Integrand size = 27, antiderivative size = 172 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b x^3 (a+b \arcsin (c x))}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b x (a+b \arcsin (c x))}{2 c^3 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{4 c^4 d^3}+\frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2 \log \left (1-c^2 x^2\right )}{3 c^4 d^3} \]
1/12*b^2/c^4/d^3/(-c^2*x^2+1)-1/6*b*x^3*(a+b*arcsin(c*x))/c/d^3/(-c^2*x^2+ 1)^(3/2)-1/4*(a+b*arcsin(c*x))^2/c^4/d^3+1/4*x^4*(a+b*arcsin(c*x))^2/d^3/( -c^2*x^2+1)^2+1/3*b^2*ln(-c^2*x^2+1)/c^4/d^3+1/2*b*x*(a+b*arcsin(c*x))/c^3 /d^3/(-c^2*x^2+1)^(1/2)
Time = 0.49 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-3 a^2+b^2+6 a^2 c^2 x^2-b^2 c^2 x^2+6 a b c x \sqrt {1-c^2 x^2}-8 a b c^3 x^3 \sqrt {1-c^2 x^2}+2 b \left (b c x \left (3-4 c^2 x^2\right ) \sqrt {1-c^2 x^2}+a \left (-3+6 c^2 x^2\right )\right ) \arcsin (c x)+3 b^2 \left (-1+2 c^2 x^2\right ) \arcsin (c x)^2+4 b^2 \left (-1+c^2 x^2\right )^2 \log \left (1-c^2 x^2\right )}{12 c^4 d^3 \left (-1+c^2 x^2\right )^2} \]
(-3*a^2 + b^2 + 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 6*a*b*c*x*Sqrt[1 - c^2*x^2] - 8*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*(b*c*x*(3 - 4*c^2*x^2)*Sqrt[1 - c^ 2*x^2] + a*(-3 + 6*c^2*x^2))*ArcSin[c*x] + 3*b^2*(-1 + 2*c^2*x^2)*ArcSin[c *x]^2 + 4*b^2*(-1 + c^2*x^2)^2*Log[1 - c^2*x^2])/(12*c^4*d^3*(-1 + c^2*x^2 )^2)
Time = 0.76 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5186, 5206, 243, 49, 2009, 5206, 240, 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.
| \(\displaystyle \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5186 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \int \frac {x^4 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (-\frac {\int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c^2}-\frac {b \int \frac {x^3}{\left (1-c^2 x^2\right )^2}dx}{3 c}+\frac {x^3 (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (-\frac {\int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c^2}-\frac {b \int \frac {x^2}{\left (1-c^2 x^2\right )^2}dx^2}{6 c}+\frac {x^3 (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (-\frac {\int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c^2}-\frac {b \int \left (\frac {1}{c^2 \left (c^2 x^2-1\right )}+\frac {1}{c^2 \left (c^2 x^2-1\right )^2}\right )dx^2}{6 c}+\frac {x^3 (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (-\frac {\int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c^2}+\frac {x^3 (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (-\frac {-\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{c^2}-\frac {b \int \frac {x}{1-c^2 x^2}dx}{c}+\frac {x (a+b \arcsin (c x))}{c^2 \sqrt {1-c^2 x^2}}}{c^2}+\frac {x^3 (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (-\frac {-\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{c^2}+\frac {x (a+b \arcsin (c x))}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3}}{c^2}+\frac {x^3 (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {b \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {x^4 (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x^3 (a+b \arcsin (c x))}{3 c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {-\frac {(a+b \arcsin (c x))^2}{2 b c^3}+\frac {x (a+b \arcsin (c x))}{c^2 \sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3}}{c^2}-\frac {b \left (\frac {1}{c^4 \left (1-c^2 x^2\right )}+\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{6 c}\right )}{2 d^3}\) |
(x^4*(a + b*ArcSin[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) - (b*c*((x^3*(a + b*Ar cSin[c*x]))/(3*c^2*(1 - c^2*x^2)^(3/2)) - (b*(1/(c^4*(1 - c^2*x^2)) + Log[ 1 - c^2*x^2]/c^4))/(6*c) - ((x*(a + b*ArcSin[c*x]))/(c^2*Sqrt[1 - c^2*x^2] ) - (a + b*ArcSin[c*x])^2/(2*b*c^3) + (b*Log[1 - c^2*x^2])/(2*c^3))/c^2))/ (2*d^3)
3.3.2.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp [b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.24
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {3}{16 \left (c x -1\right )}-\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}\right )}{d^{3}}-\frac {b^{2} \left (\frac {4 i \arcsin \left (c x \right )}{3}-\frac {8 i \arcsin \left (c x \right ) x^{4} c^{4}-8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-16 i \arcsin \left (c x \right ) x^{2} c^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -3 \arcsin \left (c x \right )^{2}+8 i \arcsin \left (c x \right )-c^{2} x^{2}+1}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{16 \left (c x -1\right )^{2}}-\frac {3 \arcsin \left (c x \right )}{16 \left (c x -1\right )}-\frac {\arcsin \left (c x \right )}{16 \left (c x +1\right )^{2}}+\frac {3 \arcsin \left (c x \right )}{16 \left (c x +1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{6 c x +6}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{6 c x -6}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}\right )}{d^{3}}}{c^{4}}\) | \(386\) |
| default | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {3}{16 \left (c x -1\right )}-\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}\right )}{d^{3}}-\frac {b^{2} \left (\frac {4 i \arcsin \left (c x \right )}{3}-\frac {8 i \arcsin \left (c x \right ) x^{4} c^{4}-8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-16 i \arcsin \left (c x \right ) x^{2} c^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -3 \arcsin \left (c x \right )^{2}+8 i \arcsin \left (c x \right )-c^{2} x^{2}+1}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{16 \left (c x -1\right )^{2}}-\frac {3 \arcsin \left (c x \right )}{16 \left (c x -1\right )}-\frac {\arcsin \left (c x \right )}{16 \left (c x +1\right )^{2}}+\frac {3 \arcsin \left (c x \right )}{16 \left (c x +1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{6 c x +6}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{6 c x -6}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}\right )}{d^{3}}}{c^{4}}\) | \(386\) |
| parts | \(-\frac {a^{2} \left (-\frac {1}{16 c^{4} \left (c x -1\right )^{2}}-\frac {3}{16 c^{4} \left (c x -1\right )}-\frac {1}{16 c^{4} \left (c x +1\right )^{2}}+\frac {3}{16 c^{4} \left (c x +1\right )}\right )}{d^{3}}-\frac {b^{2} \left (\frac {4 i \arcsin \left (c x \right )}{3}-\frac {8 i \arcsin \left (c x \right ) x^{4} c^{4}-8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-16 i \arcsin \left (c x \right ) x^{2} c^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -3 \arcsin \left (c x \right )^{2}+8 i \arcsin \left (c x \right )-c^{2} x^{2}+1}{12 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3} c^{4}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{16 \left (c x -1\right )^{2}}-\frac {3 \arcsin \left (c x \right )}{16 \left (c x -1\right )}-\frac {\arcsin \left (c x \right )}{16 \left (c x +1\right )^{2}}+\frac {3 \arcsin \left (c x \right )}{16 \left (c x +1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{6 c x +6}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{6 c x -6}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}\right )}{d^{3} c^{4}}\) | \(400\) |
1/c^4*(-a^2/d^3*(-1/16/(c*x-1)^2-3/16/(c*x-1)-1/16/(c*x+1)^2+3/16/(c*x+1)) -b^2/d^3*(4/3*I*arcsin(c*x)-1/12*(8*I*arcsin(c*x)*x^4*c^4-8*(-c^2*x^2+1)^( 1/2)*arcsin(c*x)*c^3*x^3+6*arcsin(c*x)^2*x^2*c^2-16*I*arcsin(c*x)*x^2*c^2+ 6*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-3*arcsin(c*x)^2+8*I*arcsin(c*x)-c^2*x ^2+1)/(c^4*x^4-2*c^2*x^2+1)-2/3*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2))-2*a*b/ d^3*(-1/16*arcsin(c*x)/(c*x-1)^2-3/16*arcsin(c*x)/(c*x-1)-1/16*arcsin(c*x) /(c*x+1)^2+3/16*arcsin(c*x)/(c*x+1)-1/48/(c*x+1)^2*(-(c*x+1)^2+2*c*x+2)^(1 /2)+1/6/(c*x+1)*(-(c*x+1)^2+2*c*x+2)^(1/2)+1/6/(c*x-1)*(-(c*x-1)^2-2*c*x+2 )^(1/2)+1/48/(c*x-1)^2*(-(c*x-1)^2-2*c*x+2)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {{\left (6 \, a^{2} - b^{2}\right )} c^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - 3 \, a^{2} + b^{2} + 6 \, {\left (2 \, a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right ) + 4 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) - 2 \, {\left (4 \, a b c^{3} x^{3} - 3 \, a b c x + {\left (4 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
1/12*((6*a^2 - b^2)*c^2*x^2 + 3*(2*b^2*c^2*x^2 - b^2)*arcsin(c*x)^2 - 3*a^ 2 + b^2 + 6*(2*a*b*c^2*x^2 - a*b)*arcsin(c*x) + 4*(b^2*c^4*x^4 - 2*b^2*c^2 *x^2 + b^2)*log(c^2*x^2 - 1) - 2*(4*a*b*c^3*x^3 - 3*a*b*c*x + (4*b^2*c^3*x ^3 - 3*b^2*c*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/(c^8*d^3*x^4 - 2*c^6*d^3* x^2 + c^4*d^3)
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2} x^{3}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{3} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
-(Integral(a**2*x**3/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Int egral(b**2*x**3*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**3*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
1/4*(2*c^2*x^2 - 1)*a^2/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) + 1/4*((2* b^2*c^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 4*(c^8*d ^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3)*integrate(-1/2*(4*a*b*c^3*x^3*arctan2(c* x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (2*b^2*c^2*x^2 - b^2)*sqrt(c*x + 1)*sqr t(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^9*d^3*x^6 - 3*c ^7*d^3*x^4 + 3*c^5*d^3*x^2 - c^3*d^3), x))/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (154) = 308\).
Time = 0.41 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.85 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^{2} x^{4} \arcsin \left (c x\right )^{2}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a b x^{4} \arcsin \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a^{2} x^{4}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {b^{2} x^{3} \arcsin \left (c x\right )}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {a b x^{3}}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} c d^{3}} - \frac {b^{2} x^{2}}{12 \, {\left (c^{2} x^{2} - 1\right )} c^{2} d^{3}} + \frac {b^{2} x \arcsin \left (c x\right )}{2 \, \sqrt {-c^{2} x^{2} + 1} c^{3} d^{3}} - \frac {b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{4} d^{3}} + \frac {a b x}{2 \, \sqrt {-c^{2} x^{2} + 1} c^{3} d^{3}} - \frac {a b \arcsin \left (c x\right )}{2 \, c^{4} d^{3}} + \frac {2 \, b^{2} \log \left (2\right )}{3 \, c^{4} d^{3}} + \frac {b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{3 \, c^{4} d^{3}} - \frac {a^{2}}{4 \, c^{4} d^{3}} + \frac {b^{2}}{12 \, c^{4} d^{3}} \]
1/4*b^2*x^4*arcsin(c*x)^2/((c^2*x^2 - 1)^2*d^3) + 1/2*a*b*x^4*arcsin(c*x)/ ((c^2*x^2 - 1)^2*d^3) + 1/4*a^2*x^4/((c^2*x^2 - 1)^2*d^3) + 1/6*b^2*x^3*ar csin(c*x)/((c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*c*d^3) + 1/6*a*b*x^3/((c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*c*d^3) - 1/12*b^2*x^2/((c^2*x^2 - 1)*c^2*d^3) + 1 /2*b^2*x*arcsin(c*x)/(sqrt(-c^2*x^2 + 1)*c^3*d^3) - 1/4*b^2*arcsin(c*x)^2/ (c^4*d^3) + 1/2*a*b*x/(sqrt(-c^2*x^2 + 1)*c^3*d^3) - 1/2*a*b*arcsin(c*x)/( c^4*d^3) + 2/3*b^2*log(2)/(c^4*d^3) + 1/3*b^2*log(abs(-c^2*x^2 + 1))/(c^4* d^3) - 1/4*a^2/(c^4*d^3) + 1/12*b^2/(c^4*d^3)
Timed out. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]