### 3.470 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-\frac{c}{a x})^{7/2}} \, dx$$

Optimal. Leaf size=277 $\frac{a^2 x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{35 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{16 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{5 a \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{4 \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{5 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{115 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{16 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{7/2}}$

[Out]

(-5*a*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])/(4*(a - x^(-1))^2*(c - c/(a*x))^(7/2)) - (35*(1 - 1/(a*x))^(7/2)*
Sqrt[1 + 1/(a*x)])/(16*(a - x^(-1))*(c - c/(a*x))^(7/2)) + (a^2*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x)/((a -
x^(-1))^2*(c - c/(a*x))^(7/2)) + (5*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) -
(115*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(16*Sqrt[2]*a*(c - c/(a*x))^(7/2))

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Rubi [A]  time = 0.169433, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.417, Rules used = {6182, 6179, 103, 21, 99, 151, 156, 63, 208, 206} $\frac{a^2 x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{35 \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{16 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{5 a \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{7/2}}{4 \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{5 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{115 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{16 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*a*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])/(4*(a - x^(-1))^2*(c - c/(a*x))^(7/2)) - (35*(1 - 1/(a*x))^(7/2)*
Sqrt[1 + 1/(a*x)])/(16*(a - x^(-1))*(c - c/(a*x))^(7/2)) + (a^2*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x)/((a -
x^(-1))^2*(c - c/(a*x))^(7/2)) + (5*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) -
(115*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(16*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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 99

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 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^{-\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^{-\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 )^3 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{7/2}}\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^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{-\frac{5}{2 a}-\frac{5 x}{2 a^2}}{x \left (1-\frac{x}{a}\right )^3 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (c-\frac{c}{a x}\right )^{7/2}}\\ &=\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (5 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x \left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{2 a \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=-\frac{5 a \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{\left (5 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{-2-\frac{3 x}{2 a}}{x \left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{4 a \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=-\frac{5 a \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{35 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{16 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (5 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{4}{a}+\frac{7 x}{4 a^2}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=-\frac{5 a \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{35 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{16 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (115 \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 )}{32 a^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (5 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \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{5 a \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{35 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{16 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{\left (5 \left (1-\frac{1}{a x}\right )^{7/2}\right ) \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 (115 \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 )}{16 a \left (c-\frac{c}{a x}\right )^{7/2}}\\ &=-\frac{5 a \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{4 \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{35 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}}}{16 \left (a-\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{a^2 \left (1-\frac{1}{a x}\right )^{7/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{5 \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{115 \left (1-\frac{1}{a x}\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{16 \sqrt{2} a \left (c-\frac{c}{a x}\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.155664, size = 135, normalized size = 0.49 $\frac{\sqrt{1-\frac{1}{a x}} \left (2 a x \sqrt{\frac{1}{a x}+1} \left (16 a^2 x^2-55 a x+35\right )+160 (a x-1)^2 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )-115 \sqrt{2} (a x-1)^2 \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )\right )}{32 a c^3 (a x-1)^2 \sqrt{c-\frac{c}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(35 - 55*a*x + 16*a^2*x^2) + 160*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(
a*x)]] - 115*Sqrt[2]*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(32*a*c^3*Sqrt[c - c/(a*x)]*(-1 + a*x)^
2)

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Maple [A]  time = 0.186, size = 371, normalized size = 1.3 \begin{align*}{\frac{ \left ( ax+1 \right ) x}{64\,{c}^{4} \left ( ax-1 \right ) ^{3}}\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 64\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}{x}^{2}-115\,{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}-220\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x+160\,\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}+230\,{a}^{3/2}\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-320\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){a}^{2}\sqrt{{a}^{-1}}x+140\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}+160\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}}-115\,\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))^(1/2)/(c-c/a/x)^(7/2),x)

[Out]

1/64*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(64*a^(7/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*x^2-115
*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^2-220*a^(5/2)*(1/a)^(1/2)*(
(a*x+1)*x)^(1/2)*x+160*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+230*a^(3/2)*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-320*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/
2)+2*a*x+1)/a^(1/2))*a^2*(1/a)^(1/2)*x+140*((a*x+1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)+160*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)-115*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/(a*x-1)^3/((a*x+1)*x)^(1/2)/(1/a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\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))^(1/2)/(c-c/a/x)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 3.156, size = 1467, normalized size = 5.3 \begin{align*} \left [\frac{115 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, 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 ) + 160 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, 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 (16 \, a^{4} x^{4} - 39 \, a^{3} x^{3} - 20 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{128 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, \frac{115 \, \sqrt{2}{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, 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 ) - 160 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, 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 (16 \, a^{4} x^{4} - 39 \, a^{3} x^{3} - 20 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{64 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, 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))^(1/2)/(c-c/a/x)^(7/2),x, algorithm="fricas")

[Out]

[1/128*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*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)) + 160*(a^3*x^3 - 3*a^2*x^2 + 3*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)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(16*a^
4*x^4 - 39*a^3*x^3 - 20*a^2*x^2 + 35*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*
a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4), 1/64*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*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)) -
160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqr
t((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 4*(16*a^4*x^4 - 39*a^3*x^3 - 20*a^2*x^2 + 35*a*x)*sqrt((a*x
- 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*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))**(1/2)/(c-c/a/x)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\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))^(1/2)/(c-c/a/x)^(7/2),x, algorithm="giac")

[Out]

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