\(\int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) [497]

Optimal result

Integrand size = 25, antiderivative size = 159 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {16 a^4 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {16 a^4 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \sqrt {c-\frac {c}{a x}}}-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac {c}{a x}\right )^{3/2} x^3}+\frac {4 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 \left (c-\frac {c}{a x}\right )^{3/2} x^2} \]

[Out]

-16/315*a^4*c^2*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(3/2)-2/9*a*c^2*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(3/2)/x^3+4/21*a^2
*c^2*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(3/2)/x^2+16/105*a^4*c*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6313, 885, 809, 663} \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {4 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 x^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 x^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {16 a^4 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {16 a^4 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \sqrt {c-\frac {c}{a x}}} \]

[In]

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

[Out]

(-16*a^4*c^2*(1 - 1/(a^2*x^2))^(3/2))/(315*(c - c/(a*x))^(3/2)) + (16*a^4*c*(1 - 1/(a^2*x^2))^(3/2))/(105*Sqrt
[c - c/(a*x)]) - (2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(9*(c - c/(a*x))^(3/2)*x^3) + (4*a^2*c^2*(1 - 1/(a^2*x^2))^
(3/2))/(21*(c - c/(a*x))^(3/2)*x^2)

Rule 663

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rule 809

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*
((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rule 885

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d +
e*x)^(m - 1)*(f + g*x)^n*((a + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Dist[n*((e*f + d*g)/(e*(m - n - 1))), In
t[(d + e*x)^m*(f + g*x)^(n - 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g,
0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2
*p] || IntegerQ[n])

Rule 6313

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^n, Subst[Int[(c +
d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {x^3 \sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac {c}{a x}\right )^{3/2} x^3}+\frac {1}{3} (2 a c) \text {Subst}\left (\int \frac {x^2 \sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac {c}{a x}\right )^{3/2} x^3}+\frac {4 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 \left (c-\frac {c}{a x}\right )^{3/2} x^2}-\frac {1}{21} \left (8 a^2 c\right ) \text {Subst}\left (\int \frac {x \sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {16 a^4 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \sqrt {c-\frac {c}{a x}}}-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac {c}{a x}\right )^{3/2} x^3}+\frac {4 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 \left (c-\frac {c}{a x}\right )^{3/2} x^2}-\frac {1}{105} \left (8 a^3 c\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {16 a^4 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {16 a^4 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \sqrt {c-\frac {c}{a x}}}-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 \left (c-\frac {c}{a x}\right )^{3/2} x^3}+\frac {4 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 \left (c-\frac {c}{a x}\right )^{3/2} x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.47 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (-35-5 a x+6 a^2 x^2-8 a^3 x^3+16 a^4 x^4\right )}{315 x^3 (-1+a x)} \]

[In]

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

[Out]

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-35 - 5*a*x + 6*a^2*x^2 - 8*a^3*x^3 + 16*a^4*x^4))/(315*x^3*(-1
+ a*x))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.40

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \left (16 a^{3} x^{3}-24 a^{2} x^{2}+30 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \sqrt {\frac {a x -1}{a x +1}}}\)	\(63\)
default	\(\frac {2 \left (a x +1\right ) \left (16 a^{3} x^{3}-24 a^{2} x^{2}+30 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \sqrt {\frac {a x -1}{a x +1}}}\)	\(63\)
risch	\(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (16 a^{5} x^{5}+8 a^{4} x^{4}-2 a^{3} x^{3}+a^{2} x^{2}-40 a x -35\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x^{4}}\)	\(80\)

[In]

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

[Out]

2/315*(a*x+1)*(16*a^3*x^3-24*a^2*x^2+30*a*x-35)*(c*(a*x-1)/a/x)^(1/2)/x^4/((a*x-1)/(a*x+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \, {\left (16 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2} - 40 \, a x - 35\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{315 \, {\left (a x^{5} - x^{4}\right )}} \]

[In]

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

[Out]

2/315*(16*a^5*x^5 + 8*a^4*x^4 - 2*a^3*x^3 + a^2*x^2 - 40*a*x - 35)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/
(a*x))/(a*x^5 - x^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}\,\left (16\,a^4\,x^4+24\,a^3\,x^3+22\,a^2\,x^2+23\,a\,x-17\right )}{315\,x^4}-\frac {104\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{315\,x^4\,\left (a\,x-1\right )} \]

[In]

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

[Out]

(2*((a*x - 1)/(a*x + 1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2)*(23*a*x + 22*a^2*x^2 + 24*a^3*x^3 + 16*a^4*x^4 - 17
))/(315*x^4) - (104*((a*x - 1)/(a*x + 1))^(1/2)*((c*(a*x - 1))/(a*x))^(1/2))/(315*x^4*(a*x - 1))