\(\int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx\) [309]

Optimal result

Integrand size = 23, antiderivative size = 110 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]

[Out]

75/64*a^4*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)+1/4*(-a*c*x+c)^(1/2)/x^4+5/8*a*(-a*c*x+c)^(1/2)/x^3+25/32*
a^2*(-a*c*x+c)^(1/2)/x^2+75/64*a^3*(-a*c*x+c)^(1/2)/x

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6302, 6265, 21, 79, 44, 65, 214} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3} \]

[In]

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

[Out]

Sqrt[c - a*c*x]/(4*x^4) + (5*a*Sqrt[c - a*c*x])/(8*x^3) + (25*a^2*Sqrt[c - a*c*x])/(32*x^2) + (75*a^3*Sqrt[c -
a*c*x])/(64*x) + (75*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/64

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 6265

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

Rule 6302

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]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx \\ & = -\int \frac {(1+a x) \sqrt {c-a c x}}{x^5 (1-a x)} \, dx \\ & = -\left (c \int \frac {1+a x}{x^5 \sqrt {c-a c x}} \, dx\right ) \\ & = \frac {\sqrt {c-a c x}}{4 x^4}-\frac {1}{8} (15 a c) \int \frac {1}{x^4 \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {1}{16} \left (25 a^2 c\right ) \int \frac {1}{x^3 \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {1}{64} \left (75 a^3 c\right ) \int \frac {1}{x^2 \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}-\frac {1}{128} \left (75 a^4 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {1}{64} \left (75 a^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right ) \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.65 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x} \left (16+40 a x+50 a^2 x^2+75 a^3 x^3\right )}{64 x^4}+\frac {75}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right ) \]

[In]

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

[Out]

(Sqrt[c - a*c*x]*(16 + 40*a*x + 50*a^2*x^2 + 75*a^3*x^3))/(64*x^4) + (75*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/S
qrt[c]])/64

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64

method	result	size
risch	\(-\frac {\left (75 a^{4} x^{4}-25 a^{3} x^{3}-10 a^{2} x^{2}-24 a x -16\right ) c}{64 x^{4} \sqrt {-c \left (a x -1\right )}}+\frac {75 a^{4} \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}}{64}\)	\(70\)
pseudoelliptic	\(\frac {\frac {\sqrt {-c \left (a x -1\right )}\, \left (75 a^{3} x^{3}+50 a^{2} x^{2}+40 a x +16\right ) \sqrt {c}}{64}+\frac {75 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right ) c \,a^{4} x^{4}}{64}}{\sqrt {c}\, x^{4}}\)	\(70\)
derivativedivides	\(2 c^{4} a^{4} \left (\frac {-\frac {75 \left (-a c x +c \right )^{\frac {7}{2}}}{128 c^{3}}+\frac {275 \left (-a c x +c \right )^{\frac {5}{2}}}{128 c^{2}}-\frac {365 \left (-a c x +c \right )^{\frac {3}{2}}}{128 c}+\frac {181 \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {75 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 c^{\frac {7}{2}}}\right )\)	\(93\)
default	\(2 c^{4} a^{4} \left (\frac {-\frac {75 \left (-a c x +c \right )^{\frac {7}{2}}}{128 c^{3}}+\frac {275 \left (-a c x +c \right )^{\frac {5}{2}}}{128 c^{2}}-\frac {365 \left (-a c x +c \right )^{\frac {3}{2}}}{128 c}+\frac {181 \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {75 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 c^{\frac {7}{2}}}\right )\)	\(93\)

[In]

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

[Out]

-1/64*(75*a^4*x^4-25*a^3*x^3-10*a^2*x^2-24*a*x-16)/x^4/(-c*(a*x-1))^(1/2)*c+75/64*a^4*arctanh((-a*c*x+c)^(1/2)
/c^(1/2))*c^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.35 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {75 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{128 \, x^{4}}, -\frac {75 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{64 \, x^{4}}\right ] \]

[In]

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

[Out]

[1/128*(75*a^4*sqrt(c)*x^4*log((a*c*x - 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) + 2*(75*a^3*x^3 + 50*a^2*x^2 + 40
*a*x + 16)*sqrt(-a*c*x + c))/x^4, -1/64*(75*a^4*sqrt(-c)*x^4*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (75*a^3*x^3
+ 50*a^2*x^2 + 40*a*x + 16)*sqrt(-a*c*x + c))/x^4]

Sympy [F]

\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )}{x^{5} \left (a x - 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.48 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {1}{128} \, a^{4} c^{4} {\left (\frac {2 \, {\left (75 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 275 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 181 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {75 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]

[In]

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

[Out]

-1/128*a^4*c^4*(2*(75*(-a*c*x + c)^(7/2) - 275*(-a*c*x + c)^(5/2)*c + 365*(-a*c*x + c)^(3/2)*c^2 - 181*sqrt(-a
*c*x + c)*c^3)/((a*c*x - c)^4*c^3 + 4*(a*c*x - c)^3*c^4 + 6*(a*c*x - c)^2*c^5 + 4*(a*c*x - c)*c^6 + c^7) + 75*
log((sqrt(-a*c*x + c) - sqrt(c))/(sqrt(-a*c*x + c) + sqrt(c)))/c^(7/2))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {\frac {75 \, a^{5} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {75 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{5} c + 275 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{5} c^{2} - 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{5} c^{3} + 181 \, \sqrt {-a c x + c} a^{5} c^{4}}{a^{4} c^{4} x^{4}}}{64 \, a} \]

[In]

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

[Out]

-1/64*(75*a^5*c*arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - (75*(a*c*x - c)^3*sqrt(-a*c*x + c)*a^5*c + 275*(a
*c*x - c)^2*sqrt(-a*c*x + c)*a^5*c^2 - 365*(-a*c*x + c)^(3/2)*a^5*c^3 + 181*sqrt(-a*c*x + c)*a^5*c^4)/(a^4*c^4
*x^4))/a

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {181\,\sqrt {c-a\,c\,x}}{64\,x^4}-\frac {365\,{\left (c-a\,c\,x\right )}^{3/2}}{64\,c\,x^4}+\frac {275\,{\left (c-a\,c\,x\right )}^{5/2}}{64\,c^2\,x^4}-\frac {75\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}-\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,75{}\mathrm {i}}{64} \]

[In]

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

[Out]

(181*(c - a*c*x)^(1/2))/(64*x^4) - (a^4*c^(1/2)*atan(((c - a*c*x)^(1/2)*1i)/c^(1/2))*75i)/64 - (365*(c - a*c*x
)^(3/2))/(64*c*x^4) + (275*(c - a*c*x)^(5/2))/(64*c^2*x^4) - (75*(c - a*c*x)^(7/2))/(64*c^3*x^4)