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

Optimal. Leaf size=78 \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (\frac{1}{2} (-n-3),-\frac{n}{2},\frac{1}{2} (-n-1),\frac{1}{2} (1-a x)\right )}{a c (n+3) (c-a c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0671078, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6130, 23, 69} \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (\frac{1}{2} (-n-3),-\frac{n}{2};\frac{1}{2} (-n-1);\frac{1}{2} (1-a x)\right )}{a c (n+3) (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6130

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

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

Mathematica [A]  time = 0.0316857, size = 80, normalized size = 1.03 \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-\frac{n}{2}-1} \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{3}{2},-\frac{n}{2},-\frac{n}{2}-\frac{1}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a c^2 (n+3) \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.219, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(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)