3.6.25 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\) [525]

Optimal. Leaf size=119 \[ \frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}} \]

[Out]

9/5/a/(c-c/a/x)^(5/2)+3/a/c/(c-c/a/x)^(3/2)-x/(c-c/a/x)^(5/2)-9*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a/c^(5/2)+9/a
/c^2/(c-c/a/x)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6268, 25, 528, 382, 79, 53, 65, 214} \begin {gather*} -\frac {9 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}+\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - c/(a*x))^(5/2),x]

[Out]

9/(5*a*(c - c/(a*x))^(5/2)) + 3/(a*c*(c - c/(a*x))^(3/2)) + 9/(a*c^2*Sqrt[c - c/(a*x)]) - x/(c - c/(a*x))^(5/2
) - (9*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(5/2))

Rule 25

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

Rule 53

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

Rule 65

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 79

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

Rule 214

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

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[(a + b/x^n)^p*((c +
d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 528

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 6268

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 59, normalized size = 0.50 \begin {gather*} -\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {9 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-\frac {1}{a x}\right )}{5 a \left (c-\frac {c}{a x}\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - c/(a*x))^(5/2),x]

[Out]

-(x/(c - c/(a*x))^(5/2)) + (9*Hypergeometric2F1[-5/2, 1, -3/2, 1 - 1/(a*x)])/(5*a*(c - c/(a*x))^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(327\) vs. \(2(103)=206\).
time = 0.99, size = 328, normalized size = 2.76

method	result	size
risch	\(-\frac {a x -1}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (\frac {9 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {4 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{5 a^{7} c \left (x -\frac {1}{a}\right )^{3}}-\frac {18 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{5 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {54 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{5 a^{5} c \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {a c x \left (a x -1\right )}}{c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\)	\(249\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (90 x^{4} \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}}-80 a^{\frac {7}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x^{2}-360 x^{3} \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}+132 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x +45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{4}+540 x^{2} \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}}-180 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}-60 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}-360 x \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}}+270 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}-180 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x +90 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{10 \sqrt {\left (a x -1\right ) x}\, c^{3} \left (a x -1\right )^{4} \sqrt {a}}\)	\(328\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/10*(c*(a*x-1)/a/x)^(1/2)*x*(90*x^4*((a*x-1)*x)^(1/2)*a^(9/2)-80*a^(7/2)*((a*x-1)*x)^(3/2)*x^2-360*x^3*((a*x
-1)*x)^(1/2)*a^(7/2)+132*a^(5/2)*((a*x-1)*x)^(3/2)*x+45*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*
a^4*x^4+540*x^2*((a*x-1)*x)^(1/2)*a^(5/2)-180*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*a^3*x^3-60
*a^(3/2)*((a*x-1)*x)^(3/2)-360*x*((a*x-1)*x)^(1/2)*a^(3/2)+270*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^
(1/2))*a^2*x^2-180*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*a*x+90*((a*x-1)*x)^(1/2)*a^(1/2)+45*l
n(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2)))/((a*x-1)*x)^(1/2)/c^3/(a*x-1)^4/a^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 294, normalized size = 2.47 \begin {gather*} \left [\frac {45 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{10 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {45 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/10*(45*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c)
- 2*(5*a^4*x^4 - 69*a^3*x^3 + 105*a^2*x^2 - 45*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*
a^2*c^3*x - a*c^3), 1/5*(45*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))
/c) - (5*a^4*x^4 - 69*a^3*x^3 + 105*a^2*x^2 - 45*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 +
3*a^2*c^3*x - a*c^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a x}{a c^{2} x \sqrt {c - \frac {c}{a x}} - 3 c^{2} \sqrt {c - \frac {c}{a x}} + \frac {3 c^{2} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}}}\, dx - \int \frac {1}{a c^{2} x \sqrt {c - \frac {c}{a x}} - 3 c^{2} \sqrt {c - \frac {c}{a x}} + \frac {3 c^{2} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (103) = 206\).
time = 0.57, size = 317, normalized size = 2.66 \begin {gather*} -\frac {9 \, \log \left (c^{2} {\left | a \right |} {\left | c \right |}^{\frac {3}{2}}\right ) \mathrm {sgn}\left (x\right )}{14 \, a c^{\frac {5}{2}}} + \frac {9 \, \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} {\left | a \right |} + 13 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a \sqrt {c} - 36 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} c {\left | a \right |} + 55 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a c^{\frac {3}{2}} - 50 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c^{2} {\left | a \right |} + 27 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{\frac {5}{2}} - 8 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{3} {\left | a \right |} + a c^{\frac {7}{2}} \right |}\right ) \mathrm {sgn}\left (x\right )}{14 \, a c^{\frac {5}{2}}} - \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/14*log(c^2*abs(a)*abs(c)^(3/2))*sgn(x)/(a*c^(5/2)) + 9/14*log(abs(-2*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*
x))^7*abs(a) + 13*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^6*a*sqrt(c) - 36*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 -
a*c*x))^5*c*abs(a) + 55*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^4*a*c^(3/2) - 50*(sqrt(a^2*c)*x - sqrt(a^2*
c*x^2 - a*c*x))^3*c^2*abs(a) + 27*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^2*a*c^(5/2) - 8*(sqrt(a^2*c)*x - s
qrt(a^2*c*x^2 - a*c*x))*c^3*abs(a) + a*c^(7/2)))*sgn(x)/(a*c^(5/2)) - sqrt(a^2*c*x^2 - a*c*x)*abs(a)*sgn(x)/(a
^2*c^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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