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

Optimal. Leaf size=49 \[ -\frac{8 a^3 \log (a-b x)}{b}-4 a^2 x-\frac{a (a+b x)^2}{b}-\frac{(a+b x)^3}{3 b} \]

[Out]

-4*a^2*x - (a*(a + b*x)^2)/b - (a + b*x)^3/(3*b) - (8*a^3*Log[a - b*x])/b

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Rubi [A]  time = 0.056197, antiderivative size = 49, 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 \[ -\frac{8 a^3 \log (a-b x)}{b}-4 a^2 x-\frac{a (a+b x)^2}{b}-\frac{(a+b x)^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

-4*a^2*x - (a*(a + b*x)^2)/b - (a + b*x)^3/(3*b) - (8*a^3*Log[a - b*x])/b

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Rubi in Sympy [A]  time = 12.751, size = 41, normalized size = 0.84 \[ - \frac{8 a^{3} \log{\left (a - b x \right )}}{b} - 4 a^{2} x - \frac{a \left (a + b x\right )^{2}}{b} - \frac{\left (a + b x\right )^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0102532, size = 39, normalized size = 0.8 \[ -\frac{8 a^3 \log (a-b x)}{b}-7 a^2 x-2 a b x^2-\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

-7*a^2*x - 2*a*b*x^2 - (b^2*x^3)/3 - (8*a^3*Log[a - b*x])/b

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Maple [A]  time = 0.004, size = 39, normalized size = 0.8 \[ -{\frac{{b}^{2}{x}^{3}}{3}}-2\,ab{x}^{2}-7\,{a}^{2}x-8\,{\frac{{a}^{3}\ln \left ( bx-a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4/(-b^2*x^2+a^2),x)

[Out]

-1/3*b^2*x^3-2*a*b*x^2-7*a^2*x-8/b*a^3*ln(b*x-a)

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Maxima [A]  time = 0.681688, size = 51, normalized size = 1.04 \[ -\frac{1}{3} \, b^{2} x^{3} - 2 \, a b x^{2} - 7 \, a^{2} x - \frac{8 \, a^{3} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b^2*x^3 - 2*a*b*x^2 - 7*a^2*x - 8*a^3*log(b*x - a)/b

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Fricas [A]  time = 0.21123, size = 57, normalized size = 1.16 \[ -\frac{b^{3} x^{3} + 6 \, a b^{2} x^{2} + 21 \, a^{2} b x + 24 \, a^{3} \log \left (b x - a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b^3*x^3 + 6*a*b^2*x^2 + 21*a^2*b*x + 24*a^3*log(b*x - a))/b

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Sympy [A]  time = 1.25461, size = 37, normalized size = 0.76 \[ - \frac{8 a^{3} \log{\left (- a + b x \right )}}{b} - 7 a^{2} x - 2 a b x^{2} - \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a**3*log(-a + b*x)/b - 7*a**2*x - 2*a*b*x**2 - b**2*x**3/3

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GIAC/XCAS [A]  time = 0.216897, size = 66, normalized size = 1.35 \[ -\frac{8 \, a^{3}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{b^{5} x^{3} + 6 \, a b^{4} x^{2} + 21 \, a^{2} b^{3} x}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a^3*ln(abs(b*x - a))/b - 1/3*(b^5*x^3 + 6*a*b^4*x^2 + 21*a^2*b^3*x)/b^3