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

Optimal. Leaf size=81 \[ -\frac{2^{\frac{n}{2}+1} (c-a c x)^{5/2} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (\frac{5-n}{2},-\frac{n}{2},\frac{7-n}{2},\frac{1}{2} (1-a x)\right )}{a c (5-n)} \]

[Out]

-((2^(1 + n/2)*(c - a*c*x)^(5/2)*Hypergeometric2F1[(5 - n)/2, -n/2, (7 - n)/2, (1 - a*x)/2])/(a*c*(5 - n)*(1 -
a*x)^(n/2)))

________________________________________________________________________________________

Rubi [A]  time = 0.0679431, antiderivative size = 81, 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} \[ -\frac{2^{\frac{n}{2}+1} (c-a c x)^{5/2} (1-a x)^{-n/2} \, _2F_1\left (\frac{5-n}{2},-\frac{n}{2};\frac{7-n}{2};\frac{1}{2} (1-a x)\right )}{a c (5-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(1 + n/2)*(c - a*c*x)^(5/2)*Hypergeometric2F1[(5 - n)/2, -n/2, (7 - n)/2, (1 - a*x)/2])/(a*c*(5 - n)*(1 -
a*x)^(n/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 e^{n \tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\int (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} (c-a c x)^{5/2} \, _2F_1\left (\frac{5-n}{2},-\frac{n}{2};\frac{7-n}{2};\frac{1}{2} (1-a x)\right )}{a c (5-n)}\\ \end{align*}

Mathematica [A]  time = 0.0353, size = 78, normalized size = 0.96 \[ \frac{c 2^{\frac{n}{2}+1} \sqrt{c-a c x} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (\frac{5}{2}-\frac{n}{2},-\frac{n}{2},\frac{7}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a (n-5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(1 + n/2)*c*(1 - a*x)^(2 - n/2)*Sqrt[c - a*c*x]*Hypergeometric2F1[5/2 - n/2, -n/2, 7/2 - n/2, 1/2 - (a*x)/2
])/(a*(-5 + n))

________________________________________________________________________________________

Maple [F]  time = 0.23, 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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a c x + c\right )}^{\frac{3}{2}} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{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*c*x + c)^(3/2)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a c x - c\right )} \sqrt{-a c x + c} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}, 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(-(a*c*x - c)*sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

________________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a c x + c\right )}^{\frac{3}{2}} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}\,{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*c*x + c)^(3/2)*((a*x + 1)/(a*x - 1))^(1/2*n), x)