3.740 \(\int \frac{(a+b x)^2}{a^2-b^2 x^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{2 a \log (a-b x)}{b}-x \]

[Out]

-x - (2*a*Log[a - b*x])/b

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Rubi [A]  time = 0.0360339, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 a \log (a-b x)}{b}-x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/(a^2 - b^2*x^2),x]

[Out]

-x - (2*a*Log[a - b*x])/b

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Rubi in Sympy [A]  time = 8.58288, size = 14, normalized size = 0.82 \[ - \frac{2 a \log{\left (a - b x \right )}}{b} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/(-b**2*x**2+a**2),x)

[Out]

-2*a*log(a - b*x)/b - x

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Mathematica [A]  time = 0.00494374, size = 17, normalized size = 1. \[ -\frac{2 a \log (a-b x)}{b}-x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^2/(a^2 - b^2*x^2),x]

[Out]

-x - (2*a*Log[a - b*x])/b

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Maple [A]  time = 0.004, size = 19, normalized size = 1.1 \[ -x-2\,{\frac{a\ln \left ( bx-a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(-b^2*x^2+a^2),x)

[Out]

-x-2*a/b*ln(b*x-a)

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Maxima [A]  time = 0.684891, size = 24, normalized size = 1.41 \[ -x - \frac{2 \, a \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x - 2*a*log(b*x - a)/b

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Fricas [A]  time = 0.209087, size = 27, normalized size = 1.59 \[ -\frac{b x + 2 \, a \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x + 2*a*log(b*x - a))/b

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Sympy [A]  time = 1.11245, size = 14, normalized size = 0.82 \[ - \frac{2 a \log{\left (- a + b x \right )}}{b} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/(-b**2*x**2+a**2),x)

[Out]

-2*a*log(-a + b*x)/b - x

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GIAC/XCAS [A]  time = 0.213995, size = 26, normalized size = 1.53 \[ -x - \frac{2 \, a{\rm ln}\left ({\left | b x - a \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x - 2*a*ln(abs(b*x - a))/b