3.103 \(\int e^{-\frac{3}{2} \tanh ^{-1}(a x)} x \, dx\)

Optimal. Leaf size=255 \[ -\frac{\sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a^2}-\frac{3 \sqrt [4]{a x+1} (1-a x)^{3/4}}{4 a^2}+\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}-\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a^2}+\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a^2} \]

[Out]

(-3*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*a^2) - ((1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(2*a^2) - (9*ArcTan[1 - (Sqrt
[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (9*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/
4)])/(4*Sqrt[2]*a^2) + (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8
*Sqrt[2]*a^2) - (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2
]*a^2)

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Rubi [A]  time = 0.169892, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6126, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{a x+1} (1-a x)^{7/4}}{2 a^2}-\frac{3 \sqrt [4]{a x+1} (1-a x)^{3/4}}{4 a^2}+\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt{2} a^2}-\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{4 \sqrt{2} a^2}+\frac{9 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{4 \sqrt{2} a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/E^((3*ArcTanh[a*x])/2),x]

[Out]

(-3*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*a^2) - ((1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(2*a^2) - (9*ArcTan[1 - (Sqrt
[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (9*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/
4)])/(4*Sqrt[2]*a^2) + (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8
*Sqrt[2]*a^2) - (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2
]*a^2)

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int e^{-\frac{3}{2} \tanh ^{-1}(a x)} x \, dx &=\int \frac{x (1-a x)^{3/4}}{(1+a x)^{3/4}} \, dx\\ &=-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}-\frac{3 \int \frac{(1-a x)^{3/4}}{(1+a x)^{3/4}} \, dx}{4 a}\\ &=-\frac{3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}-\frac{9 \int \frac{1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx}{8 a}\\ &=-\frac{3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{2 a^2}\\ &=-\frac{3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a^2}\\ &=-\frac{3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a^2}\\ &=-\frac{3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}\\ &=-\frac{3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}+\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}\\ &=-\frac{3 (1-a x)^{3/4} \sqrt [4]{1+a x}}{4 a^2}-\frac{(1-a x)^{7/4} \sqrt [4]{1+a x}}{2 a^2}-\frac{9 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}+\frac{9 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 \sqrt{2} a^2}+\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}-\frac{9 \log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt{2} a^2}\\ \end{align*}

Mathematica [C]  time = 0.0202954, size = 56, normalized size = 0.22 \[ \frac{(1-a x)^{7/4} \left (3 \sqrt [4]{2} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-a x)\right )-7 \sqrt [4]{a x+1}\right )}{14 a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/E^((3*ArcTanh[a*x])/2),x]

[Out]

((1 - a*x)^(7/4)*(-7*(1 + a*x)^(1/4) + 3*2^(1/4)*Hypergeometric2F1[3/4, 7/4, 11/4, (1 - a*x)/2]))/(14*a^2)

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{x \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x)

[Out]

int(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm="maxima")

[Out]

integrate(x/((a*x + 1)/sqrt(-a^2*x^2 + 1))^(3/2), x)

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Fricas [B]  time = 1.84449, size = 1312, normalized size = 5.15 \begin{align*} \frac{36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{6} \sqrt{\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} +{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - 1\right ) + 36 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} a^{6} \sqrt{-\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} -{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} - \sqrt{2} a^{6} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{3}{4}} + 1\right ) - 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} +{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} - \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) + 9 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a^{3} x - a^{2}\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} \frac{1}{a^{8}}^{\frac{1}{4}} -{\left (a^{5} x - a^{4}\right )} \sqrt{\frac{1}{a^{8}}} + \sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \,{\left (2 \, a^{2} x^{2} - 7 \, a x + 5\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{16 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm="fricas")

[Out]

1/16*(36*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^6*sqrt((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a
*x - 1))*(a^(-8))^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)
*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3/4) - 1) + 36*sqrt(2)*a^2*(a^(-8))^(1/4)*arctan(sqrt(2)*a^
6*sqrt(-(sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8))
+ sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-8))^(3/4) - sqrt(2)*a^6*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(3
/4) + 1) - 9*sqrt(2)*a^2*(a^(-8))^(1/4)*log((sqrt(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8)
)^(1/4) + (a^5*x - a^4)*sqrt(a^(-8)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 9*sqrt(2)*a^2*(a^(-8))^(1/4)*log(-(sqr
t(2)*(a^3*x - a^2)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-8))^(1/4) - (a^5*x - a^4)*sqrt(a^(-8)) + sqrt(-a^2
*x^2 + 1))/(a*x - 1)) - 4*(2*a^2*x^2 - 7*a*x + 5)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2),x)

[Out]

Integral(x/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((a*x+1)/(-a^2*x^2+1)^(1/2))^(3/2),x, algorithm="giac")

[Out]

integrate(x/((a*x + 1)/sqrt(-a^2*x^2 + 1))^(3/2), x)