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

Optimal. Leaf size=59 \[ -\frac{(a e+c d) \log (a-c x)}{2 a^3}+\frac{(c d-a e) \log (a+c x)}{2 a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0539944, antiderivative size = 59, 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} \[ -\frac{(a e+c d) \log (a-c x)}{2 a^3}+\frac{(c d-a e) \log (a+c x)}{2 a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0128025, size = 51, normalized size = 0.86 \[ -\frac{e \log \left (a^2-c^2 x^2\right )}{2 a^2}+\frac{c d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 72, normalized size = 1.2 \begin{align*} -{\frac{d}{{a}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( cx+a \right ) e}{2\,{a}^{2}}}+{\frac{\ln \left ( cx+a \right ) cd}{2\,{a}^{3}}}-{\frac{\ln \left ( cx-a \right ) e}{2\,{a}^{2}}}-{\frac{\ln \left ( cx-a \right ) cd}{2\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08224, size = 76, normalized size = 1.29 \begin{align*} \frac{e \log \left (x\right )}{a^{2}} + \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a^{3}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{3}} - \frac{d}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6317, size = 130, normalized size = 2.2 \begin{align*} \frac{2 \, a e x \log \left (x\right ) +{\left (c d - a e\right )} x \log \left (c x + a\right ) -{\left (c d + a e\right )} x \log \left (c x - a\right ) - 2 \, a d}{2 \, a^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.78309, size = 221, normalized size = 3.75 \begin{align*} - \frac{d}{a^{2} x} + \frac{e \log{\left (x \right )}}{a^{2}} - \frac{\left (a e - c d\right ) \log{\left (x + \frac{6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e - c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e - c d\right )^{2} + a c^{2} d^{2} \left (a e - c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e + c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e + c d\right )^{2} + a c^{2} d^{2} \left (a e + c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/(a**2*x) + e*log(x)/a**2 - (a*e - c*d)*log(x + (6*a**4*e**3 - 3*a**3*e**2*(a*e - c*d) + 2*a**2*c**2*d**2*e
- 3*a**2*e*(a*e - c*d)**2 + a*c**2*d**2*(a*e - c*d))/(9*a**2*c**2*d*e**2 - c**4*d**3))/(2*a**3) - (a*e + c*d)*
log(x + (6*a**4*e**3 - 3*a**3*e**2*(a*e + c*d) + 2*a**2*c**2*d**2*e - 3*a**2*e*(a*e + c*d)**2 + a*c**2*d**2*(a
*e + c*d))/(9*a**2*c**2*d*e**2 - c**4*d**3))/(2*a**3)

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Giac [A]  time = 1.14551, size = 100, normalized size = 1.69 \begin{align*} \frac{e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{d}{a^{2} x} + \frac{{\left (c^{2} d - a c e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{3} c} - \frac{{\left (c^{2} d + a c e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{3} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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