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

Optimal. Leaf size=84 \[ \frac{d+e x}{2 a^2 \left (a^2-c^2 x^2\right )}-\frac{(a e+2 c d) \log (a-c x)}{4 a^4 c}-\frac{(2 c d-a e) \log (a+c x)}{4 a^4 c}+\frac{d \log (x)}{a^4} \]

[Out]

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

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Rubi [A]  time = 0.0745372, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {823, 801} \[ \frac{d+e x}{2 a^2 \left (a^2-c^2 x^2\right )}-\frac{(a e+2 c d) \log (a-c x)}{4 a^4 c}-\frac{(2 c d-a e) \log (a+c x)}{4 a^4 c}+\frac{d \log (x)}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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

Mathematica [A]  time = 0.0684567, size = 65, normalized size = 0.77 \[ \frac{\frac{a^2 (d+e x)}{a^2-c^2 x^2}-d \log \left (a^2-c^2 x^2\right )+\frac{a e \tanh ^{-1}\left (\frac{c x}{a}\right )}{c}+2 d \log (x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 129, normalized size = 1.5 \begin{align*}{\frac{d\ln \left ( x \right ) }{{a}^{4}}}+{\frac{\ln \left ( cx+a \right ) e}{4\,c{a}^{3}}}-{\frac{\ln \left ( cx+a \right ) d}{2\,{a}^{4}}}-{\frac{e}{4\,{a}^{2}c \left ( cx+a \right ) }}+{\frac{d}{4\,{a}^{3} \left ( cx+a \right ) }}-{\frac{\ln \left ( cx-a \right ) e}{4\,c{a}^{3}}}-{\frac{\ln \left ( cx-a \right ) d}{2\,{a}^{4}}}-{\frac{e}{4\,{a}^{2}c \left ( cx-a \right ) }}-{\frac{d}{4\,{a}^{3} \left ( cx-a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05087, size = 108, normalized size = 1.29 \begin{align*} -\frac{e x + d}{2 \,{\left (a^{2} c^{2} x^{2} - a^{4}\right )}} + \frac{d \log \left (x\right )}{a^{4}} - \frac{{\left (2 \, c d - a e\right )} \log \left (c x + a\right )}{4 \, a^{4} c} - \frac{{\left (2 \, c d + a e\right )} \log \left (c x - a\right )}{4 \, a^{4} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6273, size = 282, normalized size = 3.36 \begin{align*} -\frac{2 \, a^{2} c e x + 2 \, a^{2} c d -{\left (2 \, a^{2} c d - a^{3} e -{\left (2 \, c^{3} d - a c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) -{\left (2 \, a^{2} c d + a^{3} e -{\left (2 \, c^{3} d + a c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right ) - 4 \,{\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (x\right )}{4 \,{\left (a^{4} c^{3} x^{2} - a^{6} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a^2*c*e*x + 2*a^2*c*d - (2*a^2*c*d - a^3*e - (2*c^3*d - a*c^2*e)*x^2)*log(c*x + a) - (2*a^2*c*d + a^3*
e - (2*c^3*d + a*c^2*e)*x^2)*log(c*x - a) - 4*(c^3*d*x^2 - a^2*c*d)*log(x))/(a^4*c^3*x^2 - a^6*c)

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Sympy [B]  time = 2.82555, size = 230, normalized size = 2.74 \begin{align*} - \frac{d + e x}{- 2 a^{4} + 2 a^{2} c^{2} x^{2}} + \frac{d \log{\left (x \right )}}{a^{4}} + \frac{\left (a e - 2 c d\right ) \log{\left (x + \frac{- 4 a^{2} d e^{2} + \frac{a^{2} e^{2} \left (a e - 2 c d\right )}{c} - 48 c^{2} d^{3} - 12 c d^{2} \left (a e - 2 c d\right ) + 6 d \left (a e - 2 c d\right )^{2}}{a^{2} e^{3} - 36 c^{2} d^{2} e} \right )}}{4 a^{4} c} - \frac{\left (a e + 2 c d\right ) \log{\left (x + \frac{- 4 a^{2} d e^{2} - \frac{a^{2} e^{2} \left (a e + 2 c d\right )}{c} - 48 c^{2} d^{3} + 12 c d^{2} \left (a e + 2 c d\right ) + 6 d \left (a e + 2 c d\right )^{2}}{a^{2} e^{3} - 36 c^{2} d^{2} e} \right )}}{4 a^{4} 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)**2,x)

[Out]

-(d + e*x)/(-2*a**4 + 2*a**2*c**2*x**2) + d*log(x)/a**4 + (a*e - 2*c*d)*log(x + (-4*a**2*d*e**2 + a**2*e**2*(a
*e - 2*c*d)/c - 48*c**2*d**3 - 12*c*d**2*(a*e - 2*c*d) + 6*d*(a*e - 2*c*d)**2)/(a**2*e**3 - 36*c**2*d**2*e))/(
4*a**4*c) - (a*e + 2*c*d)*log(x + (-4*a**2*d*e**2 - a**2*e**2*(a*e + 2*c*d)/c - 48*c**2*d**3 + 12*c*d**2*(a*e
+ 2*c*d) + 6*d*(a*e + 2*c*d)**2)/(a**2*e**3 - 36*c**2*d**2*e))/(4*a**4*c)

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Giac [A]  time = 1.18903, size = 127, normalized size = 1.51 \begin{align*} \frac{d \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (2 \, c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{4} c} - \frac{{\left (2 \, c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{4} c} - \frac{a^{2} x e + a^{2} d}{2 \,{\left (c x + a\right )}{\left (c x - a\right )} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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