\(\int \frac {x^m (a+b \arcsin (c x))^2}{(d-c^2 d x^2)^3} \, dx\) [274]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Int}\left (\frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3},x\right ) \] Output:

Defer(Int)(x^m*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x)
 

Mathematica [N/A]

Not integrable

Time = 7.90 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 1.99 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

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

\(\Big \downarrow \) 5208

\(\displaystyle -\frac {b c \int \frac {x^{m+1} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {(3-m) \int \frac {x^m (a+b \arcsin (c x))^2}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {x^{m+1} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {(3-m) \int \frac {x^m (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5208

\(\displaystyle -\frac {b c \left (\frac {1}{3} (1-m) \int \frac {x^{m+1} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {x^{m+2}}{\left (1-c^2 x^2\right )^2}dx+\frac {x^{m+2} (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}+\frac {(3-m) \left (-b c \int \frac {x^{m+1} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {b c \left (\frac {1}{3} (1-m) \int \frac {x^{m+1} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x^{m+2} (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {(3-m) \left (-b c \int \frac {x^{m+1} (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5208

\(\displaystyle -\frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-b c \int \frac {x^{m+2}}{1-c^2 x^2}dx+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {(3-m) \left (-b c \left (-(m+1) \int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-b c \int \frac {x^{m+2}}{1-c^2 x^2}dx+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {(3-m) \left (-b c \left (-(m+1) \int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arcsin (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5220

\(\displaystyle \frac {(3-m) \left (\frac {1}{2} (1-m) \int \frac {x^m (a+b \arcsin (c x))^2}{1-c^2 x^2}dx-b c \left (-(m+1) \left (\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+1} (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \left (\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5234

\(\displaystyle \frac {(3-m) \left (\frac {1}{2} (1-m) \int \frac {x^m (a+b \arcsin (c x))^2}{1-c^2 x^2}dx-b c \left (-(m+1) \left (\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+1} (a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \left (\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+2} (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {x^{m+1} (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\)

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:

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

Output:

integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*x^m/(c^6*d^3*x^6 - 
 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

Sympy [N/A]

Not integrable

Time = 100.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2} x^{m}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{m} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \] Input:

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

Output:

-(Integral(a**2*x**m/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Int 
egral(b**2*x**m*asin(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), 
x) + Integral(2*a*b*x**m*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 
- 1), x))/d**3
 

Maxima [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.74 \[ \int \frac {x^m (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-\left (\int \frac {x^{m}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a^{2}-2 \left (\int \frac {x^{m} \mathit {asin} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b -\left (\int \frac {x^{m} \mathit {asin} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2}}{d^{3}} \] Input:

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

Output:

( - int(x**m/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a**2 - 2*int(( 
x**m*asin(c*x))/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b - int(( 
x**m*asin(c*x)**2)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2)/d* 
*3