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

Optimal. Leaf size=50 \[ -\frac{(a e+c d) \log (a-c x)}{2 c^3}-\frac{(c d-a e) \log (a+c x)}{2 c^3}-\frac{e x}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.0384204, antiderivative size = 50, 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 = {774, 633, 31} \[ -\frac{(a e+c d) \log (a-c x)}{2 c^3}-\frac{(c d-a e) \log (a+c x)}{2 c^3}-\frac{e x}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(a^2 - c^2*x^2),x]

[Out]

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

Rule 774

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

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0092586, size = 42, normalized size = 0.84 \[ -\frac{d \log \left (a^2-c^2 x^2\right )}{2 c^2}+\frac{a e \tanh ^{-1}\left (\frac{c x}{a}\right )}{c^3}-\frac{e x}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 63, normalized size = 1.3 \begin{align*} -{\frac{ex}{{c}^{2}}}+{\frac{\ln \left ( cx+a \right ) ae}{2\,{c}^{3}}}-{\frac{\ln \left ( cx+a \right ) d}{2\,{c}^{2}}}-{\frac{\ln \left ( cx-a \right ) ae}{2\,{c}^{3}}}-{\frac{\ln \left ( cx-a \right ) d}{2\,{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07019, size = 63, normalized size = 1.26 \begin{align*} -\frac{e x}{c^{2}} - \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, c^{3}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.5489, size = 100, normalized size = 2. \begin{align*} -\frac{2 \, c e x +{\left (c d - a e\right )} \log \left (c x + a\right ) +{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.633854, size = 60, normalized size = 1.2 \begin{align*} - \frac{e x}{c^{2}} + \frac{\left (a e - c d\right ) \log{\left (x + \frac{d + \frac{a e - c d}{c}}{e} \right )}}{2 c^{3}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{d - \frac{a e + c d}{c}}{e} \right )}}{2 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),x)

[Out]

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

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Giac [A]  time = 1.14182, size = 70, normalized size = 1.4 \begin{align*} -\frac{x e}{c^{2}} - \frac{{\left (c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{3}} - \frac{{\left (c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, 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),x, algorithm="giac")

[Out]

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