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

Optimal. Leaf size=93 \[ \frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}-\frac{(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac{(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac{3 d}{2 a^4 x}+\frac{e \log (x)}{a^4} \]

[Out]

(-3*d)/(2*a^4*x) + (d + e*x)/(2*a^2*x*(a^2 - c^2*x^2)) + (e*Log[x])/a^4 - ((3*c*d + 2*a*e)*Log[a - c*x])/(4*a^
5) + ((3*c*d - 2*a*e)*Log[a + c*x])/(4*a^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0948619, antiderivative size = 93, 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} \[ \frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}-\frac{(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac{(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac{3 d}{2 a^4 x}+\frac{e \log (x)}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*d)/(2*a^4*x) + (d + e*x)/(2*a^2*x*(a^2 - c^2*x^2)) + (e*Log[x])/a^4 - ((3*c*d + 2*a*e)*Log[a - c*x])/(4*a^
5) + ((3*c*d - 2*a*e)*Log[a + c*x])/(4*a^5)

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^2 \left (a^2-c^2 x^2\right )^2} \, dx &=\frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}+\frac{\int \frac{3 a^2 c^2 d+2 a^2 c^2 e x}{x^2 \left (a^2-c^2 x^2\right )} \, dx}{2 a^4 c^2}\\ &=\frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}+\frac{\int \left (\frac{3 c^2 d}{x^2}+\frac{2 c^2 e}{x}+\frac{c^3 (3 c d+2 a e)}{2 a (a-c x)}-\frac{c^3 (-3 c d+2 a e)}{2 a (a+c x)}\right ) \, dx}{2 a^4 c^2}\\ &=-\frac{3 d}{2 a^4 x}+\frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}+\frac{e \log (x)}{a^4}-\frac{(3 c d+2 a e) \log (a-c x)}{4 a^5}+\frac{(3 c d-2 a e) \log (a+c x)}{4 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0824386, size = 77, normalized size = 0.83 \[ \frac{\frac{a^3 e+a c^2 d x}{a^2-c^2 x^2}-a e \log \left (a^2-c^2 x^2\right )+3 c d \tanh ^{-1}\left (\frac{c x}{a}\right )-\frac{2 a d}{x}+2 a e \log (x)}{2 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a*d)/x + (a^3*e + a*c^2*d*x)/(a^2 - c^2*x^2) + 3*c*d*ArcTanh[(c*x)/a] + 2*a*e*Log[x] - a*e*Log[a^2 - c^2*
x^2])/(2*a^5)

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 130, normalized size = 1.4 \begin{align*}{\frac{e\ln \left ( x \right ) }{{a}^{4}}}-{\frac{d}{{a}^{4}x}}-{\frac{\ln \left ( cx+a \right ) e}{2\,{a}^{4}}}+{\frac{3\,\ln \left ( cx+a \right ) cd}{4\,{a}^{5}}}+{\frac{e}{4\,{a}^{3} \left ( cx+a \right ) }}-{\frac{cd}{4\,{a}^{4} \left ( cx+a \right ) }}-{\frac{\ln \left ( cx-a \right ) e}{2\,{a}^{4}}}-{\frac{3\,\ln \left ( cx-a \right ) cd}{4\,{a}^{5}}}-{\frac{e}{4\,{a}^{3} \left ( cx-a \right ) }}-{\frac{cd}{4\,{a}^{4} \left ( cx-a \right ) }} \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)^2,x)

[Out]

e*ln(x)/a^4-d/a^4/x-1/2/a^4*ln(c*x+a)*e+3/4/a^5*ln(c*x+a)*c*d+1/4/a^3/(c*x+a)*e-1/4/a^4/(c*x+a)*c*d-1/2/a^4*ln
(c*x-a)*e-3/4/a^5*ln(c*x-a)*c*d-1/4/a^3/(c*x-a)*e-1/4/a^4/(c*x-a)*c*d

________________________________________________________________________________________

Maxima [A]  time = 1.04167, size = 126, normalized size = 1.35 \begin{align*} -\frac{3 \, c^{2} d x^{2} + a^{2} e x - 2 \, a^{2} d}{2 \,{\left (a^{4} c^{2} x^{3} - a^{6} x\right )}} + \frac{e \log \left (x\right )}{a^{4}} + \frac{{\left (3 \, c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a^{5}} - \frac{{\left (3 \, c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a^{5}} \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)^2,x, algorithm="maxima")

[Out]

-1/2*(3*c^2*d*x^2 + a^2*e*x - 2*a^2*d)/(a^4*c^2*x^3 - a^6*x) + e*log(x)/a^4 + 1/4*(3*c*d - 2*a*e)*log(c*x + a)
/a^5 - 1/4*(3*c*d + 2*a*e)*log(c*x - a)/a^5

________________________________________________________________________________________

Fricas [A]  time = 1.64361, size = 323, normalized size = 3.47 \begin{align*} -\frac{6 \, a c^{2} d x^{2} + 2 \, a^{3} e x - 4 \, a^{3} d -{\left ({\left (3 \, c^{3} d - 2 \, a c^{2} e\right )} x^{3} -{\left (3 \, a^{2} c d - 2 \, a^{3} e\right )} x\right )} \log \left (c x + a\right ) +{\left ({\left (3 \, c^{3} d + 2 \, a c^{2} e\right )} x^{3} -{\left (3 \, a^{2} c d + 2 \, a^{3} e\right )} x\right )} \log \left (c x - a\right ) - 4 \,{\left (a c^{2} e x^{3} - a^{3} e x\right )} \log \left (x\right )}{4 \,{\left (a^{5} c^{2} x^{3} - a^{7} x\right )}} \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)^2,x, algorithm="fricas")

[Out]

-1/4*(6*a*c^2*d*x^2 + 2*a^3*e*x - 4*a^3*d - ((3*c^3*d - 2*a*c^2*e)*x^3 - (3*a^2*c*d - 2*a^3*e)*x)*log(c*x + a)
+ ((3*c^3*d + 2*a*c^2*e)*x^3 - (3*a^2*c*d + 2*a^3*e)*x)*log(c*x - a) - 4*(a*c^2*e*x^3 - a^3*e*x)*log(x))/(a^5
*c^2*x^3 - a^7*x)

________________________________________________________________________________________

Sympy [B]  time = 2.19835, size = 291, normalized size = 3.13 \begin{align*} - \frac{- 2 a^{2} d + a^{2} e x + 3 c^{2} d x^{2}}{- 2 a^{6} x + 2 a^{4} c^{2} x^{3}} + \frac{e \log{\left (x \right )}}{a^{4}} - \frac{\left (2 a e - 3 c d\right ) \log{\left (x + \frac{16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e - 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e - 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e - 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} - \frac{\left (2 a e + 3 c d\right ) \log{\left (x + \frac{16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e + 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e + 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e + 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} \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)**2,x)

[Out]

-(-2*a**2*d + a**2*e*x + 3*c**2*d*x**2)/(-2*a**6*x + 2*a**4*c**2*x**3) + e*log(x)/a**4 - (2*a*e - 3*c*d)*log(x
+ (16*a**4*e**3 - 4*a**3*e**2*(2*a*e - 3*c*d) + 12*a**2*c**2*d**2*e - 2*a**2*e*(2*a*e - 3*c*d)**2 + 3*a*c**2*
d**2*(2*a*e - 3*c*d))/(36*a**2*c**2*d*e**2 - 9*c**4*d**3))/(4*a**5) - (2*a*e + 3*c*d)*log(x + (16*a**4*e**3 -
4*a**3*e**2*(2*a*e + 3*c*d) + 12*a**2*c**2*d**2*e - 2*a**2*e*(2*a*e + 3*c*d)**2 + 3*a*c**2*d**2*(2*a*e + 3*c*d
))/(36*a**2*c**2*d*e**2 - 9*c**4*d**3))/(4*a**5)

________________________________________________________________________________________

Giac [A]  time = 1.14878, size = 151, normalized size = 1.62 \begin{align*} \frac{e \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{3 \, c^{2} d x^{2} + a^{2} x e - 2 \, a^{2} d}{2 \,{\left (c^{2} x^{3} - a^{2} x\right )} a^{4}} + \frac{{\left (3 \, c^{2} d - 2 \, a c e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{5} c} - \frac{{\left (3 \, c^{2} d + 2 \, a c e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{5} 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)^2,x, algorithm="giac")

[Out]

e*log(abs(x))/a^4 - 1/2*(3*c^2*d*x^2 + a^2*x*e - 2*a^2*d)/((c^2*x^3 - a^2*x)*a^4) + 1/4*(3*c^2*d - 2*a*c*e)*lo
g(abs(c*x + a))/(a^5*c) - 1/4*(3*c^2*d + 2*a*c*e)*log(abs(c*x - a))/(a^5*c)