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

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

[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)*Log[a + c*x])/(4*a^4*c)

_______________________________________________________________________________________

Rubi [A]  time = 0.179209, 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 \[ -\frac{(a e+2 c d) \log (a-c x)}{4 a^4 c}-\frac{(2 c d-a e) \log (a+c x)}{4 a^4 c}+\frac{d \log (x)}{a^4}+\frac{d+e x}{2 a^2 \left (a^2-c^2 x^2\right )} \]

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)*Log[a + c*x])/(4*a^4*c)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.7921, size = 71, normalized size = 0.85 \[ \frac{d + e x}{2 a^{2} \left (a^{2} - c^{2} x^{2}\right )} + \frac{d \log{\left (x \right )}}{a^{4}} + \frac{\left (a e - 2 c d\right ) \log{\left (a + c x \right )}}{4 a^{4} c} - \frac{\left (a e + 2 c d\right ) \log{\left (a - c x \right )}}{4 a^{4} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/x/(-c**2*x**2+a**2)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.127916, 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)

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 129, normalized size = 1.5 \[{\frac{d\ln \left ( x \right ) }{{a}^{4}}}+{\frac{\ln \left ( cx+a \right ) e}{4\,{a}^{3}c}}-{\frac{\ln \left ( cx+a \right ) d}{2\,{a}^{4}}}-{\frac{e}{4\,{a}^{2}c \left ( cx+a \right ) }}+{\frac{d}{4\,{a}^{3} \left ( cx+a \right ) }}-{\frac{\ln \left ( cx-a \right ) e}{4\,{a}^{3}c}}-{\frac{\ln \left ( cx-a \right ) d}{2\,{a}^{4}}}-{\frac{e}{4\,{a}^{2}c \left ( cx-a \right ) }}-{\frac{d}{4\,{a}^{3} \left ( cx-a \right ) }} \]

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*d/
a^3/(c*x+a)-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*d/
a^3/(c*x-a)

_______________________________________________________________________________________

Maxima [A]  time = 0.686897, size = 108, normalized size = 1.29 \[ -\frac{e x + d}{2 \,{\left (a^{2} c^{2} x^{2} - a^{4}\right )}} + \frac{d \log \left (x\right )}{a^{4}} - \frac{{\left (2 \, c d - a e\right )} \log \left (c x + a\right )}{4 \, a^{4} c} - \frac{{\left (2 \, c d + a e\right )} \log \left (c x - a\right )}{4 \, a^{4} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Fricas [A]  time = 0.300777, size = 188, normalized size = 2.24 \[ -\frac{2 \, a^{2} c e x + 2 \, a^{2} c d -{\left (2 \, a^{2} c d - a^{3} e -{\left (2 \, c^{3} d - a c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) -{\left (2 \, a^{2} c d + a^{3} e -{\left (2 \, c^{3} d + a c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right ) - 4 \,{\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (x\right )}{4 \,{\left (a^{4} c^{3} x^{2} - a^{6} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/((c^2*x^2 - a^2)^2*x),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)*lo
g(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)

_______________________________________________________________________________________

Sympy [A]  time = 5.6523, size = 230, normalized size = 2.74 \[ - \frac{d + e x}{- 2 a^{4} + 2 a^{2} c^{2} x^{2}} + \frac{d \log{\left (x \right )}}{a^{4}} + \frac{\left (a e - 2 c d\right ) \log{\left (x + \frac{- 4 a^{2} d e^{2} + \frac{a^{2} e^{2} \left (a e - 2 c d\right )}{c} - 48 c^{2} d^{3} - 12 c d^{2} \left (a e - 2 c d\right ) + 6 d \left (a e - 2 c d\right )^{2}}{a^{2} e^{3} - 36 c^{2} d^{2} e} \right )}}{4 a^{4} c} - \frac{\left (a e + 2 c d\right ) \log{\left (x + \frac{- 4 a^{2} d e^{2} - \frac{a^{2} e^{2} \left (a e + 2 c d\right )}{c} - 48 c^{2} d^{3} + 12 c d^{2} \left (a e + 2 c d\right ) + 6 d \left (a e + 2 c d\right )^{2}}{a^{2} e^{3} - 36 c^{2} d^{2} e} \right )}}{4 a^{4} c} \]

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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284554, size = 127, normalized size = 1.51 \[ \frac{d{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (2 \, c d - a e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{4 \, a^{4} c} - \frac{{\left (2 \, c d + a e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{4 \, a^{4} c} - \frac{a^{2} x e + a^{2} d}{2 \,{\left (c x + a\right )}{\left (c x - a\right )} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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