3.318 \(\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx\)

Optimal. Leaf size=322 \[ \frac{11 a^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{24 x^2 \sqrt{1-\frac{1}{a x}}}+\frac{21 a^3 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{32 x \sqrt{1-\frac{1}{a x}}}+\frac{107 a^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{64 x \sqrt{1-\frac{1}{a x}}}+\frac{363 a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{64 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{4 x^3 \sqrt{1-\frac{1}{a x}}} \]

[Out]

(a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x^3) + (11*a^2*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x
])/(24*Sqrt[1 - 1/(a*x)]*x^2) + (107*a^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(64*Sqrt[1 - 1/(a*x)]*x) + (21*a^3
*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(32*Sqrt[1 - 1/(a*x)]*x) + (363*a^(7/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*Arc
Sinh[Sqrt[x^(-1)]/Sqrt[a]])/(64*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^(7/2)*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 = 0.30185, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{11 a^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{24 x^2 \sqrt{1-\frac{1}{a x}}}+\frac{21 a^3 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{32 x \sqrt{1-\frac{1}{a x}}}+\frac{107 a^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{64 x \sqrt{1-\frac{1}{a x}}}+\frac{363 a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{64 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{4 x^3 \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^5,x]

[Out]

(a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x^3) + (11*a^2*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x
])/(24*Sqrt[1 - 1/(a*x)]*x^2) + (107*a^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(64*Sqrt[1 - 1/(a*x)]*x) + (21*a^3
*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(32*Sqrt[1 - 1/(a*x)]*x) + (363*a^(7/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*Arc
Sinh[Sqrt[x^(-1)]/Sqrt[a]])/(64*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^(7/2)*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^5} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^{9/2}} \, 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^{5/2} \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}}{4 \sqrt{1-\frac{1}{a x}} x^3}-\frac{\left (a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{x^{3/2} \sqrt{1+\frac{x}{a}} \left (\frac{5}{2}+\frac{11 x}{2 a}\right )}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^3}+\frac{11 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^2}+\frac{\left (a^3 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \left (-\frac{33}{4 a}-\frac{63 x}{4 a^2}\right ) \sqrt{1+\frac{x}{a}}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{12 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^3}+\frac{11 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^2}+\frac{21 a^3 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{32 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (a^5 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{63}{8 a^2}+\frac{321 x}{8 a^3}\right ) \sqrt{1+\frac{x}{a}}}{\sqrt{x} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{24 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^3}+\frac{11 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^2}+\frac{107 a^3 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{64 \sqrt{1-\frac{1}{a x}} x}+\frac{21 a^3 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{32 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (a^6 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{447}{16 a^3}-\frac{1089 x}{16 a^4}}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{24 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^3}+\frac{11 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^2}+\frac{107 a^3 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{64 \sqrt{1-\frac{1}{a x}} x}+\frac{21 a^3 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{32 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (363 a^3 \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 )}{128 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 a^3 \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}}{4 \sqrt{1-\frac{1}{a x}} x^3}+\frac{11 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^2}+\frac{107 a^3 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{64 \sqrt{1-\frac{1}{a x}} x}+\frac{21 a^3 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{32 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (363 a^3 \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 )}{64 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 a^3 \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}}{4 \sqrt{1-\frac{1}{a x}} x^3}+\frac{11 a^2 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^2}+\frac{107 a^3 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{64 \sqrt{1-\frac{1}{a x}} x}+\frac{21 a^3 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{32 \sqrt{1-\frac{1}{a x}} x}+\frac{363 a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{64 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{7/2} \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 = 0.250251, size = 148, normalized size = 0.46 \[ \frac{\sqrt{c-a c x} \left (\sqrt{\frac{1}{a x}+1} \left (447 a^3 x^3+214 a^2 x^2+136 a x+48\right )+\frac{1089 a^{7/2} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{7/2}}-\frac{768 \sqrt{2} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\left (\frac{1}{x}\right )^{7/2}}\right )}{192 x^4 \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^5,x]

[Out]

(Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*(48 + 136*a*x + 214*a^2*x^2 + 447*a^3*x^3) + (1089*a^(7/2)*ArcSinh[Sqrt[x^
(-1)]/Sqrt[a]])/(x^(-1))^(7/2) - (768*Sqrt[2]*a^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)
])])/(x^(-1))^(7/2)))/(192*Sqrt[1 - 1/(a*x)]*x^4)

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Maple [A]  time = 0.196, size = 186, normalized size = 0.6 \begin{align*} -{\frac{ax-1}{ \left ( 192\,ax+192 \right ){x}^{4}}\sqrt{-c \left ( ax-1 \right ) } \left ( 768\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{4}{a}^{4}c-1089\,c\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{4}{a}^{4}-447\,{x}^{3}{a}^{3}\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}-214\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}-136\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}-48\,\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^5,x)

[Out]

-1/192*(a*x-1)*(-c*(a*x-1))^(1/2)*(768*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x^4*a^4*c-1089*c
*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x^4*a^4-447*x^3*a^3*(-c*(a*x+1))^(1/2)*c^(1/2)-214*x^2*a^2*(-c*(a*x+1))^(1
/2)*c^(1/2)-136*x*a*(-c*(a*x+1))^(1/2)*c^(1/2)-48*(-c*(a*x+1))^(1/2)*c^(1/2))/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/
c^(1/2)/(-c*(a*x+1))^(1/2)/x^4

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

[Out]

integrate(sqrt(-a*c*x + c)/(x^5*((a*x - 1)/(a*x + 1))^(3/2)), x)

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Fricas [A]  time = 1.71726, size = 1079, normalized size = 3.35 \begin{align*} \left [\frac{768 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\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 ) + 1089 \,{\left (a^{5} x^{5} - a^{4} x^{4}\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 (447 \, a^{4} x^{4} + 661 \, a^{3} x^{3} + 350 \, a^{2} x^{2} + 184 \, a x + 48\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{384 \,{\left (a x^{5} - x^{4}\right )}}, -\frac{768 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\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 ) - 1089 \,{\left (a^{5} x^{5} - a^{4} x^{4}\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 (447 \, a^{4} x^{4} + 661 \, a^{3} x^{3} + 350 \, a^{2} x^{2} + 184 \, a x + 48\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{192 \,{\left (a x^{5} - x^{4}\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^5,x, algorithm="fricas")

[Out]

[1/384*(768*sqrt(2)*(a^5*x^5 - a^4*x^4)*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)) + 1089*(a^5*x^5 - a^4*x^4)*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*(
447*a^4*x^4 + 661*a^3*x^3 + 350*a^2*x^2 + 184*a*x + 48)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^5 - x
^4), -1/192*(768*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a
*x + 1))/(a*c*x - c)) - 1089*(a^5*x^5 - a^4*x^4)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x +
1))/(a*c*x - c)) - (447*a^4*x^4 + 661*a^3*x^3 + 350*a^2*x^2 + 184*a*x + 48)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(
a*x + 1)))/(a*x^5 - x^4)]

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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**5,x)

[Out]

Timed out

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Giac [C]  time = 1.3485, size = 315, normalized size = 0.98 \begin{align*} -\frac{1}{192} \, a^{4} c^{4}{\left (\frac{768 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{c^{\frac{7}{2}} \mathrm{sgn}\left (-a c x - c\right )} - \frac{1089 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{7}{2}} \mathrm{sgn}\left (-a c x - c\right )} - \frac{447 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 1127 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 1049 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} - 321 \, \sqrt{-a c x - c} c^{3}}{a^{4} c^{7} x^{4} \mathrm{sgn}\left (-a c x - c\right )}\right )} - \frac{768 i \, \sqrt{2} a^{4} \sqrt{-c} \arctan \left (-i\right ) - 1089 i \, a^{4} \sqrt{-c} \arctan \left (-i \, \sqrt{2}\right ) - 845 \, \sqrt{2} a^{4} \sqrt{-c}}{192 \, \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^5,x, algorithm="giac")

[Out]

-1/192*a^4*c^4*(768*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/(c^(7/2)*sgn(-a*c*x - c)) - 1089*arct
an(sqrt(-a*c*x - c)/sqrt(c))/(c^(7/2)*sgn(-a*c*x - c)) - (447*(a*c*x + c)^3*sqrt(-a*c*x - c) - 1127*(a*c*x + c
)^2*sqrt(-a*c*x - c)*c - 1049*(-a*c*x - c)^(3/2)*c^2 - 321*sqrt(-a*c*x - c)*c^3)/(a^4*c^7*x^4*sgn(-a*c*x - c))
) - 1/192*(768*I*sqrt(2)*a^4*sqrt(-c)*arctan(-I) - 1089*I*a^4*sqrt(-c)*arctan(-I*sqrt(2)) - 845*sqrt(2)*a^4*sq
rt(-c))/sgn(c)