3.750 \(\int \frac{(a+b x)^3}{\left (a^2-b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=26 \[ \frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b} \]

[Out]

(2*a)/(b*(a - b*x)) + Log[a - b*x]/b

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Rubi [A]  time = 0.0441525, antiderivative size = 26, 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 \[ \frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/(a^2 - b^2*x^2)^2,x]

[Out]

(2*a)/(b*(a - b*x)) + Log[a - b*x]/b

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Rubi in Sympy [A]  time = 8.82452, size = 17, normalized size = 0.65 \[ \frac{2 a}{b \left (a - b x\right )} + \frac{\log{\left (a - b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/(-b**2*x**2+a**2)**2,x)

[Out]

2*a/(b*(a - b*x)) + log(a - b*x)/b

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Mathematica [A]  time = 0.0136674, size = 23, normalized size = 0.88 \[ \frac{\frac{2 a}{a-b x}+\log (a-b x)}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/(a^2 - b^2*x^2)^2,x]

[Out]

((2*a)/(a - b*x) + Log[a - b*x])/b

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Maple [A]  time = 0.008, size = 29, normalized size = 1.1 \[{\frac{\ln \left ( bx-a \right ) }{b}}-2\,{\frac{a}{b \left ( bx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(-b^2*x^2+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.677029, size = 38, normalized size = 1.46 \[ -\frac{2 \, a}{b^{2} x - a b} + \frac{\log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(b^2*x^2 - a^2)^2,x, algorithm="maxima")

[Out]

-2*a/(b^2*x - a*b) + log(b*x - a)/b

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Fricas [A]  time = 0.219035, size = 45, normalized size = 1.73 \[ \frac{{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} x - a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(b^2*x^2 - a^2)^2,x, algorithm="fricas")

[Out]

((b*x - a)*log(b*x - a) - 2*a)/(b^2*x - a*b)

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Sympy [A]  time = 1.29904, size = 19, normalized size = 0.73 \[ - \frac{2 a}{- a b + b^{2} x} + \frac{\log{\left (- a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/(-b**2*x**2+a**2)**2,x)

[Out]

-2*a/(-a*b + b**2*x) + log(-a + b*x)/b

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GIAC/XCAS [A]  time = 0.216616, size = 39, normalized size = 1.5 \[ \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{2 \, a}{{\left (b x - a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(b^2*x^2 - a^2)^2,x, algorithm="giac")

[Out]

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