\(\int (1-a^2-2 a b x-b^2 x^2)^{3/2} \arcsin (a+b x)^3 \, dx\) [195]

Optimal result

Integrand size = 33, antiderivative size = 251 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\frac {45 (a+b x)^2}{128 b}-\frac {3 \left (1-(a+b x)^2\right )^2}{128 b}-\frac {45 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{64 b}-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)}{32 b}+\frac {27 \arcsin (a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \arcsin (a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{4 b}+\frac {3 \arcsin (a+b x)^4}{32 b} \] Output:

45/128*(b*x+a)^2/b-3/128*(1-(b*x+a)^2)^2/b-45/64*(b*x+a)*(1-(b*x+a)^2)^(1/ 
2)*arcsin(b*x+a)/b-3/32*(b*x+a)*(1-(b*x+a)^2)^(3/2)*arcsin(b*x+a)/b+27/128 
*arcsin(b*x+a)^2/b-9/16*(b*x+a)^2*arcsin(b*x+a)^2/b+3/16*(1-(b*x+a)^2)^2*a 
rcsin(b*x+a)^2/b+3/8*(b*x+a)*(1-(b*x+a)^2)^(1/2)*arcsin(b*x+a)^3/b+1/4*(b* 
x+a)*(1-(b*x+a)^2)^(3/2)*arcsin(b*x+a)^3/b+3/32*arcsin(b*x+a)^4/b
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.08 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\frac {6 a \left (17-2 a^2\right ) b x+3 \left (17-6 a^2\right ) b^2 x^2-12 a b^3 x^3-3 b^4 x^4+6 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-17 a+2 a^3-17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \arcsin (a+b x)+3 \left (17+8 a^4+32 a^3 b x-40 b^2 x^2+8 b^4 x^4+16 a b x \left (-5+2 b^2 x^2\right )+8 a^2 \left (-5+6 b^2 x^2\right )\right ) \arcsin (a+b x)^2-16 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-5 a+2 a^3-5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \arcsin (a+b x)^3+12 \arcsin (a+b x)^4}{128 b} \] Input:

Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3,x]
 

Output:

(6*a*(17 - 2*a^2)*b*x + 3*(17 - 6*a^2)*b^2*x^2 - 12*a*b^3*x^3 - 3*b^4*x^4 
+ 6*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-17*a + 2*a^3 - 17*b*x + 6*a^2*b*x 
+ 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x] + 3*(17 + 8*a^4 + 32*a^3*b*x - 
40*b^2*x^2 + 8*b^4*x^4 + 16*a*b*x*(-5 + 2*b^2*x^2) + 8*a^2*(-5 + 6*b^2*x^2 
))*ArcSin[a + b*x]^2 - 16*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-5*a + 2*a^3 
- 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x]^3 + 12*ArcS 
in[a + b*x]^4)/(128*b)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.25, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5306, 5158, 5156, 5138, 5152, 5182, 5158, 244, 2009, 5156, 15, 5152, 5210, 15, 5152}

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.

\(\displaystyle \int \left (-a^2-2 a b x-b^2 x^2+1\right )^{3/2} \arcsin (a+b x)^3 \, dx\)

\(\Big \downarrow \) 5306

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )-\frac {3}{4} \int (a+b x) \left (1-(a+b x)^2\right ) \arcsin (a+b x)^2d(a+b x)+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{2} \int \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)d(a+b x)-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 5158

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{2} \left (\frac {3}{4} \int \sqrt {1-(a+b x)^2} \arcsin (a+b x)d(a+b x)-\frac {1}{4} \int (a+b x) \left (1-(a+b x)^2\right )d(a+b x)+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{2} \left (\frac {3}{4} \int \sqrt {1-(a+b x)^2} \arcsin (a+b x)d(a+b x)-\frac {1}{4} \int \left (-(a+b x)^3+a+b x\right )d(a+b x)+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )-\frac {3}{4} \left (\frac {1}{2} \left (\frac {3}{4} \int \sqrt {1-(a+b x)^2} \arcsin (a+b x)d(a+b x)+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 5156

\(\displaystyle \frac {-\frac {3}{4} \left (\frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )+\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {-\frac {3}{4} \left (\frac {1}{2} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)+\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )+\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{b}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2-\int \frac {(a+b x)^2 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3-\frac {3}{4} \left (\frac {1}{2} \left (\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {3}{4} \left (\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)+\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )}{b}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (-\frac {1}{2} \int \frac {\arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3-\frac {3}{4} \left (\frac {1}{2} \left (\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {3}{4} \left (\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)+\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\frac {3}{4} \left (-\frac {3}{2} \left (-\frac {1}{2} \int \frac {\arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)+\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2+\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)-\frac {1}{4} (a+b x)^2\right )+\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3\right )+\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3-\frac {3}{4} \left (\frac {1}{2} \left (\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {3}{4} \left (\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)+\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )}{b}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3+\frac {3}{4} \left (\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2+\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)-\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )\right )-\frac {3}{4} \left (\frac {1}{2} \left (\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {3}{4} \left (\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)+\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )}{b}\)

Input:

Int[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3,x]
 

Output:

(((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^3)/4 + (3*(((a + b*x)* 
Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^3)/2 + ArcSin[a + b*x]^4/8 - (3*(-1/ 
4*(a + b*x)^2 + ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/2 - ArcS 
in[a + b*x]^2/4 + ((a + b*x)^2*ArcSin[a + b*x]^2)/2))/2))/4 - (3*(-1/4*((1 
 - (a + b*x)^2)^2*ArcSin[a + b*x]^2) + ((-1/2*(a + b*x)^2 + (a + b*x)^4/4) 
/4 + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x])/4 + (3*(-1/4*(a + 
 b*x)^2 + ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/2 + ArcSin[a + 
 b*x]^2/4))/4)/2))/4)/b
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 5138

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

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5156

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcSin[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 5158

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], 
x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

rule 5182

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

rule 5210

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

rule 5306

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + ( 
C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/d   Subst[Int[(-C/d^2 + (C/d^2)*x^2 
)^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C, 
 n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(627\) vs. \(2(223)=446\).

Time = 0.38 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.50

method	result	size
default	\(\frac {-75+96 \arcsin \left (b x +a \right )^{2} b^{4} x^{4}-128 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a^{3}+48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a^{3}+320 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} b x -960 \arcsin \left (b x +a \right )^{2} a b x -408 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) b x -12 a^{4}+204 a^{2}+48 \arcsin \left (b x +a \right )^{4}+408 a b x +96 \arcsin \left (b x +a \right )^{2} a^{4}-12 b^{4} x^{4}+204 \arcsin \left (b x +a \right )^{2}-480 \arcsin \left (b x +a \right )^{2} a^{2}+204 b^{2} x^{2}-480 \arcsin \left (b x +a \right )^{2} b^{2} x^{2}+320 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a -408 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a +144 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a^{2} b x +384 \arcsin \left (b x +a \right )^{2} a \,b^{3} x^{3}+576 \arcsin \left (b x +a \right )^{2} a^{2} b^{2} x^{2}+384 \arcsin \left (b x +a \right )^{2} a^{3} b x +48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) b^{3} x^{3}-128 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} b^{3} x^{3}-384 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a \,b^{2} x^{2}-384 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a^{2} b x +144 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a \,b^{2} x^{2}-48 a \,b^{3} x^{3}-72 a^{2} b^{2} x^{2}-48 a^{3} b x}{512 b}\)	\(628\)

Input:

int((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x,method=_RETURNVERBOSE 
)
                                                                                    
                                                                                    
 

Output:

1/512*(-75+96*arcsin(b*x+a)^2*b^4*x^4-128*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a 
rcsin(b*x+a)^3*a^3+48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)*a^3+320 
*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*b*x-960*arcsin(b*x+a)^2*a* 
b*x-408*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)*b*x-12*a^4+204*a^2+48 
*arcsin(b*x+a)^4+408*a*b*x+96*arcsin(b*x+a)^2*a^4-12*b^4*x^4+204*arcsin(b* 
x+a)^2-480*arcsin(b*x+a)^2*a^2+204*b^2*x^2-480*arcsin(b*x+a)^2*b^2*x^2+320 
*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*a-408*(-b^2*x^2-2*a*b*x-a^ 
2+1)^(1/2)*arcsin(b*x+a)*a+144*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a 
)*a^2*b*x+384*arcsin(b*x+a)^2*a*b^3*x^3+576*arcsin(b*x+a)^2*a^2*b^2*x^2+38 
4*arcsin(b*x+a)^2*a^3*b*x+48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)* 
b^3*x^3-128*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*b^3*x^3-384*(-b 
^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*a*b^2*x^2-384*(-b^2*x^2-2*a*b* 
x-a^2+1)^(1/2)*arcsin(b*x+a)^3*a^2*b*x+144*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)* 
arcsin(b*x+a)*a*b^2*x^2-48*a*b^3*x^3-72*a^2*b^2*x^2-48*a^3*b*x)/b
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.97 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=-\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} - 17\right )} b^{2} x^{2} - 12 \, \arcsin \left (b x + a\right )^{4} + 6 \, {\left (2 \, a^{3} - 17 \, a\right )} b x - 3 \, {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} - 5 \, a\right )} b x - 40 \, a^{2} + 17\right )} \arcsin \left (b x + a\right )^{2} + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (8 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 17\right )} b x - 17 \, a\right )} \arcsin \left (b x + a\right )\right )}}{128 \, b} \] Input:

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="fri 
cas")
 

Output:

-1/128*(3*b^4*x^4 + 12*a*b^3*x^3 + 3*(6*a^2 - 17)*b^2*x^2 - 12*arcsin(b*x 
+ a)^4 + 6*(2*a^3 - 17*a)*b*x - 3*(8*b^4*x^4 + 32*a*b^3*x^3 + 8*(6*a^2 - 5 
)*b^2*x^2 + 8*a^4 + 16*(2*a^3 - 5*a)*b*x - 40*a^2 + 17)*arcsin(b*x + a)^2 
+ 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(8*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 
 + (6*a^2 - 5)*b*x - 5*a)*arcsin(b*x + a)^3 - 3*(2*b^3*x^3 + 6*a*b^2*x^2 + 
 2*a^3 + (6*a^2 - 17)*b*x - 17*a)*arcsin(b*x + a)))/b
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (226) = 452\).

Time = 1.08 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.76 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx =\text {Too large to display} \] Input:

integrate((-b**2*x**2-2*a*b*x-a**2+1)**(3/2)*asin(b*x+a)**3,x)
 

Output:

Piecewise((3*a**4*asin(a + b*x)**2/(16*b) + 3*a**3*x*asin(a + b*x)**2/4 - 
3*a**3*x/32 - a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/ 
(4*b) + 3*a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(32*b) 
+ 9*a**2*b*x**2*asin(a + b*x)**2/8 - 9*a**2*b*x**2/64 - 3*a**2*x*sqrt(-a** 
2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/4 + 9*a**2*x*sqrt(-a**2 - 2* 
a*b*x - b**2*x**2 + 1)*asin(a + b*x)/32 - 15*a**2*asin(a + b*x)**2/(16*b) 
+ 3*a*b**2*x**3*asin(a + b*x)**2/4 - 3*a*b**2*x**3/32 - 3*a*b*x**2*sqrt(-a 
**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/4 + 9*a*b*x**2*sqrt(-a**2 
- 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/32 - 15*a*x*asin(a + b*x)**2/8 + 
51*a*x/64 + 5*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/(8* 
b) - 51*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(64*b) + 3*b 
**3*x**4*asin(a + b*x)**2/16 - 3*b**3*x**4/128 - b**2*x**3*sqrt(-a**2 - 2* 
a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/4 + 3*b**2*x**3*sqrt(-a**2 - 2*a*b 
*x - b**2*x**2 + 1)*asin(a + b*x)/32 - 15*b*x**2*asin(a + b*x)**2/16 + 51* 
b*x**2/128 + 5*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/8 
- 51*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/64 + 3*asin(a + 
 b*x)**4/(32*b) + 51*asin(a + b*x)**2/(128*b), Ne(b, 0)), (x*(1 - a**2)**( 
3/2)*asin(a)**3, True))
 

Maxima [F]

\[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\int { {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{3} \,d x } \] Input:

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="max 
ima")
 

Output:

integrate((-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*arcsin(b*x + a)^3, x)
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.18 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{4 \, b} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{8 \, b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )^{2}}{16 \, b} + \frac {3 \, \arcsin \left (b x + a\right )^{4}}{32 \, b} - \frac {3 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{32 \, b} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{16 \, b} - \frac {45 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{64 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{128 \, b} - \frac {45 \, \arcsin \left (b x + a\right )^{2}}{128 \, b} + \frac {45 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{128 \, b} + \frac {189}{1024 \, b} \] Input:

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="gia 
c")
 

Output:

1/4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)^3/b + 3 
/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)^3/b + 3/16 
*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2*arcsin(b*x + a)^2/b + 3/32*arcsin(b*x + a 
)^4/b - 3/32*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a 
)/b - 9/16*(b^2*x^2 + 2*a*b*x + a^2 - 1)*arcsin(b*x + a)^2/b - 45/64*sqrt( 
-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)/b - 3/128*(b^2*x^2 
 + 2*a*b*x + a^2 - 1)^2/b - 45/128*arcsin(b*x + a)^2/b + 45/128*(b^2*x^2 + 
 2*a*b*x + a^2 - 1)/b + 189/1024/b
 

Mupad [F(-1)]

Timed out. \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\int {\mathrm {asin}\left (a+b\,x\right )}^3\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2} \,d x \] Input:

int(asin(a + b*x)^3*(1 - b^2*x^2 - 2*a*b*x - a^2)^(3/2),x)
 

Output:

int(asin(a + b*x)^3*(1 - b^2*x^2 - 2*a*b*x - a^2)^(3/2), x)
 

Reduce [F]

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

int((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*asin(b*x+a)^3,x)
 

Output:

 - int(sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3*x**2,x)*b* 
*2 - 2*int(sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3*x,x)*a 
*b - int(sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3,x)*a**2 
+ int(sqrt( - a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3,x)