3.1014 \(\int e^{\tanh ^{-1}(a x)} x (c-a^2 c x^2)^p \, dx\)

Optimal. Leaf size=96 \[ \frac{1}{3} a x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{2}-p,\frac{5}{2},a^2 x^2\right )-\frac{\sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^2 (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.109345, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6153, 6148, 764, 261, 364} \[ \frac{1}{3} a x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{3}{2},\frac{1}{2}-p;\frac{5}{2};a^2 x^2\right )-\frac{\sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^2 (2 p+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*x*(c - a^2*c*x^2)^p,x]

[Out]

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

Rule 6153

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

Rule 6148

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

Rule 764

Int[(x_)^(m_.)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[f, Int[x^m*(a + c*x^2)^p, x]
, x] + Dist[g, Int[x^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && IntegerQ[m] &&  !IntegerQ[2
*p]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0343258, size = 88, normalized size = 0.92 \[ \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac{1}{3} a x^3 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{2}-p,\frac{5}{2},a^2 x^2\right )-\frac{\left (1-a^2 x^2\right )^{p+\frac{1}{2}}}{2 a^2 \left (p+\frac{1}{2}\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]*x*(c - a^2*c*x^2)^p,x]

[Out]

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

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Maple [F]  time = 0.313, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) x \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c^p*integrate(x^2*e^(p*log(a*x + 1) + p*log(-a*x + 1))/(sqrt(a*x + 1)*sqrt(-a*x + 1)), x) + (a^2*c^p*x^2 - c
^p)*(-a^2*x^2 + 1)^p/(sqrt(-a^2*x^2 + 1)*a^2*(2*p + 1))

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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^{2} c x^{2} + c\right )}^{p} x}{a x - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 13.5348, size = 314, normalized size = 3.27 \begin{align*} - \frac{a a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac{3}{2}\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{3}F_{2}\left (\begin{matrix} \frac{1}{2}, 1, p + \frac{3}{2} \\ p + 1, p + \frac{5}{2} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt{\pi } \Gamma \left (- p - \frac{1}{2}\right ) \Gamma \left (p + 1\right )} - \frac{a a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac{3}{2}\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{3}F_{2}\left (\begin{matrix} 1, - p, - p - \frac{3}{2} \\ \frac{1}{2}, - p - \frac{1}{2} \end{matrix}\middle |{\frac{1}{a^{2} x^{2}}} \right )}}{2 \sqrt{\pi } \Gamma \left (- p - \frac{1}{2}\right ) \Gamma \left (p + 1\right )} - \frac{a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, 1 \\ p + 2 \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt{\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac{a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, - p - 1 \\ \frac{1}{2} \end{matrix}\middle |{\frac{1}{a^{2} x^{2}}} \right )}}{2 \sqrt{\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*a**(2*p)*c**p*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 3/2), (p + 1, p +
5/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - a*a**(2*p)*c**p*x**3*x**(2*p)*
exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1, -p, -p - 3/2), (1/2, -p - 1/2), 1/(a**2*x**2))/(2*sqrt(pi
)*gamma(-p - 1/2)*gamma(p + 1)) - a**(2*p)*c**p*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((
1/2, 1), (p + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*x**2*x**(2
*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p)*g
amma(p + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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