3.894 $$\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx$$

Optimal. Leaf size=181 $\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{3 a^5 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{4 \sqrt{1-a x} \sqrt{a x+1}}$

[Out]

(6*a^4*Sqrt[c - c/(a^2*x^2)])/5 + Sqrt[c - c/(a^2*x^2)]/(5*x^4) + (a*Sqrt[c - c/(a^2*x^2)])/(2*x^3) + (3*a^2*S
qrt[c - c/(a^2*x^2)])/(5*x^2) + (3*a^3*Sqrt[c - c/(a^2*x^2)])/(4*x) + (3*a^5*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[S
qrt[1 - a*x]*Sqrt[1 + a*x]])/(4*Sqrt[1 - a*x]*Sqrt[1 + a*x])

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Rubi [A]  time = 0.57542, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.296, Rules used = {6167, 6159, 6129, 98, 151, 12, 92, 208} $\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{3 a^5 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{4 \sqrt{1-a x} \sqrt{a x+1}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*a^4*Sqrt[c - c/(a^2*x^2)])/5 + Sqrt[c - c/(a^2*x^2)]/(5*x^4) + (a*Sqrt[c - c/(a^2*x^2)])/(2*x^3) + (3*a^2*S
qrt[c - c/(a^2*x^2)])/(5*x^2) + (3*a^3*Sqrt[c - c/(a^2*x^2)])/(4*x) + (3*a^5*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[S
qrt[1 - a*x]*Sqrt[1 + a*x]])/(4*Sqrt[1 - a*x]*Sqrt[1 + a*x])

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 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{1-a x} \sqrt{1+a x}}{x^6} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1+a x)^{3/2}}{x^6 \sqrt{1-a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{-10 a-9 a^2 x}{x^5 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{5 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{36 a^2+30 a^3 x}{x^4 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{20 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{-90 a^3-72 a^4 x}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{60 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{144 a^4+90 a^5 x}{x^2 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{120 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int -\frac{90 a^5}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{120 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}-\frac{\left (3 a^5 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{4 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{\left (3 a^6 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{4 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}+\frac{3 a^5 \sqrt{c-\frac{c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{4 \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.100253, size = 102, normalized size = 0.56 $\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (24 a^4 x^4+15 a^3 x^3+12 a^2 x^2+10 a x+4\right )-15 a^5 x^5 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{20 x^4 \sqrt{a^2 x^2-1}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(4 + 10*a*x + 12*a^2*x^2 + 15*a^3*x^3 + 24*a^4*x^4) - 15*a^5*x^5*Ar
cTan[1/Sqrt[-1 + a^2*x^2]]))/(20*x^4*Sqrt[-1 + a^2*x^2])

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Maple [B]  time = 0.178, size = 447, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}}{20\,{x}^{4}c}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -40\,\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}\sqrt{-{\frac{c}{{a}^{2}}}}{x}^{6}{a}^{4}c+40\, \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}\sqrt{-{\frac{c}{{a}^{2}}}}{x}^{4}{a}^{4}+15\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{5}{a}^{3}c+40\,\sqrt{-{\frac{c}{{a}^{2}}}}{c}^{3/2}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ){x}^{5}{a}^{2}-40\,\sqrt{-{\frac{c}{{a}^{2}}}}{c}^{3/2}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{c}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}+cx \right ) } \right ){x}^{5}{a}^{2}-40\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{x}^{5}{a}^{3}c+25\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{3}{a}^{3}+15\,\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}-c \right ) } \right ){x}^{5}a{c}^{2}+16\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{2}+10\,a \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}x\sqrt{-{\frac{c}{{a}^{2}}}}+4\, \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}\sqrt{-{\frac{c}{{a}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{c}{{a}^{2}}}}}}} \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)^(1/2)/x^5,x)

[Out]

-1/20*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x^4*a^2*(-40*(c*(a^2*x^2-1)/a^2)^(1/2)*(-c/a^2)^(1/2)*x^6*a^4*c+40*(c*(a^2
*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2)*x^4*a^4+15*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^5*a^3*c+40*(-c/a^2)^(1
/2)*c^(3/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*x^5*a^2-40*(-c/a^2)^(1/2)*c^(3/2)*ln((c^(1/2)*((a*x-1)*(a*
x+1)*c/a^2)^(1/2)+c*x)/c^(1/2))*x^5*a^2-40*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^5*a^3*c+25*(-c/a^2)^
(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^3*a^3+15*ln(2*((-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/x/a^2)*x^5*a*
c^2+16*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^2*a^2+10*a*(c*(a^2*x^2-1)/a^2)^(3/2)*x*(-c/a^2)^(1/2)+4*(c*(
a^2*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2))/(c*(a^2*x^2-1)/a^2)^(1/2)/c/(-c/a^2)^(1/2)

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

[Out]

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

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Fricas [A]  time = 1.63842, size = 522, normalized size = 2.88 \begin{align*} \left [\frac{15 \, a^{4} \sqrt{-c} x^{4} \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} x^{4} + 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 10 \, a x + 4\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, -\frac{15 \, a^{4} \sqrt{c} x^{4} \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 ) -{\left (24 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 10 \, a x + 4\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\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)^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/40*(15*a^4*sqrt(-c)*x^4*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*(2
4*a^4*x^4 + 15*a^3*x^3 + 12*a^2*x^2 + 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4, -1/20*(15*a^4*sqrt(c)*
x^4*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (24*a^4*x^4 + 15*a^3*x^3 + 12*a^2*x^
2 + 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4]

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

[Out]

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

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Giac [B]  time = 3.28731, size = 489, normalized size = 2.7 \begin{align*} \frac{1}{10} \,{\left (15 \, a^{3} \sqrt{c} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right ) - \frac{15 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{9} a^{3} c \mathrm{sgn}\left (x\right ) + 70 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{7} a^{3} c^{2} \mathrm{sgn}\left (x\right ) - 40 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{6} a^{2} c^{\frac{5}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 200 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{4} a^{2} c^{\frac{7}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 70 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} a^{3} c^{4} \mathrm{sgn}\left (x\right ) - 120 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a^{2} c^{\frac{9}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 15 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} a^{3} c^{5} \mathrm{sgn}\left (x\right ) - 24 \, a^{2} c^{\frac{11}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{5}}\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)^(1/2)/x^5,x, algorithm="giac")

[Out]

1/10*(15*a^3*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x) - (15*(sqrt(a^2*c)*x - sqrt
(a^2*c*x^2 - c))^9*a^3*c*sgn(x) + 70*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*a^3*c^2*sgn(x) - 40*(sqrt(a^2*c)*
x - sqrt(a^2*c*x^2 - c))^6*a^2*c^(5/2)*abs(a)*sgn(x) - 200*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a^2*c^(7/2)
*abs(a)*sgn(x) - 70*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*a^3*c^4*sgn(x) - 120*(sqrt(a^2*c)*x - sqrt(a^2*c*x
^2 - c))^2*a^2*c^(9/2)*abs(a)*sgn(x) - 15*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*a^3*c^5*sgn(x) - 24*a^2*c^(11/
2)*abs(a)*sgn(x))/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^5)*abs(a)