3.759 \(\int \frac{(a+b x)^5}{\left (a^2-b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 a^2}{b (a-b x)^2}-\frac{4 a}{b (a-b x)}-\frac{\log (a-b x)}{b} \]

[Out]

(2*a^2)/(b*(a - b*x)^2) - (4*a)/(b*(a - b*x)) - Log[a - b*x]/b

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Rubi [A]  time = 0.0636689, antiderivative size = 43, 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 \[ \frac{2 a^2}{b (a-b x)^2}-\frac{4 a}{b (a-b x)}-\frac{\log (a-b x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a^2 - b^2*x^2)^3,x]

[Out]

(2*a^2)/(b*(a - b*x)^2) - (4*a)/(b*(a - b*x)) - Log[a - b*x]/b

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Rubi in Sympy [A]  time = 12.9763, size = 31, normalized size = 0.72 \[ \frac{2 a^{2}}{b \left (a - b x\right )^{2}} - \frac{4 a}{b \left (a - b x\right )} - \frac{\log{\left (a - b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(-b**2*x**2+a**2)**3,x)

[Out]

2*a**2/(b*(a - b*x)**2) - 4*a/(b*(a - b*x)) - log(a - b*x)/b

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Mathematica [A]  time = 0.0349041, size = 30, normalized size = 0.7 \[ -\frac{\frac{2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a^2 - b^2*x^2)^3,x]

[Out]

-(((2*a*(a - 2*b*x))/(a - b*x)^2 + Log[a - b*x])/b)

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Maple [A]  time = 0.009, size = 47, normalized size = 1.1 \[ -{\frac{\ln \left ( bx-a \right ) }{b}}+4\,{\frac{a}{b \left ( bx-a \right ) }}+2\,{\frac{{a}^{2}}{b \left ( bx-a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(-b^2*x^2+a^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.682952, size = 66, normalized size = 1.53 \[ \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} - \frac{\log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^5/(b^2*x^2 - a^2)^3,x, algorithm="maxima")

[Out]

2*(2*a*b*x - a^2)/(b^3*x^2 - 2*a*b^2*x + a^2*b) - log(b*x - a)/b

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Fricas [A]  time = 0.21428, size = 81, normalized size = 1.88 \[ \frac{4 \, a b x - 2 \, a^{2} -{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^5/(b^2*x^2 - a^2)^3,x, algorithm="fricas")

[Out]

(4*a*b*x - 2*a^2 - (b^2*x^2 - 2*a*b*x + a^2)*log(b*x - a))/(b^3*x^2 - 2*a*b^2*x
+ a^2*b)

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Sympy [A]  time = 1.62905, size = 39, normalized size = 0.91 \[ \frac{- 2 a^{2} + 4 a b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} - \frac{\log{\left (- a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(-b**2*x**2+a**2)**3,x)

[Out]

(-2*a**2 + 4*a*b*x)/(a**2*b - 2*a*b**2*x + b**3*x**2) - log(-a + b*x)/b

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GIAC/XCAS [A]  time = 0.219658, size = 54, normalized size = 1.26 \[ -\frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{b} + \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^5/(b^2*x^2 - a^2)^3,x, algorithm="giac")

[Out]

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