3.463 \(\int e^{\sin ^{-1}(a x)} (1-a^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=162 \[ \frac{\left (1-a^2 x^2\right )^3 e^{\sin ^{-1}(a x)}}{37 a}+\frac{6}{37} x \left (1-a^2 x^2\right )^{5/2} e^{\sin ^{-1}(a x)}+\frac{30 \left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{629 a}+\frac{120}{629} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{72 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{629 a}+\frac{144}{629} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{144 e^{\sin ^{-1}(a x)}}{629 a} \]

[Out]

(144*E^ArcSin[a*x])/(629*a) + (144*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2])/629 + (72*E^ArcSin[a*x]*(1 - a^2*x^2))/(
629*a) + (120*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2))/629 + (30*E^ArcSin[a*x]*(1 - a^2*x^2)^2)/(629*a) + (6*E^Arc
Sin[a*x]*x*(1 - a^2*x^2)^(5/2))/37 + (E^ArcSin[a*x]*(1 - a^2*x^2)^3)/(37*a)

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Rubi [A]  time = 0.430422, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4836, 6688, 6720, 4435, 2194} \[ \frac{\left (1-a^2 x^2\right )^3 e^{\sin ^{-1}(a x)}}{37 a}+\frac{6}{37} x \left (1-a^2 x^2\right )^{5/2} e^{\sin ^{-1}(a x)}+\frac{30 \left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{629 a}+\frac{120}{629} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{72 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{629 a}+\frac{144}{629} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{144 e^{\sin ^{-1}(a x)}}{629 a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSin[a*x]*(1 - a^2*x^2)^(5/2),x]

[Out]

(144*E^ArcSin[a*x])/(629*a) + (144*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2])/629 + (72*E^ArcSin[a*x]*(1 - a^2*x^2))/(
629*a) + (120*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2))/629 + (30*E^ArcSin[a*x]*(1 - a^2*x^2)^2)/(629*a) + (6*E^Arc
Sin[a*x]*x*(1 - a^2*x^2)^(5/2))/37 + (E^ArcSin[a*x]*(1 - a^2*x^2)^3)/(37*a)

Rule 4836

Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Dist[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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(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]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 4435

Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*
x))*Cos[d + e*x]^m)/(e^2*m^2 + b^2*c^2*Log[F]^2), x] + (Dist[(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] + 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]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
m, 1]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \left (1-\sin ^2(x)\right )^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \cos ^2(x)^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos ^6(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac{30 \operatorname{Subst}\left (\int e^x \cos ^4(x) \, dx,x,\sin ^{-1}(a x)\right )}{37 a}\\ &=\frac{120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac{360 \operatorname{Subst}\left (\int e^x \cos ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{629 a}\\ &=\frac{144}{629} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{72 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{629 a}+\frac{120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac{144 \operatorname{Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )}{629 a}\\ &=\frac{144 e^{\sin ^{-1}(a x)}}{629 a}+\frac{144}{629} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{72 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{629 a}+\frac{120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}\\ \end{align*}

Mathematica [A]  time = 0.333929, size = 69, normalized size = 0.43 \[ \frac{e^{\sin ^{-1}(a x)} \left (3774 \sin \left (2 \sin ^{-1}(a x)\right )+888 \sin \left (4 \sin ^{-1}(a x)\right )+102 \sin \left (6 \sin ^{-1}(a x)\right )+1887 \cos \left (2 \sin ^{-1}(a x)\right )+222 \cos \left (4 \sin ^{-1}(a x)\right )+17 \cos \left (6 \sin ^{-1}(a x)\right )+6290\right )}{20128 a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcSin[a*x]*(1 - a^2*x^2)^(5/2),x]

[Out]

(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*A
rcSin[a*x]] + 888*Sin[4*ArcSin[a*x]] + 102*Sin[6*ArcSin[a*x]]))/(20128*a)

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arcsin \left ( ax \right ) }} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arcsin(a*x))*(-a^2*x^2+1)^(5/2),x)

[Out]

int(exp(arcsin(a*x))*(-a^2*x^2+1)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(5/2),x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(5/2)*e^(arcsin(a*x)), x)

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Fricas [A]  time = 2.0783, size = 176, normalized size = 1.09 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(5/2),x, algorithm="fricas")

[Out]

-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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(asin(a*x))*(-a**2*x**2+1)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.20448, size = 193, normalized size = 1.19 \begin{align*} \frac{6}{37} \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{120}{629} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x e^{\left (\arcsin \left (a x\right )\right )} - \frac{{\left (a^{2} x^{2} - 1\right )}^{3} e^{\left (\arcsin \left (a x\right )\right )}}{37 \, a} + \frac{144}{629} \, \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{30 \,{\left (a^{2} x^{2} - 1\right )}^{2} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} - \frac{72 \,{\left (a^{2} x^{2} - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} + \frac{144 \, e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(5/2),x, algorithm="giac")

[Out]

6/37*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*x*e^(arcsin(a*x)) + 120/629*(-a^2*x^2 + 1)^(3/2)*x*e^(arcsin(a*x)) - 1
/37*(a^2*x^2 - 1)^3*e^(arcsin(a*x))/a + 144/629*sqrt(-a^2*x^2 + 1)*x*e^(arcsin(a*x)) + 30/629*(a^2*x^2 - 1)^2*
e^(arcsin(a*x))/a - 72/629*(a^2*x^2 - 1)*e^(arcsin(a*x))/a + 144/629*e^(arcsin(a*x))/a