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

Optimal. Leaf size=59 \[ -\frac{(a e+c d) \log (a-c x)}{2 a^3}+\frac{(c d-a e) \log (a+c x)}{2 a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.120407, antiderivative size = 59, 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 \[ -\frac{(a e+c d) \log (a-c x)}{2 a^3}+\frac{(c d-a e) \log (a+c x)}{2 a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.5455, size = 51, normalized size = 0.86 \[ - \frac{d}{a^{2} x} + \frac{e \log{\left (x \right )}}{a^{2}} - \frac{\left (a e - c d\right ) \log{\left (a + c x \right )}}{2 a^{3}} - \frac{\left (a e + c d\right ) \log{\left (a - c x \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(a**2*x) + e*log(x)/a**2 - (a*e - c*d)*log(a + c*x)/(2*a**3) - (a*e + c*d)*lo
g(a - c*x)/(2*a**3)

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Mathematica [A]  time = 0.0223889, size = 51, normalized size = 0.86 \[ \frac{c d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a^3}-\frac{e \log \left (a^2-c^2 x^2\right )}{2 a^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 72, normalized size = 1.2 \[ -{\frac{d}{{a}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( cx+a \right ) e}{2\,{a}^{2}}}+{\frac{\ln \left ( cx+a \right ) cd}{2\,{a}^{3}}}-{\frac{\ln \left ( cx-a \right ) e}{2\,{a}^{2}}}-{\frac{\ln \left ( cx-a \right ) cd}{2\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.691358, size = 76, normalized size = 1.29 \[ \frac{e \log \left (x\right )}{a^{2}} + \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a^{3}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{3}} - \frac{d}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.307088, size = 73, normalized size = 1.24 \[ \frac{2 \, a e x \log \left (x\right ) +{\left (c d - a e\right )} x \log \left (c x + a\right ) -{\left (c d + a e\right )} x \log \left (c x - a\right ) - 2 \, a d}{2 \, a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.07106, size = 221, normalized size = 3.75 \[ - \frac{d}{a^{2} x} + \frac{e \log{\left (x \right )}}{a^{2}} - \frac{\left (a e - c d\right ) \log{\left (x + \frac{6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e - c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e - c d\right )^{2} + a c^{2} d^{2} \left (a e - c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e + c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e + c d\right )^{2} + a c^{2} d^{2} \left (a e + c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271387, size = 100, normalized size = 1.69 \[ \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{d}{a^{2} x} + \frac{{\left (c^{2} d - a c e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{2 \, a^{3} c} - \frac{{\left (c^{2} d + a c e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{2 \, a^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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