3.591 \(\int e^{-\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^m \, dx\)

Optimal. Leaf size=116 \[ \frac{(4 m+3) x^{m+1} \sqrt{c-\frac{c}{a x}} \text{Hypergeometric2F1}\left (\frac{1}{2},m+\frac{1}{2},m+\frac{3}{2},-a x\right )}{(m+1) (2 m+1) \sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} x^{m+1} \sqrt{c-\frac{c}{a x}}}{(m+1) (1-a x)} \]

[Out]

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

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Rubi [A]  time = 0.293454, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6134, 6128, 881, 848, 64} \[ \frac{(4 m+3) x^{m+1} \sqrt{c-\frac{c}{a x}} \, _2F_1\left (\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};-a x\right )}{(m+1) (2 m+1) \sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} x^{m+1} \sqrt{c-\frac{c}{a x}}}{(m+1) (1-a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6134

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

Rule 6128

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

Rule 881

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e^2*(d +
e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] - Dist[(e*f*(p + 1) - d*g*(2*n + p
+ 3))/(g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m,
n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p - 1, 0] &&  !LtQ[n, -1]
&& IntegerQ[2*p]

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

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

Mathematica [A]  time = 0.0561967, size = 87, normalized size = 0.75 \[ -\frac{2 x^{m+1} \sqrt{c-\frac{c}{a x}} \left (a (4 m+3) x \text{Hypergeometric2F1}\left (\frac{1}{2},m+\frac{3}{2},m+\frac{5}{2},-a x\right )-(2 m+3) \sqrt{a x+1}\right )}{\left (4 m^2+8 m+3\right ) \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[c - c/(a*x)]*x^(1 + m)*(-((3 + 2*m)*Sqrt[1 + a*x]) + a*(3 + 4*m)*x*Hypergeometric2F1[1/2, 3/2 + m, 5/
2 + m, -(a*x)]))/((3 + 8*m + 4*m^2)*Sqrt[1 - a*x])

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Maple [F]  time = 0.381, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ax+1}\sqrt{c-{\frac{c}{ax}}}\sqrt{-{a}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}} x^{m}}{a x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a^2*x^2 + 1)*sqrt(c - c/(a*x))*x^m/(a*x + 1), 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} x^{m} \sqrt{\frac{a c x - c}{a x}}}{a x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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