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

Optimal. Leaf size=286 \[ -\frac{1115 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{96 x^2 \sqrt{1-\frac{1}{a x}}}+\frac{1115 a^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{64 x \sqrt{1-\frac{1}{a x}}}-\frac{1115 a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{64 \sqrt{1-\frac{1}{a x}}}+\frac{223 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{24 x^3 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x^4 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]

[Out]

(-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^4) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1
- 1/(a*x)]*x^4) + (223*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(24*Sqrt[1 - 1/(a*x)]*x^3) - (1115*a^2*Sqrt[1 + 1
/(a*x)]*Sqrt[c - a*c*x])/(96*Sqrt[1 - 1/(a*x)]*x^2) + (1115*a^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(64*Sqrt[1
- 1/(a*x)]*x) - (1115*a^(7/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(64*Sqrt[1 - 1/(a*x)
])

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Rubi [A]  time = 0.263551, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6176, 6181, 89, 80, 50, 54, 215} \[ -\frac{1115 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{96 x^2 \sqrt{1-\frac{1}{a x}}}+\frac{1115 a^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{64 x \sqrt{1-\frac{1}{a x}}}-\frac{1115 a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{64 \sqrt{1-\frac{1}{a x}}}+\frac{223 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{24 x^3 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x^4 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^4) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1
- 1/(a*x)]*x^4) + (223*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(24*Sqrt[1 - 1/(a*x)]*x^3) - (1115*a^2*Sqrt[1 + 1
/(a*x)]*Sqrt[c - a*c*x])/(96*Sqrt[1 - 1/(a*x)]*x^2) + (1115*a^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(64*Sqrt[1
- 1/(a*x)]*x) - (1115*a^(7/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(64*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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

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

Rule 50

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

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 )^2}{\left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^4}+\frac{\left (2 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{x^{5/2} \left (\frac{27}{2 a^2}-\frac{x}{2 a^3}\right )}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^4}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^4}+\frac{\left (223 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{x^{5/2}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^4}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^4}+\frac{223 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^3}-\frac{\left (1115 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{x^{3/2}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{48 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^4}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^4}+\frac{223 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^3}-\frac{1115 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{96 \sqrt{1-\frac{1}{a x}} x^2}+\frac{\left (1115 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{64 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^4}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^4}+\frac{223 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^3}-\frac{1115 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{96 \sqrt{1-\frac{1}{a x}} x^2}+\frac{1115 a^3 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{64 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (1115 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{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^4}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^4}+\frac{223 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^3}-\frac{1115 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{96 \sqrt{1-\frac{1}{a x}} x^2}+\frac{1115 a^3 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{64 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (1115 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{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^4}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x^4}+\frac{223 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{24 \sqrt{1-\frac{1}{a x}} x^3}-\frac{1115 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{96 \sqrt{1-\frac{1}{a x}} x^2}+\frac{1115 a^3 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{64 \sqrt{1-\frac{1}{a x}} x}-\frac{1115 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}}}\\ \end{align*}

Mathematica [A]  time = 0.0917659, size = 106, normalized size = 0.37 \[ -\frac{\sqrt{c-a c x} \left (-3345 a^4 x^4-1115 a^3 x^3+446 a^2 x^2+\frac{3345 a^{9/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{9/2}}-200 a x+48\right )}{192 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[c - a*c*x]*(48 - 200*a*x + 446*a^2*x^2 - 1115*a^3*x^3 - 3345*a^4*x^4 + (3345*a^(9/2)*Sqrt[1 + 1/(a*x)]*
ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(9/2)))/(192*a*Sqrt[1 - 1/(a^2*x^2)]*x^5)

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Maple [A]  time = 0.173, size = 125, normalized size = 0.4 \begin{align*}{\frac{ax+1}{192\, \left ( ax-1 \right ) ^{2}{x}^{4}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{-c \left ( ax-1 \right ) } \left ( 3345\,\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{4}{a}^{4}\sqrt{-c \left ( ax+1 \right ) }+3345\,{x}^{4}{a}^{4}\sqrt{c}+1115\,{x}^{3}{a}^{3}\sqrt{c}-446\,{x}^{2}{a}^{2}\sqrt{c}+200\,xa\sqrt{c}-48\,\sqrt{c} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(-c*(a*x-1))^(1/2)*(3345*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x^
4*a^4*(-c*(a*x+1))^(1/2)+3345*x^4*a^4*c^(1/2)+1115*x^3*a^3*c^(1/2)-446*x^2*a^2*c^(1/2)+200*x*a*c^(1/2)-48*c^(1
/2))/c^(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} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73291, size = 687, normalized size = 2.4 \begin{align*} \left [\frac{3345 \,{\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 (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, 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{3345 \,{\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 (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, 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((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="fricas")

[Out]

[1/384*(3345*(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*(3345*a^4*x^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 200*a*x - 48)*sqrt(
-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^5 - x^4), -1/192*(3345*(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)) - (3345*a^4*x^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 200*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((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**5,x)

[Out]

Timed out

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Giac [A]  time = 1.29708, size = 182, normalized size = 0.64 \begin{align*} \frac{1}{192} \, a^{4} c^{3}{\left (\frac{3345 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{7}{2}}} + \frac{1536}{\sqrt{-a c x - c} c^{3}} - \frac{1809 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 6121 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 7063 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} - 2799 \, \sqrt{-a c x - c} c^{3}}{a^{4} c^{7} x^{4}}\right )}{\left | c \right |} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="giac")

[Out]

1/192*a^4*c^3*(3345*arctan(sqrt(-a*c*x - c)/sqrt(c))/c^(7/2) + 1536/(sqrt(-a*c*x - c)*c^3) - (1809*(a*c*x + c)
^3*sqrt(-a*c*x - c) - 6121*(a*c*x + c)^2*sqrt(-a*c*x - c)*c - 7063*(-a*c*x - c)^(3/2)*c^2 - 2799*sqrt(-a*c*x -
c)*c^3)/(a^4*c^7*x^4))*abs(c)