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

Optimal. Leaf size=76 \[ \frac{c x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{m+1}+\frac{a c x^{m+2} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{m+2} \]

[Out]

(c*x^(1 + m)*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*c*x^(2 + m)*Hypergeometric2F
1[-1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)

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Rubi [A]  time = 0.0789355, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6148, 808, 364} \[ \frac{c x^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a c x^{m+2} \, _2F_1\left (-\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^(1 + m)*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*c*x^(2 + m)*Hypergeometric2F
1[-1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)

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 808

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

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^m \left (c-a^2 c x^2\right ) \, dx &=c \int x^m (1+a x) \sqrt{1-a^2 x^2} \, dx\\ &=c \int x^m \sqrt{1-a^2 x^2} \, dx+(a c) \int x^{1+m} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{c x^{1+m} \, _2F_1\left (-\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{1+m}+\frac{a c x^{2+m} \, _2F_1\left (-\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{2+m}\\ \end{align*}

Mathematica [A]  time = 0.0418694, size = 72, normalized size = 0.95 \[ c x^{m+1} \left (\frac{a x \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,a^2 x^2\right )}{m+2}+\frac{\text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.185, size = 143, normalized size = 1.9 \begin{align*} -{\frac{{a}^{3}c{x}^{4+m}}{4+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},2+{\frac{m}{2}};\,3+{\frac{m}{2}};\,{a}^{2}{x}^{2})}}+{\frac{ac{x}^{2+m}}{2+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},1+{\frac{m}{2}};\,2+{\frac{m}{2}};\,{a}^{2}{x}^{2})}}-{\frac{{a}^{2}c{x}^{3+m}}{3+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{2}}+{\frac{m}{2}};\,{\frac{5}{2}}+{\frac{m}{2}};\,{a}^{2}{x}^{2})}}+{\frac{c{x}^{1+m}}{1+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{1}{2}}+{\frac{m}{2}};\,{\frac{3}{2}}+{\frac{m}{2}};\,{a}^{2}{x}^{2})}} \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^m*(-a^2*c*x^2+c),x)

[Out]

-a^3*c/(4+m)*x^(4+m)*hypergeom([1/2,2+1/2*m],[3+1/2*m],a^2*x^2)+a*c/(2+m)*x^(2+m)*hypergeom([1/2,1+1/2*m],[2+1
/2*m],a^2*x^2)-c*a^2/(3+m)*x^(3+m)*hypergeom([1/2,3/2+1/2*m],[5/2+1/2*m],a^2*x^2)+c/(1+m)*x^(1+m)*hypergeom([1
/2,1/2+1/2*m],[3/2+1/2*m],a^2*x^2)

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

[Out]

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

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

[Out]

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

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Sympy [C]  time = 4.91502, size = 104, normalized size = 1.37 \begin{align*} \frac{a c x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{c x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\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**m*(-a**2*c*x**2+c),x)

[Out]

a*c*x**2*x**m*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 2)
) + c*x*x**m*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2
+ 3/2))

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

[Out]

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