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

Optimal. Leaf size=54 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b}+\frac{1}{4 a^2 b (a-b x)}+\frac{1}{4 a b (a-b x)^2} \]

[Out]

1/(4*a*b*(a - b*x)^2) + 1/(4*a^2*b*(a - b*x)) + ArcTanh[(b*x)/a]/(4*a^3*b)

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Rubi [A]  time = 0.0935966, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b}+\frac{1}{4 a^2 b (a-b x)}+\frac{1}{4 a b (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

1/(4*a*b*(a - b*x)^2) + 1/(4*a^2*b*(a - b*x)) + ArcTanh[(b*x)/a]/(4*a^3*b)

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Rubi in Sympy [A]  time = 22.5449, size = 39, normalized size = 0.72 \[ \frac{1}{4 a b \left (a - b x\right )^{2}} + \frac{1}{4 a^{2} b \left (a - b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{4 a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(4*a*b*(a - b*x)**2) + 1/(4*a**2*b*(a - b*x)) + atanh(b*x/a)/(4*a**3*b)

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Mathematica [A]  time = 0.0294045, size = 62, normalized size = 1.15 \[ \frac{2 a (2 a-b x)+(a-b x)^2 (-\log (a-b x))+(a-b x)^2 \log (a+b x)}{8 a^3 b (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*(2*a - b*x) - (a - b*x)^2*Log[a - b*x] + (a - b*x)^2*Log[a + b*x])/(8*a^3*b
*(a - b*x)^2)

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Maple [A]  time = 0.013, size = 66, normalized size = 1.2 \[ -{\frac{\ln \left ( bx-a \right ) }{8\,{a}^{3}b}}-{\frac{1}{4\,{a}^{2}b \left ( bx-a \right ) }}+{\frac{1}{4\,ab \left ( bx-a \right ) ^{2}}}+{\frac{\ln \left ( bx+a \right ) }{8\,{a}^{3}b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/b/a^3*ln(b*x-a)-1/4/b/a^2/(b*x-a)+1/4/b/a/(b*x-a)^2+1/8/b/a^3*ln(b*x+a)

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Maxima [A]  time = 0.683817, size = 90, normalized size = 1.67 \[ -\frac{b x - 2 \, a}{4 \,{\left (a^{2} b^{3} x^{2} - 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac{\log \left (b x + a\right )}{8 \, a^{3} b} - \frac{\log \left (b x - a\right )}{8 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(b*x - 2*a)/(a^2*b^3*x^2 - 2*a^3*b^2*x + a^4*b) + 1/8*log(b*x + a)/(a^3*b)
- 1/8*log(b*x - a)/(a^3*b)

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Fricas [A]  time = 0.211265, size = 120, normalized size = 2.22 \[ -\frac{2 \, a b x - 4 \, a^{2} -{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{8 \,{\left (a^{3} b^{3} x^{2} - 2 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(2*a*b*x - 4*a^2 - (b^2*x^2 - 2*a*b*x + a^2)*log(b*x + a) + (b^2*x^2 - 2*a*
b*x + a^2)*log(b*x - a))/(a^3*b^3*x^2 - 2*a^4*b^2*x + a^5*b)

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Sympy [A]  time = 1.85206, size = 58, normalized size = 1.07 \[ - \frac{- 2 a + b x}{4 a^{4} b - 8 a^{3} b^{2} x + 4 a^{2} b^{3} x^{2}} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{8} - \frac{\log{\left (\frac{a}{b} + x \right )}}{8}}{a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*a + b*x)/(4*a**4*b - 8*a**3*b**2*x + 4*a**2*b**3*x**2) - (log(-a/b + x)/8 -
log(a/b + x)/8)/(a**3*b)

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GIAC/XCAS [A]  time = 0.218198, size = 81, normalized size = 1.5 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{8 \, a^{3} b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{8 \, a^{3} b} - \frac{a b x - 2 \, a^{2}}{4 \,{\left (b x - a\right )}^{2} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*ln(abs(b*x + a))/(a^3*b) - 1/8*ln(abs(b*x - a))/(a^3*b) - 1/4*(a*b*x - 2*a^2
)/((b*x - a)^2*a^3*b)