3.1304 \(\int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^{9/8}} \, dx\)

Optimal. Leaf size=41 \[ \frac{4 (2-a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{3 a^3 c \sqrt [8]{c-a^2 c x^2}} \]

[Out]

(4*E^(ArcTanh[a*x]/2)*(2 - a*x))/(3*a^3*c*(c - a^2*c*x^2)^(1/8))

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Rubi [A]  time = 0.119413, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {6146} \[ \frac{4 (2-a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{3 a^3 c \sqrt [8]{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(ArcTanh[a*x]/2)*x^2)/(c - a^2*c*x^2)^(9/8),x]

[Out]

(4*E^(ArcTanh[a*x]/2)*(2 - a*x))/(3*a^3*c*(c - a^2*c*x^2)^(1/8))

Rule 6146

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

Rubi steps

\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{9/8}} \, dx &=\frac{4 e^{\frac{1}{2} \tanh ^{-1}(a x)} (2-a x)}{3 a^3 c \sqrt [8]{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0329122, size = 63, normalized size = 1.54 \[ -\frac{4 (a x-2) \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2}}{3 a^3 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(ArcTanh[a*x]/2)*x^2)/(c - a^2*c*x^2)^(9/8),x]

[Out]

(-4*(-2 + a*x)*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8))/(3*a^3*c*(1 - a*x)^(3/8)*(c - a^2*c*x^2)^(1/8))

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Maple [A]  time = 0.028, size = 54, normalized size = 1.3 \begin{align*}{\frac{ \left ( 4\,ax-4 \right ) \left ( ax+1 \right ) \left ( ax-2 \right ) }{3\,{a}^{3}}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{8}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^2/(-a^2*c*x^2+c)^(9/8),x)

[Out]

4/3*(a*x-1)*(a*x+1)*(a*x-2)*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/a^3/(-a^2*c*x^2+c)^(9/8)

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Maxima [A]  time = 2.53288, size = 38, normalized size = 0.93 \begin{align*} -\frac{4 \,{\left (a x + 1\right )}^{\frac{1}{8}}{\left (a x - 2\right )}}{3 \,{\left (-a x + 1\right )}^{\frac{3}{8}} a^{3} c^{\frac{9}{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^2/(-a^2*c*x^2+c)^(9/8),x, algorithm="maxima")

[Out]

-4/3*(a*x + 1)^(1/8)*(a*x - 2)/((-a*x + 1)^(3/8)*a^3*c^(9/8))

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Fricas [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(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*x^2/(-a^2*c*x^2+c)^(9/8),x, algorithm="fricas")

[Out]

Timed out

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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(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x**2/(-a**2*c*x**2+c)**(9/8),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{8}}}\,{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))^(1/2)*x^2/(-a^2*c*x^2+c)^(9/8),x, algorithm="giac")

[Out]

integrate(x^2*sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/(-a^2*c*x^2 + c)^(9/8), x)