Optimal. Leaf size=69 \[ -\frac{a^2 (A b-a B)}{b^4 (a+b x)}-\frac{a (2 A b-3 a B) \log (a+b x)}{b^4}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^2}{2 b^2} \]
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Rubi [A] time = 0.132283, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{a^2 (A b-a B)}{b^4 (a+b x)}-\frac{a (2 A b-3 a B) \log (a+b x)}{b^4}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^2}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(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. \[ \frac{B \int x\, dx}{b^{2}} - \frac{a^{2} \left (A b - B a\right )}{b^{4} \left (a + b x\right )} - \frac{a \left (2 A b - 3 B a\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{4 \left (A b - 2 B a\right ) \int \frac{1}{4}\, dx}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x+A)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0988488, size = 66, normalized size = 0.96 \[ \frac{\frac{2 a^2 (a B-A b)}{a+b x}+2 b x (A b-2 a B)+2 a (3 a B-2 A b) \log (a+b x)+b^2 B x^2}{2 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(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 = 84, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{2}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{aBx}{{b}^{3}}}-2\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{3}}}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{4}}}-{\frac{A{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{B{a}^{3}}{{b}^{4} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x+A)/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.689064, size = 100, normalized size = 1.45 \[ \frac{B a^{3} - A a^{2} b}{b^{5} x + a b^{4}} + \frac{B b x^{2} - 2 \,{\left (2 \, B a - A b\right )} x}{2 \, b^{3}} + \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.303061, size = 153, normalized size = 2.22 \[ \frac{B b^{3} x^{3} + 2 \, B a^{3} - 2 \, A a^{2} b -{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} - 2 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x + 2 \,{\left (3 \, B a^{3} - 2 \, A a^{2} b +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.12562, size = 66, normalized size = 0.96 \[ \frac{B x^{2}}{2 b^{2}} + \frac{a \left (- 2 A b + 3 B a\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{- A a^{2} b + B a^{3}}{a b^{4} + b^{5} x} - \frac{x \left (- A b + 2 B a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x+A)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.268799, size = 101, normalized size = 1.46 \[ \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{B b^{2} x^{2} - 4 \, B a b x + 2 \, A b^{2} x}{2 \, b^{4}} + \frac{B a^{3} - A a^{2} b}{{\left (b x + a\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")
[Out]