 
(I*(a+b*arcsin(c*x))/b)^n)-3*I*2^(-7-2*n)*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*ar 
csin(c*x))^n*GAMMA(1+n,-4*I*(a+b*arcsin(c*x))/b)/c/exp(4*I*a/b)/(-c^2*x^2+ 
1)^(1/2)/((-I*(a+b*arcsin(c*x))/b)^n)+3*I*2^(-7-2*n)*d^2*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/ 
(-c^2*x^2+1)^(1/2)/((I*(a+b*arcsin(c*x))/b)^n)-I*2^(-7-n)*3^(-1-n)*d^2*(-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/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^2*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/(-c^2*x^2+1)^(1/2)/((I*(a+b*arcsin(c*x))/b)^ 
n)
 

Mathematica [A] (verified)

Time = 3.27 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.68 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\frac {d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {120 a}{b+b n}+\frac {120 \arcsin (c x)}{1+n}-45 i 2^{-n} e^{-\frac {2 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c x))}{b}\right )+45 i 2^{-n} e^{\frac {2 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c x))}{b}\right )-9 i 4^{-n} e^{-\frac {4 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {4 i (a+b \arcsin (c x))}{b}\right )+9 i 4^{-n} e^{\frac {4 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {4 i (a+b \arcsin (c x))}{b}\right )-i 6^{-n} e^{-\frac {6 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,-\frac {6 i (a+b \arcsin (c x))}{b}\right )+i 6^{-n} e^{\frac {6 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {6 i (a+b \arcsin (c x))}{b}\right )\right )}{384 c \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n,x]
 

Output:

(d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((120*a)/(b + b*n) + (120*Arc 
Sin[c*x])/(1 + n) - ((45*I)*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/ 
(2^n*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + ((45*I)*E^(((2*I) 
*a)/b)*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*((I*(a + b*ArcSin 
[c*x]))/b)^n) - ((9*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) + ((9*I)*E^(((4*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, 
((4*I)*(a + b*ArcSin[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*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/ 
b])/(6^n*((a + b*ArcSin[c*x])^2/b^2)^n)))/(384*c*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 495, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5168, 3042, 3793, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx\)

\(\Big \downarrow \) 5168

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x))^n \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )^6d(a+b \arcsin (c x))}{b c \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {1}{32} \cos \left (\frac {6 a}{b}-\frac {6 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {3}{16} \cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {15}{32} \cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {5}{16} (a+b \arcsin (c x))^n\right )d(a+b \arcsin (c x))}{b c \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {5 (a+b \arcsin (c x))^{n+1}}{16 (n+1)}-15 i b 2^{-n-7} e^{-\frac {2 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i (a+b \arcsin (c x))}{b}\right )-3 i b 2^{-2 n-7} e^{-\frac {4 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 i (a+b \arcsin (c x))}{b}\right )-i b 2^{-n-7} 3^{-n-1} e^{-\frac {6 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 i (a+b \arcsin (c x))}{b}\right )+15 i b 2^{-n-7} e^{\frac {2 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i (a+b \arcsin (c x))}{b}\right )+3 i b 2^{-2 n-7} e^{\frac {4 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 i (a+b \arcsin (c x))}{b}\right )+i b 2^{-n-7} 3^{-n-1} e^{\frac {6 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 i (a+b \arcsin (c x))}{b}\right )\right )}{b c \sqrt {1-c^2 x^2}}\)

Input:

Int[(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n,x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((5*(a + b*ArcSin[c*x])^(1 + n))/(16*(1 + n)) - ( 
(15*I)*2^(-7 - n)*b*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcS 
in[c*x]))/b])/(E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + ((15*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 - ((3*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) + ((3*I)*2^(-7 - 2*n)*b*E^ 
(((4*I)*a)/b)*(a + b*ArcSin[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*A 
rcSin[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*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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5168

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[Int[ 
x^n*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, 0]
 

Maple [F]

\[\int \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}d x\]

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x)
 

Output:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x)
 

Fricas [F]

\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**n,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x, algorithm="maxima")
 

Output:

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)^n, x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(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. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

int((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=\sqrt {d}\, d^{2} \left (\left (\int \left (\mathit {asin} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{4}d x \right ) c^{4}-2 \left (\int \left (\mathit {asin} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{2}d x \right ) c^{2}+\int \left (\mathit {asin} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}d x \right ) \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x))^n,x)
 

Output:

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