3.317 \(\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx\)

Optimal. Leaf size=274 \[ \frac{3 a^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}+\frac{13 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{8 x \sqrt{1-\frac{1}{a x}}}+\frac{45 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{8 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 x^2 \sqrt{1-\frac{1}{a x}}} \]

[Out]

(a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]*x^2) + (13*a^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])
/(8*Sqrt[1 - 1/(a*x)]*x) + (3*a^2*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) + (45*a^(5/2)*S
qrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(8*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^(5/2)*Sqrt[x^(
-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[1 - 1/(a*x)]

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Rubi [A]  time = 0.281619, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {6176, 6181, 101, 154, 157, 54, 215, 93, 206} \[ \frac{3 a^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}+\frac{13 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{8 x \sqrt{1-\frac{1}{a x}}}+\frac{45 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{8 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{3 x^2 \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4,x]

[Out]

(a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]*x^2) + (13*a^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])
/(8*Sqrt[1 - 1/(a*x)]*x) + (3*a^2*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) + (45*a^(5/2)*S
qrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(8*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^(5/2)*Sqrt[x^(
-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[1 - 1/(a*x)]

Rule 6176

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

Rule 6181

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

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

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 54

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

Rule 215

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

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^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^{7/2}} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^2}-\frac{\left (a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \sqrt{1+\frac{x}{a}} \left (\frac{3}{2}+\frac{9 x}{2 a}\right )}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^2}+\frac{3 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (a^3 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{9}{4 a}-\frac{39 x}{4 a^2}\right ) \sqrt{1+\frac{x}{a}}}{\sqrt{x} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{6 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^2}+\frac{13 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{8 \sqrt{1-\frac{1}{a x}} x}+\frac{3 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (a^4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\frac{57}{8 a^2}+\frac{135 x}{8 a^3}}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{6 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^2}+\frac{13 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{8 \sqrt{1-\frac{1}{a x}} x}+\frac{3 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (45 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^2}+\frac{13 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{8 \sqrt{1-\frac{1}{a x}} x}+\frac{3 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (45 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,\sqrt{\frac{1}{x}}\right )}{8 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^2}+\frac{13 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{8 \sqrt{1-\frac{1}{a x}} x}+\frac{3 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{45 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{8 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.203083, size = 140, normalized size = 0.51 \[ \frac{\sqrt{c-a c x} \left (\sqrt{\frac{1}{a x}+1} \left (57 a^2 x^2+26 a x+8\right )+\frac{135 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{5/2}}-\frac{96 \sqrt{2} a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\left (\frac{1}{x}\right )^{5/2}}\right )}{24 x^3 \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4,x]

[Out]

(Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*(8 + 26*a*x + 57*a^2*x^2) + (135*a^(5/2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x
^(-1))^(5/2) - (96*Sqrt[2]*a^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(x^(-1))^(5/2)
))/(24*Sqrt[1 - 1/(a*x)]*x^3)

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Maple [A]  time = 0.191, size = 165, normalized size = 0.6 \begin{align*}{\frac{ax-1}{ \left ( 24\,ax+24 \right ){x}^{3}}\sqrt{-c \left ( ax-1 \right ) } \left ( -96\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{3}{a}^{3}c+135\,c\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{3}{a}^{3}+57\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+26\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+8\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x)

[Out]

1/24/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(-c*(a*x-1))^(1/2)*(-96*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(
1/2)/c^(1/2))*x^3*a^3*c+135*c*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x^3*a^3+57*x^2*a^2*(-c*(a*x+1))^(1/2)*c^(1/2)
+26*x*a*(-c*(a*x+1))^(1/2)*c^(1/2)+8*(-c*(a*x+1))^(1/2)*c^(1/2))/c^(1/2)/(-c*(a*x+1))^(1/2)/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c}}{x^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(-a*c*x + c)/(x^4*((a*x - 1)/(a*x + 1))^(3/2)), x)

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Fricas [A]  time = 1.74909, size = 1022, normalized size = 3.73 \begin{align*} \left [\frac{96 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 135 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \,{\left (57 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 34 \, a x + 8\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{48 \,{\left (a x^{4} - x^{3}\right )}}, -\frac{96 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 135 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (57 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 34 \, a x + 8\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{24 \,{\left (a x^{4} - x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="fricas")

[Out]

[1/48*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1
)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 135*(a^4*x^4 - a^3*x^3)*sqrt(-c)*log(-(a^
2*c*x^2 + a*c*x - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x)) + 2*(57*
a^3*x^3 + 83*a^2*x^2 + 34*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^4 - x^3), -1/24*(96*sqrt(2
)*(a^4*x^4 - a^3*x^3)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) -
135*(a^4*x^4 - a^3*x^3)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - (57*
a^3*x^3 + 83*a^2*x^2 + 34*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^4 - x^3)]

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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(1/((a*x-1)/(a*x+1))**(3/2)*(-a*c*x+c)**(1/2)/x**4,x)

[Out]

Timed out

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Giac [C]  time = 1.325, size = 282, normalized size = 1.03 \begin{align*} -\frac{1}{24} \, a^{3} c^{3}{\left (\frac{96 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{c^{\frac{5}{2}} \mathrm{sgn}\left (-a c x - c\right )} - \frac{135 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{5}{2}} \mathrm{sgn}\left (-a c x - c\right )} - \frac{57 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 88 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c + 39 \, \sqrt{-a c x - c} c^{2}}{a^{3} c^{5} x^{3} \mathrm{sgn}\left (-a c x - c\right )}\right )} - \frac{96 i \, \sqrt{2} a^{3} \sqrt{-c} \arctan \left (-i\right ) - 135 i \, a^{3} \sqrt{-c} \arctan \left (-i \, \sqrt{2}\right ) - 91 \, \sqrt{2} a^{3} \sqrt{-c}}{24 \, \mathrm{sgn}\left (c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="giac")

[Out]

-1/24*a^3*c^3*(96*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/(c^(5/2)*sgn(-a*c*x - c)) - 135*arctan(
sqrt(-a*c*x - c)/sqrt(c))/(c^(5/2)*sgn(-a*c*x - c)) - (57*(a*c*x + c)^2*sqrt(-a*c*x - c) + 88*(-a*c*x - c)^(3/
2)*c + 39*sqrt(-a*c*x - c)*c^2)/(a^3*c^5*x^3*sgn(-a*c*x - c))) - 1/24*(96*I*sqrt(2)*a^3*sqrt(-c)*arctan(-I) -
135*I*a^3*sqrt(-c)*arctan(-I*sqrt(2)) - 91*sqrt(2)*a^3*sqrt(-c))/sgn(c)