Integrand size = 21, antiderivative size = 162 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\frac {144 e^{\arcsin (a x)}}{629 a}+\frac {144}{629} e^{\arcsin (a x)} x \sqrt {1-a^2 x^2}+\frac {72 e^{\arcsin (a x)} \left (1-a^2 x^2\right )}{629 a}+\frac {120}{629} e^{\arcsin (a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac {30 e^{\arcsin (a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac {6}{37} e^{\arcsin (a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac {e^{\arcsin (a x)} \left (1-a^2 x^2\right )^3}{37 a} \]
144/629*exp(arcsin(a*x))/a+72/629*exp(arcsin(a*x))*(-a^2*x^2+1)/a+120/629* exp(arcsin(a*x))*x*(-a^2*x^2+1)^(3/2)+30/629*exp(arcsin(a*x))*(-a^2*x^2+1) ^2/a+6/37*exp(arcsin(a*x))*x*(-a^2*x^2+1)^(5/2)+1/37*exp(arcsin(a*x))*(-a^ 2*x^2+1)^3/a+144/629*exp(arcsin(a*x))*x*(-a^2*x^2+1)^(1/2)
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.43 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\frac {e^{\arcsin (a x)} (6290+1887 \cos (2 \arcsin (a x))+222 \cos (4 \arcsin (a x))+17 \cos (6 \arcsin (a x))+3774 \sin (2 \arcsin (a x))+888 \sin (4 \arcsin (a x))+102 \sin (6 \arcsin (a x)))}{20128 a} \]
(E^ArcSin[a*x]*(6290 + 1887*Cos[2*ArcSin[a*x]] + 222*Cos[4*ArcSin[a*x]] + 17*Cos[6*ArcSin[a*x]] + 3774*Sin[2*ArcSin[a*x]] + 888*Sin[4*ArcSin[a*x]] + 102*Sin[6*ArcSin[a*x]]))/(20128*a)
Time = 0.84 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5335, 7292, 7271, 4935, 4935, 4935, 2624}
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.
\(\displaystyle \int \left (1-a^2 x^2\right )^{5/2} e^{\arcsin (a x)} \, dx\) |
\(\Big \downarrow \) 5335 |
\(\displaystyle \frac {\int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^3d\arcsin (a x)}{a}\) |
\(\Big \downarrow \) 4935 |
\(\displaystyle \frac {\frac {30}{37} \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^2d\arcsin (a x)+\frac {1}{37} \left (1-a^2 x^2\right )^3 e^{\arcsin (a x)}+\frac {6}{37} a x \left (1-a^2 x^2\right )^{5/2} e^{\arcsin (a x)}}{a}\) |
\(\Big \downarrow \) 4935 |
\(\displaystyle \frac {\frac {30}{37} \left (\frac {12}{17} \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )d\arcsin (a x)+\frac {1}{17} \left (1-a^2 x^2\right )^2 e^{\arcsin (a x)}+\frac {4}{17} a x \left (1-a^2 x^2\right )^{3/2} e^{\arcsin (a x)}\right )+\frac {1}{37} \left (1-a^2 x^2\right )^3 e^{\arcsin (a x)}+\frac {6}{37} a x \left (1-a^2 x^2\right )^{5/2} e^{\arcsin (a x)}}{a}\) |
\(\Big \downarrow \) 4935 |
\(\displaystyle \frac {\frac {30}{37} \left (\frac {12}{17} \left (\frac {2}{5} \int e^{\arcsin (a x)}d\arcsin (a x)+\frac {2}{5} a x \sqrt {1-a^2 x^2} e^{\arcsin (a x)}+\frac {1}{5} \left (1-a^2 x^2\right ) e^{\arcsin (a x)}\right )+\frac {1}{17} \left (1-a^2 x^2\right )^2 e^{\arcsin (a x)}+\frac {4}{17} a x \left (1-a^2 x^2\right )^{3/2} e^{\arcsin (a x)}\right )+\frac {1}{37} \left (1-a^2 x^2\right )^3 e^{\arcsin (a x)}+\frac {6}{37} a x \left (1-a^2 x^2\right )^{5/2} e^{\arcsin (a x)}}{a}\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle \frac {\frac {1}{37} \left (1-a^2 x^2\right )^3 e^{\arcsin (a x)}+\frac {6}{37} a x \left (1-a^2 x^2\right )^{5/2} e^{\arcsin (a x)}+\frac {30}{37} \left (\frac {1}{17} \left (1-a^2 x^2\right )^2 e^{\arcsin (a x)}+\frac {4}{17} a x \left (1-a^2 x^2\right )^{3/2} e^{\arcsin (a x)}+\frac {12}{17} \left (\frac {2}{5} a x \sqrt {1-a^2 x^2} e^{\arcsin (a x)}+\frac {1}{5} \left (1-a^2 x^2\right ) e^{\arcsin (a x)}+\frac {2}{5} e^{\arcsin (a x)}\right )\right )}{a}\) |
((6*a*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(5/2))/37 + (E^ArcSin[a*x]*(1 - a^2*x^ 2)^3)/37 + (30*((4*a*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2))/17 + (E^ArcSin[a *x]*(1 - a^2*x^2)^2)/17 + (12*((2*E^ArcSin[a*x])/5 + (2*a*E^ArcSin[a*x]*x* Sqrt[1 - a^2*x^2])/5 + (E^ArcSin[a*x]*(1 - a^2*x^2))/5))/17))/37)/a
3.5.63.3.1 Defintions of rubi rules used
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbo l] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Lo g[F]^2)), x] + (Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1) /(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + Simp[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^ 2*Log[F]^2) Int[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x]) /; FreeQ[{F , a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[m, 1]
Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Simp[ 1/b Subst[Int[(u /. x -> -a/b + Sin[x]/b)*f^(c*x^n)*Cos[x], x], x, ArcSin [a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
\[\int {\mathrm e}^{\arcsin \left (a x \right )} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}d x\]
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.44 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=-\frac {{\left (17 \, a^{6} x^{6} - 81 \, a^{4} x^{4} + 183 \, a^{2} x^{2} - 6 \, {\left (17 \, a^{5} x^{5} - 54 \, a^{3} x^{3} + 61 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} - 263\right )} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} \]
-1/629*(17*a^6*x^6 - 81*a^4*x^4 + 183*a^2*x^2 - 6*(17*a^5*x^5 - 54*a^3*x^3 + 61*a*x)*sqrt(-a^2*x^2 + 1) - 263)*e^(arcsin(a*x))/a
Time = 13.73 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\begin {cases} - \frac {a^{5} x^{6} e^{\operatorname {asin}{\left (a x \right )}}}{37} + \frac {6 a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{37} + \frac {81 a^{3} x^{4} e^{\operatorname {asin}{\left (a x \right )}}}{629} - \frac {324 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{629} - \frac {183 a x^{2} e^{\operatorname {asin}{\left (a x \right )}}}{629} + \frac {366 x \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{629} + \frac {263 e^{\operatorname {asin}{\left (a x \right )}}}{629 a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]
Piecewise((-a**5*x**6*exp(asin(a*x))/37 + 6*a**4*x**5*sqrt(-a**2*x**2 + 1) *exp(asin(a*x))/37 + 81*a**3*x**4*exp(asin(a*x))/629 - 324*a**2*x**3*sqrt( -a**2*x**2 + 1)*exp(asin(a*x))/629 - 183*a*x**2*exp(asin(a*x))/629 + 366*x *sqrt(-a**2*x**2 + 1)*exp(asin(a*x))/629 + 263*exp(asin(a*x))/(629*a), Ne( a, 0)), (x, True))
\[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} e^{\left (\arcsin \left (a x\right )\right )} \,d x } \]
Exception generated. \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{5/2} \, dx=\int {\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}\,{\left (1-a^2\,x^2\right )}^{5/2} \,d x \]