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

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

[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)

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Rubi [A]  time = 0.220097, 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 \[ -\frac{(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac{(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac{3 d}{2 a^4 x}+\frac{e \log (x)}{a^4}+\frac{d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )} \]

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)

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Rubi in Sympy [A]  time = 37.3784, size = 83, normalized size = 0.89 \[ \frac{d + e x}{2 a^{2} x \left (a^{2} - c^{2} x^{2}\right )} - \frac{3 d}{2 a^{4} x} + \frac{e \log{\left (x \right )}}{a^{4}} - \frac{\left (a e - \frac{3 c d}{2}\right ) \log{\left (a + c x \right )}}{2 a^{5}} - \frac{\left (a e + \frac{3 c d}{2}\right ) \log{\left (a - c x \right )}}{2 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.193328, size = 77, normalized size = 0.83 \[ \frac{-a e \log \left (a^2-c^2 x^2\right )+\frac{a^3 e+a c^2 d x}{a^2-c^2 x^2}+3 c d \tanh ^{-1}\left (\frac{c x}{a}\right )-\frac{2 a d}{x}+2 a e \log (x)}{2 a^5} \]

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)

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Maple [A]  time = 0.023, size = 130, normalized size = 1.4 \[ -{\frac{d}{{a}^{4}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{4}}}-{\frac{\ln \left ( cx+a \right ) e}{2\,{a}^{4}}}+{\frac{3\,\ln \left ( cx+a \right ) cd}{4\,{a}^{5}}}+{\frac{e}{4\,{a}^{3} \left ( cx+a \right ) }}-{\frac{cd}{4\,{a}^{4} \left ( cx+a \right ) }}-{\frac{\ln \left ( cx-a \right ) e}{2\,{a}^{4}}}-{\frac{3\,\ln \left ( cx-a \right ) cd}{4\,{a}^{5}}}-{\frac{e}{4\,{a}^{3} \left ( cx-a \right ) }}-{\frac{cd}{4\,{a}^{4} \left ( cx-a \right ) }} \]

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]

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

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Maxima [A]  time = 0.697689, size = 126, normalized size = 1.35 \[ -\frac{3 \, c^{2} d x^{2} + a^{2} e x - 2 \, a^{2} d}{2 \,{\left (a^{4} c^{2} x^{3} - a^{6} x\right )}} + \frac{e \log \left (x\right )}{a^{4}} + \frac{{\left (3 \, c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a^{5}} - \frac{{\left (3 \, c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.294118, size = 209, normalized size = 2.25 \[ -\frac{6 \, a c^{2} d x^{2} + 2 \, a^{3} e x - 4 \, a^{3} d -{\left ({\left (3 \, c^{3} d - 2 \, a c^{2} e\right )} x^{3} -{\left (3 \, a^{2} c d - 2 \, a^{3} e\right )} x\right )} \log \left (c x + a\right ) +{\left ({\left (3 \, c^{3} d + 2 \, a c^{2} e\right )} x^{3} -{\left (3 \, a^{2} c d + 2 \, a^{3} e\right )} x\right )} \log \left (c x - a\right ) - 4 \,{\left (a c^{2} e x^{3} - a^{3} e x\right )} \log \left (x\right )}{4 \,{\left (a^{5} c^{2} x^{3} - a^{7} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 6.08363, size = 291, normalized size = 3.13 \[ - \frac{- 2 a^{2} d + a^{2} e x + 3 c^{2} d x^{2}}{- 2 a^{6} x + 2 a^{4} c^{2} x^{3}} + \frac{e \log{\left (x \right )}}{a^{4}} - \frac{\left (2 a e - 3 c d\right ) \log{\left (x + \frac{16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e - 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e - 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e - 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} - \frac{\left (2 a e + 3 c d\right ) \log{\left (x + \frac{16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e + 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e + 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e + 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} \]

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) + 1
2*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)

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GIAC/XCAS [A]  time = 0.286393, size = 151, normalized size = 1.62 \[ \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{3 \, c^{2} d x^{2} + a^{2} x e - 2 \, a^{2} d}{2 \,{\left (c^{2} x^{3} - a^{2} x\right )} a^{4}} + \frac{{\left (3 \, c^{2} d - 2 \, a c e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{4 \, a^{5} c} - \frac{{\left (3 \, c^{2} d + 2 \, a c e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{4 \, a^{5} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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