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

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

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 286, normalized size of antiderivative = 1.00, 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 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 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 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 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 )^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*}

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Mathematica [A]  time = 0.10, size = 106, normalized size = 0.37 \[ -\frac {\sqrt {c-a c x} \left (\frac {3345 a^{9/2} \sqrt {\frac {1}{a x}+1} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{9/2}}-3345 a^4 x^4-1115 a^3 x^3+446 a^2 x^2-200 a x+48\right )}{192 a x^5 \sqrt {1-\frac {1}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/192*(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)))/(a*Sqrt[1 - 1/(a^2*x^2)]*x^5)

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fricas [A]  time = 0.60, size = 294, normalized size = 1.03 \[ \left [\frac {3345 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{384 \, {\left (a x^{5} - x^{4}\right )}}, -\frac {3345 \, {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a x^{5} - x^{4}\right )}}\right ] \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(a*x+1)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.07, size = 125, normalized size = 0.44 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (3345 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) x^{4} a^{4} \sqrt {-c \left (a x +1\right )}+3345 x^{4} a^{4} \sqrt {c}+1115 x^{3} a^{3} \sqrt {c}-446 x^{2} a^{2} \sqrt {c}+200 x a \sqrt {c}-48 \sqrt {c}\right )}{192 \left (a x -1\right )^{2} \sqrt {c}\, x^{4}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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