### 3.488 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-\frac{c}{a x})^{7/2}} \, dx$$

Optimal. Leaf size=267 $\frac{a x \left (1-\frac{1}{a x}\right )^{7/2}}{\left (a-\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{7 \left (1-\frac{1}{a x}\right )^{7/2}}{4 a \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{3 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{\left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{4 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{7/2}}$

[Out]

(7*(1 - 1/(a*x))^(7/2))/(4*a*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) - (3*(1 - 1/(a*x))^(7/2))/(2*(a - x^(-1))*
Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) + (a*(1 - 1/(a*x))^(7/2)*x)/((a - x^(-1))*Sqrt[1 + 1/(a*x)]*(c - c/(a*x
))^(7/2)) + ((1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) - (11*(1 - 1/(a*x))^(7/2)
*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(4*Sqrt[2]*a*(c - c/(a*x))^(7/2))

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Rubi [A]  time = 0.171721, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {6182, 6179, 103, 151, 152, 156, 63, 208, 206} $\frac{a x \left (1-\frac{1}{a x}\right )^{7/2}}{\left (a-\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{7 \left (1-\frac{1}{a x}\right )^{7/2}}{4 a \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{3 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{\left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{4 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2)),x]

[Out]

(7*(1 - 1/(a*x))^(7/2))/(4*a*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) - (3*(1 - 1/(a*x))^(7/2))/(2*(a - x^(-1))*
Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) + (a*(1 - 1/(a*x))^(7/2)*x)/((a - x^(-1))*Sqrt[1 + 1/(a*x)]*(c - c/(a*x
))^(7/2)) + ((1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) - (11*(1 - 1/(a*x))^(7/2)
*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(4*Sqrt[2]*a*(c - c/(a*x))^(7/2))

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

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

Rule 152

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

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{7/2}} \, dx &=\frac{\left (1-\frac{1}{a x}\right )^{7/2} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{7/2}} \, dx}{\left (c-\frac{c}{a x}\right )^{7/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{7/2}}\\ &=\frac{a \left (1-\frac{1}{a x}\right )^{7/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{\left (1-\frac{1}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{-\frac{1}{2 a}-\frac{5 x}{2 a^2}}{x \left (1-\frac{x}{a}\right )^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{7/2}}\\ &=-\frac{3 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a \left (1-\frac{1}{a x}\right )^{7/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (a \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{a^2}+\frac{9 x}{2 a^3}}{x \left (1-\frac{x}{a}\right ) \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=\frac{7 \left (1-\frac{1}{a x}\right )^{7/2}}{4 a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{3 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a \left (1-\frac{1}{a x}\right )^{7/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (a^2 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{a^3}+\frac{7 x}{4 a^4}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=\frac{7 \left (1-\frac{1}{a x}\right )^{7/2}}{4 a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{3 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a \left (1-\frac{1}{a x}\right )^{7/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (11 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (1-\frac{1}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=\frac{7 \left (1-\frac{1}{a x}\right )^{7/2}}{4 a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{3 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a \left (1-\frac{1}{a x}\right )^{7/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (1-\frac{1}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (11 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{4 a \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=\frac{7 \left (1-\frac{1}{a x}\right )^{7/2}}{4 a \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{3 \left (1-\frac{1}{a x}\right )^{7/2}}{2 \left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a \left (1-\frac{1}{a x}\right )^{7/2} x}{\left (a-\frac{1}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{\left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{4 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0808455, size = 121, normalized size = 0.45 $\frac{\sqrt{1-\frac{1}{a x}} \left (11 (a x-1) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{a+\frac{1}{x}}{2 a}\right )+(4-4 a x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{1}{a x}+1\right )+2 a x (2 a x-3)\right )}{4 a c^3 \sqrt{\frac{1}{a x}+1} (a x-1) \sqrt{c-\frac{c}{a x}}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2)),x]

[Out]

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

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Maple [A]  time = 0.191, size = 290, normalized size = 1.1 \begin{align*}{\frac{ \left ( ax+1 \right ) x}{16\, \left ( ax-1 \right ) ^{3}{c}^{4}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}{x}^{2}-11\,{a}^{5/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ){x}^{2}+4\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x+8\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){a}^{3}\sqrt{{a}^{-1}}{x}^{2}-28\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-8\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}}+11\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(7/2),x)

[Out]

1/16*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^3*(c*(a*x-1)/a/x)^(1/2)*x*(16*a^(7/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/
2)*x^2-11*a^(5/2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*x^2+4*a^(5/2)*(1/a)^
(1/2)*((a*x+1)*x)^(1/2)*x+8*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^3*(1/a)^(1/2)*x^2-28*((a*x
+1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)-8*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2)+11*2^(1
/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*a^(1/2))/a^(3/2)/c^4/(1/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 (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(7/2),x, algorithm="maxima")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^(7/2), x)

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Fricas [A]  time = 2.69819, size = 1301, normalized size = 4.87 \begin{align*} \left [\frac{11 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2}{\left (3 \, a^{3} x^{3} + 4 \, 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^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 8 \,{\left (a^{2} x^{2} - 2 \, a x + 1\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 ) + 8 \,{\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{32 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac{11 \, \sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{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}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 8 \,{\left (a^{2} x^{2} - 2 \, a x + 1\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 ) + 4 \,{\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{16 \,{\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(7/2),x, algorithm="fricas")

[Out]

[1/32*(11*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3
*x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 +
3*a*x - 1)) + 8*(a^2*x^2 - 2*a*x + 1)*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)*sq
rt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt
((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4), 1/16*(11*sqrt(2)*(a^2*x^2
- 2*a*x + 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*
x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 8*(a^2*x^2 - 2*a*x + 1)*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)) + 4*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt((a
*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)]

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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(((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**(7/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError