Integrand size = 24, antiderivative size = 219 \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}} \] Output:
3/16*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3/8*a*x^2 *(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)/(-a^2*x^2+1)^(1/2)+1/2*x*(-a^2*c*x ^2+c)^(1/2)*arcsin(a*x)^(3/2)+1/5*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(5/2)/a /(-a^2*x^2+1)^(1/2)-3/32*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*FresnelC(2*arcsin(a *x)^(1/2)/Pi^(1/2))/a/(-a^2*x^2+1)^(1/2)
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.66 \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (32 \arcsin (a x)^2 \left (5 a x \sqrt {1-a^2 x^2}+2 \arcsin (a x)\right )+15 i \sqrt {2} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-2 i \arcsin (a x)\right )-15 i \sqrt {2} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},2 i \arcsin (a x)\right )\right )}{320 a \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \] Input:
Integrate[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2),x]
Output:
(Sqrt[c - a^2*c*x^2]*(32*ArcSin[a*x]^2*(5*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin [a*x]) + (15*I)*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a* x]] - (15*I)*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (2*I)*ArcSin[a*x]]))/( 320*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])
Time = 0.90 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.75, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5156, 5140, 5152, 5224, 3042, 3793, 2009}
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.
\(\displaystyle \int \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 5156 |
\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \int x \sqrt {\arcsin (a x)}dx}{4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 5140 |
\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin (a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin (a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin (a x)}-\frac {\int \frac {a^2 x^2}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin (a x)}-\frac {\int \frac {\sin (\arcsin (a x))^2}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin (a x)}-\frac {\int \left (\frac {1}{2 \sqrt {\arcsin (a x)}}-\frac {\cos (2 \arcsin (a x))}{2 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin (a x)}-\frac {\sqrt {\arcsin (a x)}-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
Input:
Int[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2),x]
Output:
(x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[ a*x]^(5/2))/(5*a*Sqrt[1 - a^2*x^2]) - (3*a*Sqrt[c - a^2*c*x^2]*((x^2*Sqrt[ ArcSin[a*x]])/2 - (Sqrt[ArcSin[a*x]] - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a *x]])/Sqrt[Pi]])/2)/(4*a^2)))/(4*Sqrt[1 - a^2*x^2])
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcSin[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
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_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \sqrt {-a^{2} c \,x^{2}+c}\, \arcsin \left (a x \right )^{\frac {3}{2}}d x\]
Input:
int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x)
Output:
int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x)
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \] Input:
integrate((-a**2*c*x**2+c)**(1/2)*asin(a*x)**(3/2),x)
Output:
Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*asin(a*x)**(3/2), x)
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x, algorithm="giac")
Output:
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. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\int {\mathrm {asin}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \] Input:
int(asin(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2),x)
Output:
int(asin(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2), x)
\[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2} \, dx=\sqrt {c}\, \left (\int \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, \mathit {asin} \left (a x \right )d x \right ) \] Input:
int((-a^2*c*x^2+c)^(1/2)*asin(a*x)^(3/2),x)
Output:
sqrt(c)*int(sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*asin(a*x),x)