Integrand size = 33, antiderivative size = 245 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}-\frac {45 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{64 b}-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)}{32 b}+\frac {27 \arcsin (a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \arcsin (a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3}{4 b}+\frac {3 \arcsin (a+b x)^4}{32 b} \]
51/128*(b*x+a)^2/b-3/128*(b*x+a)^4/b-3/32*(b*x+a)*(1-(b*x+a)^2)^(3/2)*arcs in(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*arcsin(b*x+a)^2/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-45/64*(b*x+a)*arcsin(b*x+a)*(1-(b*x+a)^2 )^(1/2)/b+3/8*(b*x+a)*arcsin(b*x+a)^3*(1-(b*x+a)^2)^(1/2)/b
Time = 0.15 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.11 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\frac {6 a \left (17-2 a^2\right ) b x+3 \left (17-6 a^2\right ) b^2 x^2-12 a b^3 x^3-3 b^4 x^4+6 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-17 a+2 a^3-17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \arcsin (a+b x)+3 \left (17+8 a^4+32 a^3 b x-40 b^2 x^2+8 b^4 x^4+16 a b x \left (-5+2 b^2 x^2\right )+8 a^2 \left (-5+6 b^2 x^2\right )\right ) \arcsin (a+b x)^2-16 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-5 a+2 a^3-5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \arcsin (a+b x)^3+12 \arcsin (a+b x)^4}{128 b} \]
(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)
Time = 1.46 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.28, 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 |
| \(\displaystyle \frac {\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)^3+\frac {3}{4} \left (\frac {1}{8} \arcsin (a+b x)^4+\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^3-\frac {3}{2} \left (\frac {1}{2} (a+b x)^2 \arcsin (a+b x)^2+\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)-\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )\right )-\frac {3}{4} \left (\frac {1}{2} \left (\frac {1}{4} (a+b x) \left (1-(a+b x)^2\right )^{3/2} \arcsin (a+b x)+\frac {3}{4} \left (\frac {1}{2} \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)+\frac {1}{4} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^2\right )+\frac {1}{4} \left (\frac {1}{4} (a+b x)^4-\frac {1}{2} (a+b x)^2\right )\right )-\frac {1}{4} \left (1-(a+b x)^2\right )^2 \arcsin (a+b x)^2\right )}{b}\) |
(((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
3.4.20.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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]
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]
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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(627\) vs. \(2(217)=434\).
Time = 2.60 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.56
| method | result | size |
| default | \(\frac {-75+408 a b x +204 a^{2}+96 \arcsin \left (b x +a \right )^{2} b^{4} x^{4}-128 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}+48 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}-12 b^{4} x^{4}-480 \arcsin \left (b x +a \right )^{2} a^{2}+204 b^{2} x^{2}-12 a^{4}+48 \arcsin \left (b x +a \right )^{4}-384 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}-384 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +144 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}+144 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x -48 a \,b^{3} x^{3}-72 a^{2} b^{2} x^{2}-48 a^{3} b x +384 \arcsin \left (b x +a \right )^{2} a \,b^{3} x^{3}+576 \arcsin \left (b x +a \right )^{2} a^{2} b^{2} x^{2}+384 \arcsin \left (b x +a \right )^{2} a^{3} b x -128 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}+48 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}-480 \arcsin \left (b x +a \right )^{2} b^{2} x^{2}+320 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -408 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +320 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -960 \arcsin \left (b x +a \right )^{2} a b x -408 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x +204 \arcsin \left (b x +a \right )^{2}+96 \arcsin \left (b x +a \right )^{2} a^{4}}{512 b}\) | \(628\) |
1/512*(-75+408*a*b*x+204*a^2+96*arcsin(b*x+a)^2*b^4*x^4-128*arcsin(b*x+a)^ 3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^3+48*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^ 2+1)^(1/2)*a^3-12*b^4*x^4-480*arcsin(b*x+a)^2*a^2+204*b^2*x^2-12*a^4+48*ar csin(b*x+a)^4-384*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a*b^2*x^2 -384*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a^2*b*x+144*arcsin(b*x +a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a*b^2*x^2+144*arcsin(b*x+a)*(-b^2*x^2-2 *a*b*x-a^2+1)^(1/2)*a^2*b*x-48*a*b^3*x^3-72*a^2*b^2*x^2-48*a^3*b*x+384*arc sin(b*x+a)^2*a*b^3*x^3+576*arcsin(b*x+a)^2*a^2*b^2*x^2+384*arcsin(b*x+a)^2 *a^3*b*x-128*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b^3*x^3+48*arc sin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b^3*x^3-480*arcsin(b*x+a)^2*b^2* x^2+320*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-408*arcsin(b*x+a) *(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a+320*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^ 2+1)^(1/2)*b*x-960*arcsin(b*x+a)^2*a*b*x-408*arcsin(b*x+a)*(-b^2*x^2-2*a*b *x-a^2+1)^(1/2)*b*x+204*arcsin(b*x+a)^2+96*arcsin(b*x+a)^2*a^4)/b
Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.99 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=-\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} - 17\right )} b^{2} x^{2} - 12 \, \arcsin \left (b x + a\right )^{4} + 6 \, {\left (2 \, a^{3} - 17 \, a\right )} b x - 3 \, {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} - 5 \, a\right )} b x - 40 \, a^{2} + 17\right )} \arcsin \left (b x + a\right )^{2} + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (8 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 17\right )} b x - 17 \, a\right )} \arcsin \left (b x + a\right )\right )}}{128 \, b} \]
-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
Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (223) = 446\).
Time = 1.28 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.83 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\begin {cases} \frac {3 a^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} + \frac {3 a^{3} x \operatorname {asin}^{2}{\left (a + b x \right )}}{4} - \frac {3 a^{3} x}{32} - \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4 b} + \frac {3 a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32 b} + \frac {9 a^{2} b x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{8} - \frac {9 a^{2} b x^{2}}{64} - \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {9 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 a^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} + \frac {3 a b^{2} x^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} - \frac {3 a b^{2} x^{3}}{32} - \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {9 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 a x \operatorname {asin}^{2}{\left (a + b x \right )}}{8} + \frac {51 a x}{64} + \frac {5 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{8 b} - \frac {51 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{64 b} + \frac {3 b^{3} x^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{16} - \frac {3 b^{3} x^{4}}{128} - \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {3 b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 b x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{16} + \frac {51 b x^{2}}{128} + \frac {5 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{8} - \frac {51 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{64} + \frac {3 \operatorname {asin}^{4}{\left (a + b x \right )}}{32 b} + \frac {51 \operatorname {asin}^{2}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac {3}{2}} \operatorname {asin}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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))
\[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\int { {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{3} \,d x } \]
Time = 0.36 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.21 \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{4 \, b} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{8 \, b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )^{2}}{16 \, b} + \frac {3 \, \arcsin \left (b x + a\right )^{4}}{32 \, b} - \frac {3 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{32 \, b} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{16 \, b} - \frac {45 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{64 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{128 \, b} - \frac {45 \, \arcsin \left (b x + a\right )^{2}}{128 \, b} + \frac {45 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{128 \, b} + \frac {189}{1024 \, b} \]
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
Timed out. \[ \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \arcsin (a+b x)^3 \, dx=\int {\mathrm {asin}\left (a+b\,x\right )}^3\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2} \,d x \]