3.1350 \(\int \frac{e^{n \tanh ^{-1}(a x)} x}{(c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=133 \[ \frac{2 n (n-a x) e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{n (n-3 a x) e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

E^(n*ArcTanh[a*x])/(3*a^2*c*(c - a^2*c*x^2)^(3/2)) + (E^(n*ArcTanh[a*x])*n*(n - 3*a*x))/(3*a^2*c*(9 - n^2)*(c
- a^2*c*x^2)^(3/2)) + (2*E^(n*ArcTanh[a*x])*n*(n - a*x))/(a^2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.213821, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6145, 6136, 6135} \[ \frac{2 n (n-a x) e^{n \tanh ^{-1}(a x)}}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{n (n-3 a x) e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(n*ArcTanh[a*x])*x)/(c - a^2*c*x^2)^(5/2),x]

[Out]

E^(n*ArcTanh[a*x])/(3*a^2*c*(c - a^2*c*x^2)^(3/2)) + (E^(n*ArcTanh[a*x])*n*(n - 3*a*x))/(3*a^2*c*(9 - n^2)*(c
- a^2*c*x^2)^(3/2)) + (2*E^(n*ArcTanh[a*x])*n*(n - a*x))/(a^2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

Rule 6145

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

Rule 6136

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n + 2*a*(p + 1)*x)*(c + d*x^2
)^(p + 1)*E^(n*ArcTanh[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^2
)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p
, -1] &&  !IntegerQ[n] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]

Rule 6135

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((n - a*x)*E^(n*ArcTanh[a*x]))
/(a*c*(n^2 - 1)*Sqrt[c + d*x^2]), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}}-\frac{n \int \frac{e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{3 a}\\ &=\frac{e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{e^{n \tanh ^{-1}(a x)} n (n-3 a x)}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{(2 n) \int \frac{e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{a c \left (9-n^2\right )}\\ &=\frac{e^{n \tanh ^{-1}(a x)}}{3 a^2 c \left (c-a^2 c x^2\right )^{3/2}}+\frac{e^{n \tanh ^{-1}(a x)} n (n-3 a x)}{3 a^2 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 e^{n \tanh ^{-1}(a x)} n (n-a x)}{a^2 c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.165181, size = 114, normalized size = 0.86 \[ \frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2} (-n-3)} (a x+1)^{\frac{n-3}{2}} \left (-n^2 \left (2 a^2 x^2+1\right )+a n x \left (2 a^2 x^2-3\right )+a n^3 x+3\right )}{a^2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(n*ArcTanh[a*x])*x)/(c - a^2*c*x^2)^(5/2),x]

[Out]

((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(3 + a*n^3*x + a*n*x*(-3 + 2*a^2*x^2) - n^2*(
1 + 2*a^2*x^2)))/(a^2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.03, size = 86, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,{a}^{3}n{x}^{3}-2\,{n}^{2}{x}^{2}{a}^{2}+a{n}^{3}x-3\,nax-{n}^{2}+3 \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{a}^{2} \left ({n}^{4}-10\,{n}^{2}+9 \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^(5/2),x)

[Out]

-(a*x-1)*(a*x+1)*(2*a^3*n*x^3-2*a^2*n^2*x^2+a*n^3*x-3*a*n*x-n^2+3)*exp(n*arctanh(a*x))/a^2/(n^4-10*n^2+9)/(-a^
2*c*x^2+c)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(x*((a*x + 1)/(a*x - 1))^(1/2*n)/(-a^2*c*x^2 + c)^(5/2), x)

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Fricas [A]  time = 2.10473, size = 344, normalized size = 2.59 \begin{align*} \frac{{\left (2 \, a^{3} n x^{3} - 2 \, a^{2} n^{2} x^{2} - n^{2} +{\left (a n^{3} - 3 \, a n\right )} x + 3\right )} \sqrt{-a^{2} c x^{2} + c} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c^{3} n^{4} - 10 \, a^{2} c^{3} n^{2} + 9 \, a^{2} c^{3} +{\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{4} - 2 \,{\left (a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3}\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

(2*a^3*n*x^3 - 2*a^2*n^2*x^2 - n^2 + (a*n^3 - 3*a*n)*x + 3)*sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)
/(a^2*c^3*n^4 - 10*a^2*c^3*n^2 + 9*a^2*c^3 + (a^6*c^3*n^4 - 10*a^6*c^3*n^2 + 9*a^6*c^3)*x^4 - 2*(a^4*c^3*n^4 -
10*a^4*c^3*n^2 + 9*a^4*c^3)*x^2)

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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))*x/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*x/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(x*((a*x + 1)/(a*x - 1))^(1/2*n)/(-a^2*c*x^2 + c)^(5/2), x)