Integrand size = 26, antiderivative size = 305 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {17}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 c^2 d x^3 \sqrt {d-c^2 d x^2}+\frac {17 b^2 d \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {5 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{8 b c \sqrt {1-c^2 x^2}} \]
1/4*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-17/64*b^2*d*x*(-c^2*d*x^2+d )^(1/2)+1/32*b^2*c^2*d*x^3*(-c^2*d*x^2+d)^(1/2)+3/8*d*x*(a+b*arcsin(c*x))^ 2*(-c^2*d*x^2+d)^(1/2)+17/64*b^2*d*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^ 2*x^2+1)^(1/2)-5/8*b*c*d*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2* x^2+1)^(1/2)+1/8*b*c^3*d*x^4*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2* x^2+1)^(1/2)+1/8*d*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+ 1)^(1/2)
Time = 0.67 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.81 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (8 a^3+8 a b^2 c^2 x^2 \left (-5+c^2 x^2\right )+b^3 c x \sqrt {1-c^2 x^2} \left (-17+2 c^2 x^2\right )-8 a^2 b c x \sqrt {1-c^2 x^2} \left (-5+2 c^2 x^2\right )+b \left (24 a^2+16 a b c x \left (5-2 c^2 x^2\right ) \sqrt {1-c^2 x^2}+b^2 \left (17-40 c^2 x^2+8 c^4 x^4\right )\right ) \arcsin (c x)+8 b^2 \left (3 a+b c x \left (5-2 c^2 x^2\right ) \sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+8 b^3 \arcsin (c x)^3\right )}{64 b c \sqrt {1-c^2 x^2}} \]
(d*Sqrt[d - c^2*d*x^2]*(8*a^3 + 8*a*b^2*c^2*x^2*(-5 + c^2*x^2) + b^3*c*x*S qrt[1 - c^2*x^2]*(-17 + 2*c^2*x^2) - 8*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(-5 + 2 *c^2*x^2) + b*(24*a^2 + 16*a*b*c*x*(5 - 2*c^2*x^2)*Sqrt[1 - c^2*x^2] + b^2 *(17 - 40*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x] + 8*b^2*(3*a + b*c*x*(5 - 2*c^ 2*x^2)*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 8*b^3*ArcSin[c*x]^3))/(64*b*c*Sq rt[1 - c^2*x^2])
Time = 0.94 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5158, 5156, 5138, 262, 223, 5152, 5182, 211, 211, 223}
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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \int x (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \int \left (1-c^2 x^2\right )^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\) |
(x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*d*((x*Sqrt[d - c^2* d*x^2]*(a + b*ArcSin[c*x])^2)/2 + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) ^3)/(6*b*c*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*((x^2*(a + b*ArcS in[c*x]))/2 - (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3))) /2))/Sqrt[1 - c^2*x^2]))/4 - (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/4*((1 - c^2*x^ 2)^2*(a + b*ArcSin[c*x]))/c^2 + (b*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x*Sqr t[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[1 - c^2*x^2])
3.3.21.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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]
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 929, normalized size of antiderivative = 3.05
| method | result | size |
| default | \(\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2}}{4}+\frac {3 a^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} d}{8 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right ) d}{512 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (68 i \arcsin \left (c x \right )+56 \arcsin \left (c x \right )^{2}-31\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d}{512 c \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (20 i \arcsin \left (c x \right )+24 \arcsin \left (c x \right )^{2}-11\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d}{512 c \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 \arcsin \left (c x \right )+i\right ) d}{256 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (17 i+28 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (5 i+12 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\) | \(929\) |
| parts | \(\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2}}{4}+\frac {3 a^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} d}{8 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 i \arcsin \left (c x \right )+8 \arcsin \left (c x \right )^{2}-1\right ) d}{512 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (68 i \arcsin \left (c x \right )+56 \arcsin \left (c x \right )^{2}-31\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d}{512 c \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (20 i \arcsin \left (c x \right )+24 \arcsin \left (c x \right )^{2}-11\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d}{512 c \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (4 \arcsin \left (c x \right )+i\right ) d}{256 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (17 i+28 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (5 i+12 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\) | \(929\) |
1/4*x*(-c^2*d*x^2+d)^(3/2)*a^2+3/8*a^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a^2*d^ 2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/8*(-d *(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^3*d-1/512 *(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c ^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(4*I*arcsin (c*x)+8*arcsin(c*x)^2-1)*d/c/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*I* (-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(-2*I*arc sin(c*x)+2*arcsin(c*x)^2-1)*d/c/(c^2*x^2-1)-1/512*(-d*(c^2*x^2-1))^(1/2)*( I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(68*I*arcsin(c*x)+56*arcsin(c*x)^2-31) *cos(3*arcsin(c*x))*d/c/(c^2*x^2-1)+3/512*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2* x^2+1)^(1/2)*x*c+c^2*x^2-1)*(20*I*arcsin(c*x)+24*arcsin(c*x)^2-11)*sin(3*a rcsin(c*x))*d/c/(c^2*x^2-1))+2*a*b*(-3/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2 +1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*d-1/256*(-d*(c^2*x^2-1))^(1/2)*(-8*I *(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^ 3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(4*arcsin(c*x)+I)*d/c/(c^2*x^2-1)+1/16*( -d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^ 2+1)^(1/2)-2*c*x)*(-I+2*arcsin(c*x))*d/c/(c^2*x^2-1)-1/256*(-d*(c^2*x^2-1) )^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(17*I+28*arcsin(c*x))*cos(3*a rcsin(c*x))*d/c/(c^2*x^2-1)+3/256*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^( 1/2)*x*c+c^2*x^2-1)*(5*I+12*arcsin(c*x))*sin(3*arcsin(c*x))*d/c/(c^2*x^...
\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a rcsin(c*x)/c)*a^2 + sqrt(d)*integrate(-((b^2*c^2*d*x^2 - b^2*d)*arctan2(c* x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arctan2(c*x , sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
Exception generated. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]