Optimal. Leaf size=45 \[ \frac{a (A b-a B)}{b^3 (a+b x)}+\frac{(A b-2 a B) \log (a+b x)}{b^3}+\frac{B x}{b^2} \]
[Out]
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Rubi [A] time = 0.0882427, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{a (A b-a B)}{b^3 (a+b x)}+\frac{(A b-2 a B) \log (a+b x)}{b^3}+\frac{B x}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{4 B \int \frac{1}{4}\, dx}{b^{2}} + \frac{a \left (A b - B a\right )}{b^{3} \left (a + b x\right )} + \frac{\left (A b - 2 B a\right ) \log{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0465406, size = 41, normalized size = 0.91 \[ \frac{\frac{a (A b-a B)}{a+b x}+(A b-2 a B) \log (a+b x)+b B x}{b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [A] time = 0.01, size = 61, normalized size = 1.4 \[{\frac{Bx}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a \right ) Ba}{{b}^{3}}}+{\frac{aA}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{a}^{2}B}{{b}^{3} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.684068, size = 72, normalized size = 1.6 \[ -\frac{B a^{2} - A a b}{b^{4} x + a b^{3}} + \frac{B x}{b^{2}} - \frac{{\left (2 \, B a - A b\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.297103, size = 97, normalized size = 2.16 \[ \frac{B b^{2} x^{2} + B a b x - B a^{2} + A a b -{\left (2 \, B a^{2} - A a b +{\left (2 \, B a b - A b^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.84643, size = 44, normalized size = 0.98 \[ \frac{B x}{b^{2}} - \frac{- A a b + B a^{2}}{a b^{3} + b^{4} x} - \frac{\left (- A b + 2 B a\right ) \log{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.269429, size = 69, normalized size = 1.53 \[ \frac{B x}{b^{2}} - \frac{{\left (2 \, B a - A b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{B a^{2} - A a b}{{\left (b x + a\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")
[Out]