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

Optimal. Leaf size=56 \[ -\frac{(a e+c d) \log (a-c x)}{2 a^2 c}-\frac{(c d-a e) \log (a+c x)}{2 a^2 c}+\frac{d \log (x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0488401, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {801} \[ -\frac{(a e+c d) \log (a-c x)}{2 a^2 c}-\frac{(c d-a e) \log (a+c x)}{2 a^2 c}+\frac{d \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x*(a^2 - c^2*x^2)),x]

[Out]

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

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

Mathematica [A]  time = 0.0129546, size = 44, normalized size = 0.79 \[ -\frac{d \log \left (a^2-c^2 x^2\right )}{2 a^2}+\frac{d \log (x)}{a^2}+\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 67, normalized size = 1.2 \begin{align*}{\frac{d\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( cx+a \right ) e}{2\,ac}}-{\frac{\ln \left ( cx+a \right ) d}{2\,{a}^{2}}}-{\frac{\ln \left ( cx-a \right ) e}{2\,ac}}-{\frac{\ln \left ( cx-a \right ) d}{2\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09237, size = 72, normalized size = 1.29 \begin{align*} \frac{d \log \left (x\right )}{a^{2}} - \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a^{2} c} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57148, size = 111, normalized size = 1.98 \begin{align*} \frac{2 \, c d \log \left (x\right ) -{\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 \, a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.65768, size = 194, normalized size = 3.46 \begin{align*} \frac{d \log{\left (x \right )}}{a^{2}} + \frac{\left (a e - c d\right ) \log{\left (x + \frac{- 2 a^{2} d e^{2} + \frac{a^{2} e^{2} \left (a e - c d\right )}{c} - 6 c^{2} d^{3} - 3 c d^{2} \left (a e - c d\right ) + 3 d \left (a e - c d\right )^{2}}{a^{2} e^{3} - 9 c^{2} d^{2} e} \right )}}{2 a^{2} c} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{- 2 a^{2} d e^{2} - \frac{a^{2} e^{2} \left (a e + c d\right )}{c} - 6 c^{2} d^{3} + 3 c d^{2} \left (a e + c d\right ) + 3 d \left (a e + c d\right )^{2}}{a^{2} e^{3} - 9 c^{2} d^{2} e} \right )}}{2 a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(x)/a**2 + (a*e - c*d)*log(x + (-2*a**2*d*e**2 + a**2*e**2*(a*e - c*d)/c - 6*c**2*d**3 - 3*c*d**2*(a*e -
c*d) + 3*d*(a*e - c*d)**2)/(a**2*e**3 - 9*c**2*d**2*e))/(2*a**2*c) - (a*e + c*d)*log(x + (-2*a**2*d*e**2 - a**
2*e**2*(a*e + c*d)/c - 6*c**2*d**3 + 3*c*d**2*(a*e + c*d) + 3*d*(a*e + c*d)**2)/(a**2*e**3 - 9*c**2*d**2*e))/(
2*a**2*c)

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Giac [A]  time = 1.15739, size = 86, normalized size = 1.54 \begin{align*} \frac{d \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{{\left (c^{2} d - a c e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{2} c^{2}} - \frac{{\left (c^{2} d + a c e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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