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

Optimal. Leaf size=294 \[ \frac{25 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}-\frac{17 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{12 (1-a x)^2 (a x+1)}-\frac{5 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2}+\frac{a x^2 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{6 (1-a x)^2}+\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{4 (1-a x)}-\frac{2 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac{9 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{8 (1-a x)^{5/2} (a x+1)^{5/2}} \]

[Out]

(-5*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(8*(1 - a*x)^2) + (25*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1
+ a*x)^2) - (17*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(12*(1 - a*x)^2*(1 + a*x)) + (a*(c - c/(a^2*x^2))^(5/2)*x^2*(
1 + a*x))/(6*(1 - a*x)^2) + ((c - c/(a^2*x^2))^(5/2)*x*(1 + a*x))/(4*(1 - a*x)) - (2*a^4*(c - c/(a^2*x^2))^(5/
2)*x^5*ArcSin[a*x])/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) - (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcTanh[Sqrt[1 - a*
x]*Sqrt[1 + a*x]])/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))

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Rubi [A]  time = 0.495301, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ \frac{25 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}-\frac{17 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{12 (1-a x)^2 (a x+1)}-\frac{5 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2}+\frac{a x^2 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{6 (1-a x)^2}+\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{4 (1-a x)}-\frac{2 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac{9 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{8 (1-a x)^{5/2} (a x+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(8*(1 - a*x)^2) + (25*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1
+ a*x)^2) - (17*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(12*(1 - a*x)^2*(1 + a*x)) + (a*(c - c/(a^2*x^2))^(5/2)*x^2*(
1 + a*x))/(6*(1 - a*x)^2) + ((c - c/(a^2*x^2))^(5/2)*x*(1 + a*x))/(4*(1 - a*x)) - (2*a^4*(c - c/(a^2*x^2))^(5/
2)*x^5*ArcSin[a*x])/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) - (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcTanh[Sqrt[1 - a*
x]*Sqrt[1 + a*x]])/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))

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 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 149

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] && IntegerQ[m]

Rule 154

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] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

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

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]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

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^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \, dx\\ &=-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)^{5/2} (1+a x)^{5/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{(1-a x)^{3/2} (1+a x)^{7/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{\sqrt{1-a x} (1+a x)^{5/2} \left (2 a-5 a^2 x\right )}{x^4} \, dx}{4 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{(1+a x)^{5/2} \left (-15 a^2+13 a^3 x\right )}{x^3 \sqrt{1-a x}} \, dx}{12 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{(1+a x)^{3/2} \left (-34 a^3+41 a^4 x\right )}{x^2 \sqrt{1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}-\frac{17 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{\sqrt{1+a x} \left (-27 a^4+75 a^5 x\right )}{x \sqrt{1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac{25 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac{17 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{27 a^5-48 a^6 x}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{24 a (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac{25 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac{17 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac{\left (9 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{8 (1-a x)^{5/2} (1+a x)^{5/2}}-\frac{\left (2 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac{25 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac{17 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac{\left (9 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}-\frac{\left (2 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac{5 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac{25 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}-\frac{17 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac{2 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 \sin ^{-1}(a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}-\frac{9 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.131823, size = 134, normalized size = 0.46 \[ \frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (24 a^4 x^4-64 a^3 x^3-3 a^2 x^2+16 a x+6\right )+48 a^4 x^4 \log \left (\sqrt{a^2 x^2-1}+a x\right )+27 a^4 x^4 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{24 a^4 x^3 \sqrt{a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(6 + 16*a*x - 3*a^2*x^2 - 64*a^3*x^3 + 24*a^4*x^4) + 27*a^4*x^4
*ArcTan[1/Sqrt[-1 + a^2*x^2]] + 48*a^4*x^4*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^4*x^3*Sqrt[-1 + a^2*x^2])

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Maple [B]  time = 0.187, size = 625, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(c*(a^2*x^2-1)/a^2/x^2)^(5/2)*x/a^2*(-80*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*x^5*a^7*c+80*(-c/a^2)^
(1/2)*(c*(a^2*x^2-1)/a^2)^(7/2)*x^3*a^7+48*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*x^4*a^6*c+27*(-c/a^2)^
(1/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*x^4*a^6*c+60*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x^5*a^5*c^2-75*(-c/a
^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(7/2)*x^2*a^6+100*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^5*a^5*c^2-80*(-c/a^
2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(7/2)*x*a^5-45*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^4*a^4*c^2-90*(-c/a^2)^(
1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^5*a^3*c^3-150*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^5*a^3*c^3-30*a^4
*(c*(a^2*x^2-1)/a^2)^(7/2)*(-c/a^2)^(1/2)+150*(-c/a^2)^(1/2)*c^(7/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*x
^4*a+90*(-c/a^2)^(1/2)*c^(7/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+c*x)/c^(1/2))*x^4*a+135*(-c/a^2)^(1/2
)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^4*a^2*c^3+135*ln(2*((-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/x/a^2)*x^4*c
^4)/(-c/a^2)^(1/2)/(c*(a^2*x^2-1)/a^2)^(5/2)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.80223, size = 856, normalized size = 2.91 \begin{align*} \left [-\frac{96 \, a^{3} \sqrt{-c} c^{2} x^{3} \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 ) - 27 \, a^{3} \sqrt{-c} c^{2} x^{3} \log \left (-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \,{\left (24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, \frac{27 \, a^{3} c^{\frac{5}{2}} x^{3} \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 24 \, a^{3} c^{\frac{5}{2}} x^{3} \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 (24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(96*a^3*sqrt(-c)*c^2*x^3*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 27*
a^3*sqrt(-c)*c^2*x^3*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(24*a^4*
c^2*x^4 - 64*a^3*c^2*x^3 - 3*a^2*c^2*x^2 + 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3), 1/2
4*(27*a^3*c^(5/2)*x^3*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + 24*a^3*c^(5/2)*x^3
*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) + (24*a^4*c^2*x^4 - 64*a^3*c^2*x^3 -
3*a^2*c^2*x^2 + 16*a*c^2*x + 6*c^2)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3)]

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Sympy [C]  time = 19.4013, size = 500, normalized size = 1.7 \begin{align*} c^{2} \left (\begin{cases} \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{i \sqrt{c} \log{\left (a x \right )}}{a} + \frac{i \sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2 a} + \frac{\sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{a} + \frac{i \sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2 a} - \frac{i \sqrt{c} \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text{otherwise} \end{cases}\right ) + \frac{2 c^{2} \left (\begin{cases} - \frac{a \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + \sqrt{c} \operatorname{acosh}{\left (a x \right )} + \frac{\sqrt{c}}{a x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{i a \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - i \sqrt{c} \operatorname{asin}{\left (a x \right )} - \frac{i \sqrt{c}}{a x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right )}{a} - \frac{2 c^{2} \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{a^{2} \left (c - \frac{c}{a^{2} x^{2}}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right )}{a^{3}} - \frac{c^{2} \left (\begin{cases} \frac{i a^{3} \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{8} - \frac{i a^{2} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 i \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{4 a^{2} x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{a^{3} \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{8} + \frac{a^{2} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{\sqrt{c}}{4 a^{2} x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(c
)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I
*sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c**2*Piecewise((-a*sqrt(c)*x/sqrt(a**2*x**2 - 1) + sqrt(c
)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I
*sqrt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a - 2*c**2*Piecewise((0, Eq(c, 0)), (a**2*(c
- c/(a**2*x**2))**(3/2)/(3*c), True))/a**3 - c**2*Piecewise((I*a**3*sqrt(c)*acosh(1/(a*x))/8 - I*a**2*sqrt(c)
/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*I*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(4*a**2*x**5*sqrt(
-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**3*sqrt(c)*asin(1/(a*x))/8 + a**2*sqrt(c)/(8*x*sqrt(1 - 1/(a*
*2*x**2))) - 3*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(4*a**2*x**5*sqrt(1 - 1/(a**2*x**2))), True)
)/a**4

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Giac [A]  time = 3.20855, size = 562, normalized size = 1.91 \begin{align*} -\frac{1}{12} \,{\left (\frac{27 \, c^{\frac{5}{2}} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{24 \, c^{\frac{5}{2}} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a{\left | a \right |}} - \frac{12 \, \sqrt{a^{2} c x^{2} - c} c^{2} \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{3 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{7} c^{3}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 96 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{6} a c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) - 21 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{5} c^{4}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 192 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{4} a c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 21 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} c^{5}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 160 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 3 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} c^{6}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 64 \, a c^{\frac{13}{2}} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4} a^{2}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(27*c^(5/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 + 24*c^(5/2)*log(abs(-sqrt
(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs(a)) - 12*sqrt(a^2*c*x^2 - c)*c^2*sgn(x)/a^2 - (3*(sqrt(a^2*c)*
x - sqrt(a^2*c*x^2 - c))^7*c^3*abs(a)*sgn(x) - 96*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^6*a*c^(7/2)*sgn(x) - 2
1*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^4*abs(a)*sgn(x) - 192*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^
(9/2)*sgn(x) + 21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*c^5*abs(a)*sgn(x) - 160*(sqrt(a^2*c)*x - sqrt(a^2*c*
x^2 - c))^2*a*c^(11/2)*sgn(x) - 3*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*c^6*abs(a)*sgn(x) - 64*a*c^(13/2)*sgn(
x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^4*a^2*abs(a)))*abs(a)