3.304 \(\int \frac{d+e x}{x^3 (a^2-c^2 x^2)} \, dx\)

Optimal. Leaf size=75 \[ \frac{c^2 d \log (x)}{a^4}-\frac{c (a e+c d) \log (a-c x)}{2 a^4}-\frac{c (c d-a e) \log (a+c x)}{2 a^4}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

[Out]

-d/(2*a^2*x^2) - e/(a^2*x) + (c^2*d*Log[x])/a^4 - (c*(c*d + a*e)*Log[a - c*x])/(2*a^4) - (c*(c*d - a*e)*Log[a
+ c*x])/(2*a^4)

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Rubi [A]  time = 0.0661303, antiderivative size = 75, 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} \[ \frac{c^2 d \log (x)}{a^4}-\frac{c (a e+c d) \log (a-c x)}{2 a^4}-\frac{c (c d-a e) \log (a+c x)}{2 a^4}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(2*a^2*x^2) - e/(a^2*x) + (c^2*d*Log[x])/a^4 - (c*(c*d + a*e)*Log[a - c*x])/(2*a^4) - (c*(c*d - a*e)*Log[a
+ c*x])/(2*a^4)

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^3 \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac{d}{a^2 x^3}+\frac{e}{a^2 x^2}+\frac{c^2 d}{a^4 x}+\frac{c^2 (c d+a e)}{2 a^4 (a-c x)}+\frac{c^2 (-c d+a e)}{2 a^4 (a+c x)}\right ) \, dx\\ &=-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}+\frac{c^2 d \log (x)}{a^4}-\frac{c (c d+a e) \log (a-c x)}{2 a^4}-\frac{c (c d-a e) \log (a+c x)}{2 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0173343, size = 68, normalized size = 0.91 \[ -\frac{c^2 d \log \left (a^2-c^2 x^2\right )}{2 a^4}+\frac{c^2 d \log (x)}{a^4}+\frac{c e \tanh ^{-1}\left (\frac{c x}{a}\right )}{a^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(2*a^2*x^2) - e/(a^2*x) + (c*e*ArcTanh[(c*x)/a])/a^3 + (c^2*d*Log[x])/a^4 - (c^2*d*Log[a^2 - c^2*x^2])/(2*a
^4)

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Maple [A]  time = 0.01, size = 90, normalized size = 1.2 \begin{align*} -{\frac{d}{2\,{a}^{2}{x}^{2}}}-{\frac{e}{{a}^{2}x}}+{\frac{{c}^{2}d\ln \left ( x \right ) }{{a}^{4}}}+{\frac{c\ln \left ( cx+a \right ) e}{2\,{a}^{3}}}-{\frac{{c}^{2}\ln \left ( cx+a \right ) d}{2\,{a}^{4}}}-{\frac{c\ln \left ( cx-a \right ) e}{2\,{a}^{3}}}-{\frac{{c}^{2}\ln \left ( cx-a \right ) d}{2\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/x^3/(-c^2*x^2+a^2),x)

[Out]

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

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Maxima [A]  time = 1.03212, size = 95, normalized size = 1.27 \begin{align*} \frac{c^{2} d \log \left (x\right )}{a^{4}} - \frac{{\left (c^{2} d - a c e\right )} \log \left (c x + a\right )}{2 \, a^{4}} - \frac{{\left (c^{2} d + a c e\right )} \log \left (c x - a\right )}{2 \, a^{4}} - \frac{2 \, e x + d}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-c^2*x^2+a^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6253, size = 170, normalized size = 2.27 \begin{align*} \frac{2 \, c^{2} d x^{2} \log \left (x\right ) - 2 \, a^{2} e x -{\left (c^{2} d - a c e\right )} x^{2} \log \left (c x + a\right ) -{\left (c^{2} d + a c e\right )} x^{2} \log \left (c x - a\right ) - a^{2} d}{2 \, a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-c^2*x^2+a^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.76543, size = 236, normalized size = 3.15 \begin{align*} - \frac{d + 2 e x}{2 a^{2} x^{2}} + \frac{c^{2} d \log{\left (x \right )}}{a^{4}} + \frac{c \left (a e - c d\right ) \log{\left (x + \frac{- 2 a^{2} c^{2} d e^{2} + a^{2} c e^{2} \left (a e - c d\right ) - 6 c^{4} d^{3} - 3 c^{3} d^{2} \left (a e - c d\right ) + 3 c^{2} d \left (a e - c d\right )^{2}}{a^{2} c^{2} e^{3} - 9 c^{4} d^{2} e} \right )}}{2 a^{4}} - \frac{c \left (a e + c d\right ) \log{\left (x + \frac{- 2 a^{2} c^{2} d e^{2} - a^{2} c e^{2} \left (a e + c d\right ) - 6 c^{4} d^{3} + 3 c^{3} d^{2} \left (a e + c d\right ) + 3 c^{2} d \left (a e + c d\right )^{2}}{a^{2} c^{2} e^{3} - 9 c^{4} d^{2} e} \right )}}{2 a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.157, size = 126, normalized size = 1.68 \begin{align*} \frac{c^{2} d \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (c^{3} d - a c^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{4} c} - \frac{{\left (c^{3} d + a c^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{4} c} - \frac{2 \, a^{2} x e + a^{2} d}{2 \, a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-c^2*x^2+a^2),x, algorithm="giac")

[Out]

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