3.5.93 \(\int x (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))^n \, dx\) [493]

3.5.93.1 Optimal result

Integrand size = 27, antiderivative size = 815 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=-\frac {5 d^2 e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {5 d^2 e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{1-n} d^2 e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{1-n} d^2 e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-n} d^2 e^{-\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-n} d^2 e^{\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {7^{-1-n} d^2 e^{-\frac {7 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {7 i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {7^{-1-n} d^2 e^{\frac {7 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {7 i (a+b \arcsin (c x))}{b}\right )}{128 c^2 \sqrt {1-c^2 x^2}} \]

output

-5/128*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c*x))/b)*(-c^2*d*x 
^2+d)^(1/2)/c^2/exp(I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2) 
-5/128*d^2*exp(I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,I*(a+b*arcsin(c*x))/b) 
*(-c^2*d*x^2+d)^(1/2)/c^2/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-1 
/128*3^(1-n)*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,-3*I*(a+b*arcsin(c*x))/b)*( 
-c^2*d*x^2+d)^(1/2)/c^2/exp(3*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^ 
2+1)^(1/2)-1/128*3^(1-n)*d^2*exp(3*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,3* 
I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^2/((I*(a+b*arcsin(c*x))/b)^n 
)/(-c^2*x^2+1)^(1/2)-1/128*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,-5*I*(a+b*arc 
sin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(5^n)/c^2/exp(5*I*a/b)/((-I*(a+b*arcsin( 
c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-1/128*d^2*exp(5*I*a/b)*(a+b*arcsin(c*x))^n* 
GAMMA(1+n,5*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(5^n)/c^2/((I*(a+b 
*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-1/128*7^(-1-n)*d^2*(a+b*arcsin(c*x) 
)^n*GAMMA(1+n,-7*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^2/exp(7*I*a 
/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-1/128*7^(-1-n)*d^2*exp 
(7*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,7*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x 
^2+d)^(1/2)/c^2/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)
 

3.5.93.2 Mathematica [A] (verified)

Time = 2.82 (sec) , antiderivative size = 603, normalized size of antiderivative = 0.74 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n \, dx=-\frac {5^{-n} 21^{-1-n} d^3 e^{-\frac {7 i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{-3 n} \left (105^{1+n} e^{\frac {6 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )+\left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (105^{1+n} e^{\frac {8 i a}{b}} \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )+9\ 5^n 7^{1+n} e^{\frac {4 i a}{b}} \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{2 n} \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {3 i (a+b \arcsin (c x))}{b}\right )+9\ 5^n 7^{1+n} e^{\frac {10 i a}{b}} \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {3 i (a+b \arcsin (c x))}{b}\right )+3^{1+n} \left (7^{1+n} e^{\frac {2 i a}{b}} \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3 n} \Gamma \left (1+n,-\frac {5 i (a+b \arcsin (c x))}{b}\right )+7^{1+n} e^{\frac {12 i a}{b}} \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {5 i (a+b \arcsin (c x))}{b}\right )+5^n \left (\left (-\frac {i (a+b \arcsin (c x))}{b}\right )^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{3 n} \Gamma \left (1+n,-\frac {7 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {14 i a}{b}} \left (\frac {(a+b \arcsin (c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {7 i (a+b \arcsin (c x))}{b}\right )\right )\right )\right )\right )}{128 c^2 \sqrt {d-c^2 d x^2}} \]

input

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

output

-1/128*(21^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(105^(1 + 
n)*E^(((6*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^n*((a + b*ArcSin[c*x])^2/b^ 
2)^(2*n)*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b] + (((-I)*(a + b*ArcSin 
[c*x]))/b)^n*(105^(1 + n)*E^(((8*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n 
)*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b] + 9*5^n*7^(1 + n)*E^(((4*I)*a)/b 
)*((I*(a + b*ArcSin[c*x]))/b)^(2*n)*((a + b*ArcSin[c*x])^2/b^2)^n*Gamma[1 
+ n, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 9*5^n*7^(1 + n)*E^(((10*I)*a)/b)*(( 
a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b 
] + 3^(1 + n)*(7^(1 + n)*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n* 
((I*(a + b*ArcSin[c*x]))/b)^(3*n)*Gamma[1 + n, ((-5*I)*(a + b*ArcSin[c*x]) 
)/b] + 7^(1 + n)*E^(((12*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[ 
1 + n, ((5*I)*(a + b*ArcSin[c*x]))/b] + 5^n*((((-I)*(a + b*ArcSin[c*x]))/b 
)^n*((I*(a + b*ArcSin[c*x]))/b)^(3*n)*Gamma[1 + n, ((-7*I)*(a + b*ArcSin[c 
*x]))/b] + E^(((14*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, 
 ((7*I)*(a + b*ArcSin[c*x]))/b])))))/(5^n*c^2*E^(((7*I)*a)/b)*Sqrt[d - c^2 
*d*x^2]*((a + b*ArcSin[c*x])^2/b^2)^(3*n))
 

3.5.93.3 Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 582, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {1}{64} \sin \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {5}{64} \sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {9}{64} \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right ) (a+b \arcsin (c x))^n+\frac {5}{64} \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) (a+b \arcsin (c x))^n\right )d(a+b \arcsin (c x))}{b c^2 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {5}{128} b e^{-\frac {i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i (a+b \arcsin (c x))}{b}\right )-\frac {1}{128} b 3^{1-n} e^{-\frac {3 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 i (a+b \arcsin (c x))}{b}\right )-\frac {1}{128} b 5^{-n} e^{-\frac {5 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 i (a+b \arcsin (c x))}{b}\right )-\frac {1}{128} b 7^{-n-1} e^{-\frac {7 i a}{b}} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,-\frac {7 i (a+b \arcsin (c x))}{b}\right )-\frac {5}{128} b e^{\frac {i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {i (a+b \arcsin (c x))}{b}\right )-\frac {1}{128} b 3^{1-n} e^{\frac {3 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 i (a+b \arcsin (c x))}{b}\right )-\frac {1}{128} b 5^{-n} e^{\frac {5 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 i (a+b \arcsin (c x))}{b}\right )-\frac {1}{128} b 7^{-n-1} e^{\frac {7 i a}{b}} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (n+1,\frac {7 i (a+b \arcsin (c x))}{b}\right )\right )}{b c^2 \sqrt {1-c^2 x^2}}\)

input

Int[x*(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*b*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-I)*( 
a + b*ArcSin[c*x]))/b])/(128*E^((I*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) 
 - (5*b*E^((I*a)/b)*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c* 
x]))/b])/(128*((I*(a + b*ArcSin[c*x]))/b)^n) - (3^(1 - n)*b*(a + b*ArcSin[ 
c*x])^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b])/(128*E^(((3*I)*a)/b) 
*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (3^(1 - n)*b*E^(((3*I)*a)/b)*(a + b*A 
rcSin[c*x])^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(128*((I*(a + b 
*ArcSin[c*x]))/b)^n) - (b*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-5*I)*(a + 
b*ArcSin[c*x]))/b])/(128*5^n*E^(((5*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b 
)^n) - (b*E^(((5*I)*a)/b)*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((5*I)*(a + b 
*ArcSin[c*x]))/b])/(128*5^n*((I*(a + b*ArcSin[c*x]))/b)^n) - (7^(-1 - n)*b 
*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-7*I)*(a + b*ArcSin[c*x]))/b])/(128* 
E^(((7*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (7^(-1 - n)*b*E^(((7*I 
)*a)/b)*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((7*I)*(a + b*ArcSin[c*x]))/b]) 
/(128*((I*(a + b*ArcSin[c*x]))/b)^n)))/(b*c^2*Sqrt[1 - c^2*x^2])
 

3.5.93.3.1 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]
 

3.5.93.4 Maple [F]

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

input

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

output

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

3.5.93.5 Fricas [F]

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

input

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

output

integral((c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x)*sqrt(-c^2*d*x^2 + d)*(b*arc 
sin(c*x) + a)^n, x)
 

3.5.93.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.5.93.7 Maxima [F]

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

input

integrate(x*(-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, x)
 

3.5.93.8 Giac [F(-2)]

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

input

integrate(x*(-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
 

3.5.93.9 Mupad [F(-1)]

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

input

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

output

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