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

Optimal. Leaf size=28 \[ -\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0404711, antiderivative size = 28, 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 \[ -\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 a^{2} \log{\left (a - b x \right )}}{b} - 3 a x - b \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00933326, size = 28, normalized size = 1. \[ -\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 1. \[ -{\frac{b{x}^{2}}{2}}-3\,ax-4\,{\frac{{a}^{2}\ln \left ( bx-a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(-b^2*x^2+a^2),x)

[Out]

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

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Maxima [A]  time = 0.689342, size = 36, normalized size = 1.29 \[ -\frac{1}{2} \, b x^{2} - 3 \, a x - \frac{4 \, a^{2} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208629, size = 42, normalized size = 1.5 \[ -\frac{b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b^2*x^2 + 6*a*b*x + 8*a^2*log(b*x - a))/b

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Sympy [A]  time = 1.19004, size = 26, normalized size = 0.93 \[ - \frac{4 a^{2} \log{\left (- a + b x \right )}}{b} - 3 a x - \frac{b x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215774, size = 51, normalized size = 1.82 \[ -\frac{4 \, a^{2}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{b^{3} x^{2} + 6 \, a b^{2} x}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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