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

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

[Out]

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

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Rubi [A]  time = 0.0682523, 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} \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (\frac{1}{2} (-n-1),-\frac{n}{2};\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{a c (n+1) \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0243085, size = 78, normalized size = 1. \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{1}{2},-\frac{n}{2},\frac{1}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a c (n+1) \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.212, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

int(exp(n*arctanh(a*x))/(-a*c*x+c)^(3/2),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{3}{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)^(3/2),x, algorithm="maxima")

[Out]

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/(-a*c*x + c)^(3/2), 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^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, 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)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{3}{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)^(3/2),x, algorithm="giac")

[Out]

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