3.7.27 \(\int \frac {e^{n \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^{3/2}} \, dx\) [627]

Optimal. Leaf size=56 \[ \frac {2 x (1-a x)^{3/2} F_1\left (\frac {5}{2};\frac {3+n}{2},-\frac {n}{2};\frac {7}{2};a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}} \]

[Out]

2/5*x*(-a*x+1)^(3/2)*AppellF1(5/2,3/2+1/2*n,-1/2*n,7/2,a*x,-a*x)/(c-c/a/x)^(3/2)

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Rubi [A]
time = 0.11, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6269, 6264, 138} \begin {gather*} \frac {2 x (1-a x)^{3/2} F_1\left (\frac {5}{2};\frac {n+3}{2},-\frac {n}{2};\frac {7}{2};a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])/(c - c/(a*x))^(3/2),x]

[Out]

(2*x*(1 - a*x)^(3/2)*AppellF1[5/2, (3 + n)/2, -1/2*n, 7/2, a*x, -(a*x)])/(5*(c - c/(a*x))^(3/2))

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 6264

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

Rule 6269

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

Rubi steps

\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\frac {(1-a x)^{3/2} \int \frac {e^{n \tanh ^{-1}(a x)} x^{3/2}}{(1-a x)^{3/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {(1-a x)^{3/2} \int x^{3/2} (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{n/2} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {2 x (1-a x)^{3/2} F_1\left (\frac {5}{2};\frac {3+n}{2},-\frac {n}{2};\frac {7}{2};a x,-a x\right )}{5 \left (c-\frac {c}{a x}\right )^{3/2}}\\ \end {align*}

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Mathematica [F]
time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^(n*ArcTanh[a*x])/(c - c/(a*x))^(3/2),x]

[Out]

$Aborted

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x)

[Out]

int(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/(c-c/a/x)**(3/2),x)

[Out]

Integral(exp(n*atanh(a*x))/(-c*(-1 + 1/(a*x)))**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*atanh(a*x))/(c - c/(a*x))^(3/2),x)

[Out]

int(exp(n*atanh(a*x))/(c - c/(a*x))^(3/2), x)

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