3.749 \(\int \frac{(a+b x)^4}{\left (a^2-b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x \]

[Out]

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

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Rubi [A]  time = 0.0587777, antiderivative size = 31, 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 \[ \frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.8263, size = 24, normalized size = 0.77 \[ \frac{4 a^{2}}{b \left (a - b x\right )} + \frac{4 a \log{\left (a - b x \right )}}{b} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4/(-b**2*x**2+a**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.039443, size = 32, normalized size = 1.03 \[ -\frac{4 a^2}{b (b x-a)}+\frac{4 a \log (a-b x)}{b}+x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.009, size = 34, normalized size = 1.1 \[ x+4\,{\frac{a\ln \left ( bx-a \right ) }{b}}-4\,{\frac{{a}^{2}}{b \left ( bx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4/(-b^2*x^2+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.680048, size = 45, normalized size = 1.45 \[ -\frac{4 \, a^{2}}{b^{2} x - a b} + x + \frac{4 \, a \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(b^2*x^2 - a^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21546, size = 69, normalized size = 2.23 \[ \frac{b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \,{\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} x - a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(b^2*x^2 - a^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.35183, size = 26, normalized size = 0.84 \[ - \frac{4 a^{2}}{- a b + b^{2} x} + \frac{4 a \log{\left (- a + b x \right )}}{b} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4/(-b**2*x**2+a**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216886, size = 46, normalized size = 1.48 \[ x + \frac{4 \, a{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{4 \, a^{2}}{{\left (b x - a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(b^2*x^2 - a^2)^2,x, algorithm="giac")

[Out]

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