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

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

[Out]

1/4*a*(1+1/a/x)^(3/2)*(-a*c*x+c)^(1/2)/x^3/(1-1/a/x)^(1/2)+11/24*a^2*(1+1/a/x)^(3/2)*(-a*c*x+c)^(1/2)/x^2/(1-1
/a/x)^(1/2)+21/32*a^3*(1+1/a/x)^(3/2)*(-a*c*x+c)^(1/2)/x/(1-1/a/x)^(1/2)+107/64*a^3*(1+1/a/x)^(1/2)*(-a*c*x+c)
^(1/2)/x/(1-1/a/x)^(1/2)+363/64*a^(7/2)*arcsinh((1/x)^(1/2)/a^(1/2))*(1/x)^(1/2)*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1
/2)-4*a^(7/2)*arctanh(2^(1/2)*(1/x)^(1/2)/a^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)*(1/x)^(1/2)*(-a*c*x+c)^(1/2)/(1-1/a
/x)^(1/2)

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Rubi [A]  time = 0.30, antiderivative size = 322, normalized size of antiderivative = 1.00, 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 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 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 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.27, size = 148, normalized size = 0.46 \[ \frac {\sqrt {c-a c x} \left (\frac {1089 a^{7/2} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{7/2}}-\frac {768 \sqrt {2} a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\left (\frac {1}{x}\right )^{7/2}}+\sqrt {\frac {1}{a x}+1} \left (447 a^3 x^3+214 a^2 x^2+136 a x+48\right )\right )}{192 x^4 \sqrt {1-\frac {1}{a x}}} \]

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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fricas [A]  time = 0.66, size = 460, normalized size = 1.43 \[ \left [\frac {768 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 1089 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (447 \, a^{4} x^{4} + 661 \, a^{3} x^{3} + 350 \, a^{2} x^{2} + 184 \, a x + 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{384 \, {\left (a x^{5} - x^{4}\right )}}, -\frac {768 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 1089 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (447 \, a^{4} x^{4} + 661 \, a^{3} x^{3} + 350 \, a^{2} x^{2} + 184 \, a x + 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a x^{5} - x^{4}\right )}}\right ] \]

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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giac [C]  time = 0.22, size = 249, normalized size = 0.77 \[ -\frac {\frac {768 \, \sqrt {2} a^{5} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\mathrm {sgn}\left (-a c x - c\right )} - \frac {1089 \, a^{5} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\mathrm {sgn}\left (-a c x - c\right )} + \frac {768 i \, \sqrt {2} a^{5} \sqrt {-c} \arctan \left (-i\right ) - 1089 i \, a^{5} \sqrt {-c} \arctan \left (-i \, \sqrt {2}\right ) - 845 \, \sqrt {2} a^{5} \sqrt {-c}}{\mathrm {sgn}\relax (c)} - \frac {447 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} a^{5} c - 1127 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} a^{5} c^{2} - 1049 \, {\left (-a c x - c\right )}^{\frac {3}{2}} a^{5} c^{3} - 321 \, \sqrt {-a c x - c} a^{5} c^{4}}{a^{4} c^{4} x^{4} \mathrm {sgn}\left (-a c x - c\right )}}{192 \, a} \]

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*(768*sqrt(2)*a^5*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/sgn(-a*c*x - c) - 1089*a^5*sqrt(c
)*arctan(sqrt(-a*c*x - c)/sqrt(c))/sgn(-a*c*x - c) + (768*I*sqrt(2)*a^5*sqrt(-c)*arctan(-I) - 1089*I*a^5*sqrt(
-c)*arctan(-I*sqrt(2)) - 845*sqrt(2)*a^5*sqrt(-c))/sgn(c) - (447*(a*c*x + c)^3*sqrt(-a*c*x - c)*a^5*c - 1127*(
a*c*x + c)^2*sqrt(-a*c*x - c)*a^5*c^2 - 1049*(-a*c*x - c)^(3/2)*a^5*c^3 - 321*sqrt(-a*c*x - c)*a^5*c^4)/(a^4*c
^4*x^4*sgn(-a*c*x - c)))/a

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maple [A]  time = 0.07, size = 186, normalized size = 0.58 \[ \frac {\left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-768 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x^{4} a^{4} c +1089 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) c \,x^{4} a^{4}+447 x^{3} a^{3} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+214 x^{2} a^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+136 x a \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+48 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{192 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {c}\, \sqrt {-c \left (a x +1\right )}\, x^{4}} \]

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*a
rctan((-c*(a*x+1))^(1/2)/c^(1/2))*c*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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a c x + c}}{x^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c-a\,c\,x}}{x^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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