3.869 \(\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^{5/2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{2 (a x+1) (1-a x)^3}{15 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{2 (a x+1)^2 (13 a x+28) (1-a x)^3}{15 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{2 (1-a x)^3}{5 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{(1-a x)^2}{a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{2 (a x+1)^{5/2} (1-a x)^{5/2} \sin ^{-1}(a x)}{a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]

[Out]

-((1 - a*x)^2/(a^2*(c - c/(a^2*x^2))^(5/2)*x)) - (2*(1 - a*x)^3)/(5*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) + (2*(1 -
a*x)^3*(1 + a*x))/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) - (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 13*a*x))/(15*a^6*(c
- c/(a^2*x^2))^(5/2)*x^5) - (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*ArcSin[a*x])/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)

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Rubi [A]  time = 0.451342, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6159, 6129, 98, 150, 143, 41, 216} \[ \frac{2 (a x+1) (1-a x)^3}{15 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{2 (a x+1)^2 (13 a x+28) (1-a x)^3}{15 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{2 (1-a x)^3}{5 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{(1-a x)^2}{a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}-\frac{2 (a x+1)^{5/2} (1-a x)^{5/2} \sin ^{-1}(a x)}{a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2)),x]

[Out]

-((1 - a*x)^2/(a^2*(c - c/(a^2*x^2))^(5/2)*x)) - (2*(1 - a*x)^3)/(5*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) + (2*(1 -
a*x)^3*(1 + a*x))/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) - (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 13*a*x))/(15*a^6*(c
- c/(a^2*x^2))^(5/2)*x^5) - (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*ArcSin[a*x])/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6159

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

Rule 6129

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

Rule 98

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

Rule 150

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

Rule 143

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

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

Rubi steps

\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \, dx\\ &=-\frac{\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac{e^{-2 \tanh ^{-1}(a x)} x^5}{(1-a x)^{5/2} (1+a x)^{5/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac{x^5}{(1-a x)^{3/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^2}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}+\frac{\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac{x^3 (4+2 a x)}{\sqrt{1-a x} (1+a x)^{7/2}} \, dx}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^2}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{2 (1-a x)^3}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}+\frac{\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac{x^2 \left (6 a+8 a^2 x\right )}{\sqrt{1-a x} (1+a x)^{5/2}} \, dx}{5 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^2}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{2 (1-a x)^3}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}+\frac{2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}+\frac{\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac{x \left (-4 a^2+26 a^3 x\right )}{\sqrt{1-a x} (1+a x)^{3/2}} \, dx}{15 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^2}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{2 (1-a x)^3}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}+\frac{2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}-\frac{2 (1-a x)^3 (1+a x)^2 (28+13 a x)}{15 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (2 (1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^2}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{2 (1-a x)^3}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}+\frac{2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}-\frac{2 (1-a x)^3 (1+a x)^2 (28+13 a x)}{15 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{\left (2 (1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac{(1-a x)^2}{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}-\frac{2 (1-a x)^3}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}+\frac{2 (1-a x)^3 (1+a x)}{15 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}-\frac{2 (1-a x)^3 (1+a x)^2 (28+13 a x)}{15 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}-\frac{2 (1-a x)^{5/2} (1+a x)^{5/2} \sin ^{-1}(a x)}{a^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}\\ \end{align*}

Mathematica [A]  time = 0.0976883, size = 105, normalized size = 0.54 \[ \frac{15 a^4 x^4+76 a^3 x^3+32 a^2 x^2-30 (a x+1)^2 \sqrt{a^2 x^2-1} \log \left (\sqrt{a^2 x^2-1}+a x\right )-82 a x-56}{15 a^2 c^2 x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2)),x]

[Out]

(-56 - 82*a*x + 32*a^2*x^2 + 76*a^3*x^3 + 15*a^4*x^4 - 30*(1 + a*x)^2*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a
^2*x^2]])/(15*a^2*c^2*Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x)^2)

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Maple [B]  time = 0.184, size = 462, normalized size = 2.4 \begin{align*}{\frac{ax-1}{15\,{x}^{5}{a}^{6}} \left ( 15\,{c}^{5/2} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}{x}^{5}{a}^{5}+45\,{x}^{4}{c}^{5/2}{a}^{4} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}+16\,{c}^{5/2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{4}{a}^{4}-60\,{c}^{5/2} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}{x}^{3}{a}^{3}+16\,{c}^{5/2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{3}{a}^{3}-30\,\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}x{a}^{4}c-90\,{c}^{5/2} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{2}-24\,{c}^{5/2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{2}-30\,\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{a}^{3}c+50\,{c}^{5/2} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}xa-24\,{c}^{5/2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}xa+50\,{c}^{5/2} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}+6\,{c}^{5/2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2} \right ) \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{-{\frac{3}{2}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{c}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(15*c^(5/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x^5*a^5+45*x^4*c^(5/2)*a^4*((a*x-1)*(a*x+1)*c/a^2)^(3/2)+16*c^(
5/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^4*a^4-60*c^(5/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x^3*a^3+16*c^(5/2)*(c*(a^2*x^2
-1)/a^2)^(3/2)*x^3*a^3-30*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*(c*(a^2*x^2-1)
/a^2)^(3/2)*x*a^4*c-90*c^(5/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x^2*a^2-24*c^(5/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^2*
a^2-30*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*a^3*c+5
0*c^(5/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x*a-24*c^(5/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x*a+50*c^(5/2)*((a*x-1)*(a*x+
1)*c/a^2)^(3/2)+6*c^(5/2)*(c*(a^2*x^2-1)/a^2)^(3/2))*(a*x-1)/((a*x-1)*(a*x+1)*c/a^2)^(3/2)/x^5/(c*(a^2*x^2-1)/
a^2/x^2)^(5/2)/a^6/c^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.88524, size = 744, normalized size = 3.82 \begin{align*} \left [\frac{15 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt{c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) +{\left (15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac{30 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt{-c} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) +{\left (15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/15*(15*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(
a^2*x^2)) - c) + (15*a^5*x^5 + 76*a^4*x^4 + 32*a^3*x^3 - 82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))
/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3), 1/15*(30*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*sqrt(-c)*arct
an(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (15*a^5*x^5 + 76*a^4*x^4 + 32*a^3*x^3 -
82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{\left (- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )\right )^{\frac{5}{2}} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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