3.377 \(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\)

Optimal. Leaf size=167 \[ \frac {a x^2 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(n+3) (c-a c x)^{5/2}}-\frac {a x^2 \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{(n+3) (c-a c x)^{5/2}} \]

[Out]

-a*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)*x^2/(3+n)/(-a*c*x+c)^(5/2)+a*((a-1/x)/(a+1/x))^(3/2+1/2*n)*(1-1/a/x
)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)*x^2*hypergeom([1/2, 3/2+1/2*n],[3/2],2/(a+1/x)/x)/(3+n)/(-a*c*x+c)^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6176, 6181, 94, 132} \[ \frac {a x^2 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(n+3) (c-a c x)^{5/2}}-\frac {a x^2 \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{(n+3) (c-a c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/((3 + n)*(c - a*c*x)^(5/2))) + (a*((a - x^(-1))/
(a + x^(-1)))^((3 + n)/2)*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2*Hypergeometric2F1[1/2, (3 +
n)/2, 3/2, 2/((a + x^(-1))*x)])/((3 + n)*(c - a*c*x)^(5/2))

Rule 94

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

Rule 132

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c -
a*d)*(e + f*x)))])/(((b*e - a*f)*(m + 1))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n), x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

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^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}} \, dx}{(c-a c x)^{5/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \sqrt {x} \left (1-\frac {x}{a}\right )^{-\frac {5}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(3+n) (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{\sqrt {x}} \, dx,x,\frac {1}{x}\right )}{2 (3+n) \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=-\frac {a \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2}{(3+n) (c-a c x)^{5/2}}+\frac {a \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3+n}{2}} \left (1-\frac {1}{a x}\right )^{\frac {2-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x^2 \, _2F_1\left (\frac {1}{2},\frac {3+n}{2};\frac {3}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(3+n) (c-a c x)^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 117, normalized size = 0.70 \[ \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \left ((a x-1) \left (\frac {a x-1}{a x+1}\right )^{\frac {n+1}{2}} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2};\frac {3}{2};\frac {2}{a x+1}\right )-a x-1\right )}{a c^2 (n+3) (a x-1) \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 1/(a*x))^(n/2)*(-1 - a*x + (-1 + a*x)*((-1 + a*x)/(1 + a*x))^((1 + n)/2)*Hypergeometric2F1[1/2, (3 + n)/
2, 3/2, 2/(1 + a*x)]))/(a*c^2*(3 + n)*(1 - 1/(a*x))^(n/2)*(-1 + a*x)*Sqrt[c - a*c*x])

________________________________________________________________________________________

fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a*c*x + c)*((a*x - 1)/(a*x + 1))^(1/2*n)/(a^3*c^3*x^3 - 3*a^2*c^3*x^2 + 3*a*c^3*x - c^3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\left (-a c x +c \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))/(-a*c*x+c)^(5/2),x)

[Out]

int(exp(n*arccoth(a*x))/(-a*c*x+c)^(5/2),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a\,c\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*acoth(a*x))/(c - a*c*x)^(5/2),x)

[Out]

int(exp(n*acoth(a*x))/(c - a*c*x)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))/(-a*c*x+c)**(5/2),x)

[Out]

Integral(exp(n*acoth(a*x))/(-c*(a*x - 1))**(5/2), x)

________________________________________________________________________________________