3.216 \(\int e^{-3 \tanh ^{-1}(a x)} (c-a c x)^p \, dx\)

Optimal. Leaf size=65 \[ -\frac{(1-a x)^{3/2} (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (\frac{3}{2},p+\frac{5}{2},p+\frac{7}{2},\frac{1}{2} (1-a x)\right )}{\sqrt{2} a c (2 p+5)} \]

[Out]

-(((1 - a*x)^(3/2)*(c - a*c*x)^(1 + p)*Hypergeometric2F1[3/2, 5/2 + p, 7/2 + p, (1 - a*x)/2])/(Sqrt[2]*a*c*(5
+ 2*p)))

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Rubi [A]  time = 0.0521735, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6130, 23, 69} \[ -\frac{(1-a x)^{3/2} (c-a c x)^{p+1} \, _2F_1\left (\frac{3}{2},p+\frac{5}{2};p+\frac{7}{2};\frac{1}{2} (1-a x)\right )}{\sqrt{2} a c (2 p+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 - a*x)^(3/2)*(c - a*c*x)^(1 + p)*Hypergeometric2F1[3/2, 5/2 + p, 7/2 + p, (1 - a*x)/2])/(Sqrt[2]*a*c*(5
+ 2*p)))

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

Mathematica [A]  time = 0.0275208, size = 60, normalized size = 0.92 \[ -\frac{(1-a x)^{5/2} (c-a c x)^p \text{Hypergeometric2F1}\left (\frac{3}{2},p+\frac{5}{2},p+\frac{7}{2},\frac{1}{2}-\frac{a x}{2}\right )}{\sqrt{2} a (2 p+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 - a*x)^(5/2)*(c - a*c*x)^p*Hypergeometric2F1[3/2, 5/2 + p, 7/2 + p, 1/2 - (a*x)/2])/(Sqrt[2]*a*(5 + 2*p)
))

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Maple [F]  time = 0.474, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -acx+c \right ) ^{p}}{ \left ( ax+1 \right ) ^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

[Out]

int((-a*c*x+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(a*x - 1))**p*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^p/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

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