3.309 \(\int \frac{x (d+e x)}{(a^2-c^2 x^2)^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a c^3} \]

[Out]

(d + e*x)/(2*c^2*(a^2 - c^2*x^2)) - (e*ArcTanh[(c*x)/a])/(2*a*c^3)

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Rubi [A]  time = 0.0196568, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {778, 208} \[ \frac{d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a c^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(a^2 - c^2*x^2)^2,x]

[Out]

(d + e*x)/(2*c^2*(a^2 - c^2*x^2)) - (e*ArcTanh[(c*x)/a])/(2*a*c^3)

Rule 778

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0222976, size = 42, normalized size = 0.93 \[ \frac{\frac{c (d+e x)}{a^2-c^2 x^2}-\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{a}}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(d + e*x))/(a^2 - c^2*x^2)^2,x]

[Out]

((c*(d + e*x))/(a^2 - c^2*x^2) - (e*ArcTanh[(c*x)/a])/a)/(2*c^3)

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Maple [B]  time = 0.011, size = 96, normalized size = 2.1 \begin{align*} -{\frac{e\ln \left ( cx+a \right ) }{4\,a{c}^{3}}}-{\frac{e}{4\,{c}^{3} \left ( cx+a \right ) }}+{\frac{d}{4\,a{c}^{2} \left ( cx+a \right ) }}+{\frac{e\ln \left ( cx-a \right ) }{4\,a{c}^{3}}}-{\frac{e}{4\,{c}^{3} \left ( cx-a \right ) }}-{\frac{d}{4\,a{c}^{2} \left ( cx-a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)/(-c^2*x^2+a^2)^2,x)

[Out]

-1/4/a/c^3*e*ln(c*x+a)-1/4/c^3/(c*x+a)*e+1/4/a/c^2/(c*x+a)*d+1/4/a/c^3*e*ln(c*x-a)-1/4/c^3/(c*x-a)*e-1/4/a/c^2
/(c*x-a)*d

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Maxima [A]  time = 1.07953, size = 78, normalized size = 1.73 \begin{align*} -\frac{e x + d}{2 \,{\left (c^{4} x^{2} - a^{2} c^{2}\right )}} - \frac{e \log \left (c x + a\right )}{4 \, a c^{3}} + \frac{e \log \left (c x - a\right )}{4 \, a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(e*x + d)/(c^4*x^2 - a^2*c^2) - 1/4*e*log(c*x + a)/(a*c^3) + 1/4*e*log(c*x - a)/(a*c^3)

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Fricas [A]  time = 1.54247, size = 162, normalized size = 3.6 \begin{align*} -\frac{2 \, a c e x + 2 \, a c d +{\left (c^{2} e x^{2} - a^{2} e\right )} \log \left (c x + a\right ) -{\left (c^{2} e x^{2} - a^{2} e\right )} \log \left (c x - a\right )}{4 \,{\left (a c^{5} x^{2} - a^{3} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*c*e*x + 2*a*c*d + (c^2*e*x^2 - a^2*e)*log(c*x + a) - (c^2*e*x^2 - a^2*e)*log(c*x - a))/(a*c^5*x^2 -
a^3*c^3)

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Sympy [A]  time = 0.696135, size = 44, normalized size = 0.98 \begin{align*} - \frac{d + e x}{- 2 a^{2} c^{2} + 2 c^{4} x^{2}} + \frac{e \left (\frac{\log{\left (- \frac{a}{c} + x \right )}}{4} - \frac{\log{\left (\frac{a}{c} + x \right )}}{4}\right )}{a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)/(-c**2*x**2+a**2)**2,x)

[Out]

-(d + e*x)/(-2*a**2*c**2 + 2*c**4*x**2) + e*(log(-a/c + x)/4 - log(a/c + x)/4)/(a*c**3)

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Giac [A]  time = 1.12998, size = 85, normalized size = 1.89 \begin{align*} -\frac{x e + d}{2 \,{\left (c^{2} x^{2} - a^{2}\right )} c^{2}} - \frac{e \log \left ({\left | c x + a \right |}\right )}{4 \, a c^{3}} + \frac{e \log \left ({\left | c x - a \right |}\right )}{4 \, a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x*e + d)/((c^2*x^2 - a^2)*c^2) - 1/4*e*log(abs(c*x + a))/(a*c^3) + 1/4*e*log(abs(c*x - a))/(a*c^3)