Integrand size = 25, antiderivative size = 202 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {b^2 d x^2}{24 c^2}-\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{12 c^3}+\frac {b d x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {d (a+b \arcsin (c x))^2}{24 c^4}+\frac {1}{12} d x^4 (a+b \arcsin (c x))^2+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 \]
-1/24*b^2*d*x^2/c^2-1/72*b^2*d*x^4+1/108*b^2*c^2*d*x^6-1/24*d*(a+b*arcsin( c*x))^2/c^4+1/12*d*x^4*(a+b*arcsin(c*x))^2+1/6*d*x^4*(-c^2*x^2+1)*(a+b*arc sin(c*x))^2+1/12*b*d*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+1/18*b*d*x ^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c-1/18*b*c*d*x^5*(a+b*arcsin(c*x)) *(-c^2*x^2+1)^(1/2)
Time = 0.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {d \left (b^2 c^2 x^2 \left (9+3 c^2 x^2-2 c^4 x^4\right )+6 a b c x \sqrt {1-c^2 x^2} \left (-3-2 c^2 x^2+2 c^4 x^4\right )+9 a^2 \left (1-6 c^4 x^4+4 c^6 x^6\right )+6 b \left (b c x \sqrt {1-c^2 x^2} \left (-3-2 c^2 x^2+2 c^4 x^4\right )+3 a \left (1-6 c^4 x^4+4 c^6 x^6\right )\right ) \arcsin (c x)+9 b^2 \left (1-6 c^4 x^4+4 c^6 x^6\right ) \arcsin (c x)^2\right )}{216 c^4} \]
-1/216*(d*(b^2*c^2*x^2*(9 + 3*c^2*x^2 - 2*c^4*x^4) + 6*a*b*c*x*Sqrt[1 - c^ 2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4) + 9*a^2*(1 - 6*c^4*x^4 + 4*c^6*x^6) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4) + 3*a*(1 - 6*c^4 *x^4 + 4*c^6*x^6))*ArcSin[c*x] + 9*b^2*(1 - 6*c^4*x^4 + 4*c^6*x^6)*ArcSin[ c*x]^2))/c^4
Time = 1.23 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5202, 5138, 5198, 15, 5210, 15, 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.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {1}{3} b c d \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} d \int x^3 (a+b \arcsin (c x))^2dx+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c d \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c d \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c d \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}+\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}+\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {1}{3} d \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c d \left (\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {b x^4}{16 c}\right )-\frac {1}{36} b c x^6\right )\) |
(d*x^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/6 - (b*c*d*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/6 + ((b*x^4)/(16*c) - (x^3*Sqr t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(4*c^2) + (3*((b*x^2)/(4*c) - (x*Sqrt[ 1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3 )))/(4*c^2))/6))/3 + (d*((x^4*(a + b*ArcSin[c*x])^2)/4 - (b*c*((b*x^4)/(16 *c) - (x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(4*c^2) + (3*((b*x^2)/(4 *c) - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c* x])^2/(4*b*c^3)))/(4*c^2)))/2))/3
3.2.57.3.1 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[((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_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[ (f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[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*((a + b*ArcS in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[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/(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]
Time = 0.11 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.63
| method | result | size |
| parts | \(-d \,a^{2} \left (\frac {1}{6} c^{2} x^{6}-\frac {1}{4} x^{4}\right )-\frac {d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} x^{4} c^{4}}{4}+\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}-\frac {\arcsin \left (c x \right )^{2}}{24}+\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}+\frac {\arcsin \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {\arcsin \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-15 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {\left (c^{2} x^{2}-1\right )^{3}}{108}-\frac {13 \left (c^{2} x^{2}-1\right )^{2}}{288}-\frac {11 c^{2} x^{2}}{96}+\frac {11}{96}\right )}{c^{4}}-\frac {2 d a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}+\frac {\arcsin \left (c x \right )}{24}\right )}{c^{4}}\) | \(330\) |
| derivativedivides | \(\frac {-d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} x^{4} c^{4}}{4}+\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}-\frac {\arcsin \left (c x \right )^{2}}{24}+\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}+\frac {\arcsin \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {\arcsin \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-15 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {\left (c^{2} x^{2}-1\right )^{3}}{108}-\frac {13 \left (c^{2} x^{2}-1\right )^{2}}{288}-\frac {11 c^{2} x^{2}}{96}+\frac {11}{96}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}+\frac {\arcsin \left (c x \right )}{24}\right )}{c^{4}}\) | \(331\) |
| default | \(\frac {-d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} x^{4} c^{4}}{4}+\frac {\arcsin \left (c x \right ) \left (-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-3 c x \sqrt {-c^{2} x^{2}+1}+3 \arcsin \left (c x \right )\right )}{16}-\frac {\arcsin \left (c x \right )^{2}}{24}+\frac {\left (2 c^{2} x^{2}+3\right )^{2}}{128}+\frac {\arcsin \left (c x \right )^{2} c^{6} x^{6}}{6}-\frac {\arcsin \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-15 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {\left (c^{2} x^{2}-1\right )^{3}}{108}-\frac {13 \left (c^{2} x^{2}-1\right )^{2}}{288}-\frac {11 c^{2} x^{2}}{96}+\frac {11}{96}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}+\frac {\arcsin \left (c x \right )}{24}\right )}{c^{4}}\) | \(331\) |
-d*a^2*(1/6*c^2*x^6-1/4*x^4)-d*b^2/c^4*(-1/4*arcsin(c*x)^2*x^4*c^4+1/16*ar csin(c*x)*(-2*c^3*x^3*(-c^2*x^2+1)^(1/2)-3*c*x*(-c^2*x^2+1)^(1/2)+3*arcsin (c*x))-1/24*arcsin(c*x)^2+1/128*(2*c^2*x^2+3)^2+1/6*arcsin(c*x)^2*c^6*x^6- 1/144*arcsin(c*x)*(-8*c^5*x^5*(-c^2*x^2+1)^(1/2)-10*c^3*x^3*(-c^2*x^2+1)^( 1/2)-15*c*x*(-c^2*x^2+1)^(1/2)+15*arcsin(c*x))-1/108*(c^2*x^2-1)^3-13/288* (c^2*x^2-1)^2-11/96*c^2*x^2+11/96)-2*d*a*b/c^4*(1/6*arcsin(c*x)*c^6*x^6-1/ 4*c^4*x^4*arcsin(c*x)+1/36*c^5*x^5*(-c^2*x^2+1)^(1/2)-1/36*c^3*x^3*(-c^2*x ^2+1)^(1/2)-1/24*c*x*(-c^2*x^2+1)^(1/2)+1/24*arcsin(c*x))
Time = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.04 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {2 \, {\left (18 \, a^{2} - b^{2}\right )} c^{6} d x^{6} - 3 \, {\left (18 \, a^{2} - b^{2}\right )} c^{4} d x^{4} + 9 \, b^{2} c^{2} d x^{2} + 9 \, {\left (4 \, b^{2} c^{6} d x^{6} - 6 \, b^{2} c^{4} d x^{4} + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (4 \, a b c^{6} d x^{6} - 6 \, a b c^{4} d x^{4} + a b d\right )} \arcsin \left (c x\right ) + 6 \, {\left (2 \, a b c^{5} d x^{5} - 2 \, a b c^{3} d x^{3} - 3 \, a b c d x + {\left (2 \, b^{2} c^{5} d x^{5} - 2 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{216 \, c^{4}} \]
-1/216*(2*(18*a^2 - b^2)*c^6*d*x^6 - 3*(18*a^2 - b^2)*c^4*d*x^4 + 9*b^2*c^ 2*d*x^2 + 9*(4*b^2*c^6*d*x^6 - 6*b^2*c^4*d*x^4 + b^2*d)*arcsin(c*x)^2 + 18 *(4*a*b*c^6*d*x^6 - 6*a*b*c^4*d*x^4 + a*b*d)*arcsin(c*x) + 6*(2*a*b*c^5*d* x^5 - 2*a*b*c^3*d*x^3 - 3*a*b*c*d*x + (2*b^2*c^5*d*x^5 - 2*b^2*c^3*d*x^3 - 3*b^2*c*d*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^4
Time = 0.66 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.64 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{2} d x^{6}}{6} + \frac {a^{2} d x^{4}}{4} - \frac {a b c^{2} d x^{6} \operatorname {asin}{\left (c x \right )}}{3} - \frac {a b c d x^{5} \sqrt {- c^{2} x^{2} + 1}}{18} + \frac {a b d x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {a b d x^{3} \sqrt {- c^{2} x^{2} + 1}}{18 c} + \frac {a b d x \sqrt {- c^{2} x^{2} + 1}}{12 c^{3}} - \frac {a b d \operatorname {asin}{\left (c x \right )}}{12 c^{4}} - \frac {b^{2} c^{2} d x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{6} + \frac {b^{2} c^{2} d x^{6}}{108} - \frac {b^{2} c d x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{18} + \frac {b^{2} d x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {b^{2} d x^{4}}{72} + \frac {b^{2} d x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{18 c} - \frac {b^{2} d x^{2}}{24 c^{2}} + \frac {b^{2} d x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{12 c^{3}} - \frac {b^{2} d \operatorname {asin}^{2}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{4}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**2*d*x**6/6 + a**2*d*x**4/4 - a*b*c**2*d*x**6*asin(c*x) /3 - a*b*c*d*x**5*sqrt(-c**2*x**2 + 1)/18 + a*b*d*x**4*asin(c*x)/2 + a*b*d *x**3*sqrt(-c**2*x**2 + 1)/(18*c) + a*b*d*x*sqrt(-c**2*x**2 + 1)/(12*c**3) - a*b*d*asin(c*x)/(12*c**4) - b**2*c**2*d*x**6*asin(c*x)**2/6 + b**2*c**2 *d*x**6/108 - b**2*c*d*x**5*sqrt(-c**2*x**2 + 1)*asin(c*x)/18 + b**2*d*x** 4*asin(c*x)**2/4 - b**2*d*x**4/72 + b**2*d*x**3*sqrt(-c**2*x**2 + 1)*asin( c*x)/(18*c) - b**2*d*x**2/(24*c**2) + b**2*d*x*sqrt(-c**2*x**2 + 1)*asin(c *x)/(12*c**3) - b**2*d*asin(c*x)**2/(24*c**4), Ne(c, 0)), (a**2*d*x**4/4, True))
\[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
-1/6*a^2*c^2*d*x^6 + 1/4*a^2*d*x^4 - 1/144*(48*x^6*arcsin(c*x) + (8*sqrt(- c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b*c^2*d + 1/16*(8*x^4*arcsin(c*x) + (2 *sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c ^5)*c)*a*b*d - 1/12*(2*b^2*c^2*d*x^6 - 3*b^2*d*x^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - integrate(1/6*(2*b^2*c^3*d*x^6 - 3*b^2*c*d*x^4)*s qrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^ 2*x^2 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (177) = 354\).
Time = 0.33 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.87 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {1}{6} \, a^{2} c^{2} d x^{6} + \frac {1}{4} \, a^{2} d x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right )}{18 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d \arcsin \left (c x\right )^{2}}{6 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d x}{18 \, c^{3}} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d x \arcsin \left (c x\right )}{18 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} a b d \arcsin \left (c x\right )}{3 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d \arcsin \left (c x\right )^{2}}{4 \, c^{4}} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d x}{18 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right )}{12 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d}{108 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a b d \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b d x}{12 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d}{72 \, c^{4}} + \frac {b^{2} d \arcsin \left (c x\right )^{2}}{24 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d}{24 \, c^{4}} + \frac {a b d \arcsin \left (c x\right )}{12 \, c^{4}} - \frac {5 \, b^{2} d}{216 \, c^{4}} \]
-1/6*a^2*c^2*d*x^6 + 1/4*a^2*d*x^4 - 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)/c^3 - 1/6*(c^2*x^2 - 1)^3*b^2*d*arcsin(c*x)^2/c^4 - 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d*x/c^3 + 1/18*(-c^2*x^2 + 1) ^(3/2)*b^2*d*x*arcsin(c*x)/c^3 - 1/3*(c^2*x^2 - 1)^3*a*b*d*arcsin(c*x)/c^4 - 1/4*(c^2*x^2 - 1)^2*b^2*d*arcsin(c*x)^2/c^4 + 1/18*(-c^2*x^2 + 1)^(3/2) *a*b*d*x/c^3 + 1/12*sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)/c^3 + 1/108*(c^ 2*x^2 - 1)^3*b^2*d/c^4 - 1/2*(c^2*x^2 - 1)^2*a*b*d*arcsin(c*x)/c^4 + 1/12* sqrt(-c^2*x^2 + 1)*a*b*d*x/c^3 + 1/72*(c^2*x^2 - 1)^2*b^2*d/c^4 + 1/24*b^2 *d*arcsin(c*x)^2/c^4 - 1/24*(c^2*x^2 - 1)*b^2*d/c^4 + 1/12*a*b*d*arcsin(c* x)/c^4 - 5/216*b^2*d/c^4
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right ) \,d x \]