### 3.455 $$\int e^{3 \coth ^{-1}(a x)} (c-\frac{c}{a x})^{9/2} \, dx$$

Optimal. Leaf size=268 $\frac{9 \left (a-\frac{1}{x}\right )^2 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}}$

[Out]

(3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(a*(1 - 1/(a*x))^(9/2)) + (3*(28*a - 17/x)*(1 + 1/(a*x))^(3/2)*(c -
c/(a*x))^(9/2))/(35*a^2*(1 - 1/(a*x))^(9/2)) + (9*(a - x^(-1))^2*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2))/(7*a
^3*(1 - 1/(a*x))^(9/2)) + ((a - x^(-1))^3*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2)*x)/(a^3*(1 - 1/(a*x))^(9/2))
- (3*(c - c/(a*x))^(9/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(9/2))

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Rubi [A]  time = 0.163274, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {6182, 6179, 97, 153, 147, 50, 63, 208} $\frac{9 \left (a-\frac{1}{x}\right )^2 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{x \left (a-\frac{1}{x}\right )^3 \left (\frac{1}{a x}+1\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]

[Out]

(3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(a*(1 - 1/(a*x))^(9/2)) + (3*(28*a - 17/x)*(1 + 1/(a*x))^(3/2)*(c -
c/(a*x))^(9/2))/(35*a^2*(1 - 1/(a*x))^(9/2)) + (9*(a - x^(-1))^2*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2))/(7*a
^3*(1 - 1/(a*x))^(9/2)) + ((a - x^(-1))^3*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2)*x)/(a^3*(1 - 1/(a*x))^(9/2))
- (3*(c - c/(a*x))^(9/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(9/2))

Rule 6182

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Dist[(c + d/x)^p/(1 + d/(c*x))^
p, Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&
!IntegerQ[n/2] &&  !(IntegerQ[p] || GtQ[c, 0])

Rule 6179

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> -Dist[c^p, Subst[Int[((1 + (d*x)/c)^
p*(1 + x/a)^(n/2))/(x^2*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0
] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0])

Rule 97

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

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 208

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

Rubi steps

\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{9/2} \, dx &=\frac{\left (c-\frac{c}{a x}\right )^{9/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{9/2} \, dx}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{\left (c-\frac{c}{a x}\right )^{9/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3 \left (1+\frac{x}{a}\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (c-\frac{c}{a x}\right )^{9/2} \operatorname{Subst}\left (\int \frac{\left (-\frac{3}{2 a}-\frac{9 x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (2 a \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{21}{4 a^2}-\frac{51 x}{4 a^3}\right ) \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{7 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (3 \left (c-\frac{c}{a x}\right )^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{9/2}}{a \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{3 \left (28 a-\frac{17}{x}\right ) \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{9 \left (a-\frac{1}{x}\right )^2 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \left (1+\frac{1}{a x}\right )^{3/2} \left (c-\frac{c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{3 \left (c-\frac{c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (1-\frac{1}{a x}\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.127988, size = 109, normalized size = 0.41 $\frac{c^4 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (35 a^4 x^4+164 a^3 x^3-12 a^2 x^2-26 a x+10\right )-105 a^3 x^3 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{35 a^4 x^3 \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]

[Out]

(c^4*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(10 - 26*a*x - 12*a^2*x^2 + 164*a^3*x^3 + 35*a^4*x^4) - 105*a^3*x^3*
ArcTanh[Sqrt[1 + 1/(a*x)]]))/(35*a^4*Sqrt[1 - 1/(a*x)]*x^3)

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Maple [A]  time = 0.181, size = 178, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{4}}{ \left ( 70\,ax+70 \right ){x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -70\,{a}^{9/2}\sqrt{ \left ( ax+1 \right ) x}{x}^{4}+105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{4}{a}^{4}-328\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}+24\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+52\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-20\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(9/2),x)

[Out]

-1/70/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)/x^3*c^4/a^(9/2)*(-70*a^(9/2)*((a*x+1)*x)^(
1/2)*x^4+105*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*x^4*a^4-328*a^(7/2)*x^3*((a*x+1)*x)^(1/2)+2
4*a^(5/2)*x^2*((a*x+1)*x)^(1/2)+52*a^(3/2)*x*((a*x+1)*x)^(1/2)-20*((a*x+1)*x)^(1/2)*a^(1/2))/((a*x+1)*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(9/2),x, algorithm="maxima")

[Out]

integrate((c - c/(a*x))^(9/2)/((a*x - 1)/(a*x + 1))^(3/2), x)

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Fricas [A]  time = 2.32026, size = 919, normalized size = 3.43 \begin{align*} \left [\frac{105 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (35 \, a^{5} c^{4} x^{5} + 199 \, a^{4} c^{4} x^{4} + 152 \, a^{3} c^{4} x^{3} - 38 \, a^{2} c^{4} x^{2} - 16 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{140 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac{105 \,{\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (35 \, a^{5} c^{4} x^{5} + 199 \, a^{4} c^{4} x^{4} + 152 \, a^{3} c^{4} x^{3} - 38 \, a^{2} c^{4} x^{2} - 16 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{70 \,{\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(9/2),x, algorithm="fricas")

[Out]

[1/140*(105*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*
sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(35*a^5*c^4*x^5 + 199*a^4*c^4*x^
4 + 152*a^3*c^4*x^3 - 38*a^2*c^4*x^2 - 16*a*c^4*x + 10*c^4)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))
/(a^5*x^4 - a^4*x^3), 1/70*(105*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a
*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(35*a^5*c^4*x^5 + 199*a^4*c^4*x^4 +
152*a^3*c^4*x^3 - 38*a^2*c^4*x^2 - 16*a*c^4*x + 10*c^4)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^
5*x^4 - a^4*x^3)]

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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(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{9}{2}}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(9/2),x, algorithm="giac")

[Out]

integrate((c - c/(a*x))^(9/2)/((a*x - 1)/(a*x + 1))^(3/2), x)