Optimal. Leaf size=66 \[ \frac{a^2 (A b-a B) \log (a+b x)}{b^4}-\frac{a x (A b-a B)}{b^3}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^3}{3 b} \]
[Out]
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Rubi [A] time = 0.111041, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a^2 (A b-a B) \log (a+b x)}{b^4}-\frac{a x (A b-a B)}{b^3}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^3}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(A + B*x))/(a + b*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B x^{3}}{3 b} + \frac{a^{2} \left (A b - B a\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{\left (A b - B a\right ) \int x\, dx}{b^{2}} - \frac{\left (A b - B a\right ) \int a\, dx}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x+A)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0352871, size = 61, normalized size = 0.92 \[ \frac{b x \left (6 a^2 B-3 a b (2 A+B x)+b^2 x (3 A+2 B x)\right )+6 a^2 (A b-a B) \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(A + B*x))/(a + b*x),x]
[Out]
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Maple [A] time = 0.004, size = 76, normalized size = 1.2 \[{\frac{B{x}^{3}}{3\,b}}+{\frac{A{x}^{2}}{2\,b}}-{\frac{B{x}^{2}a}{2\,{b}^{2}}}-{\frac{aAx}{{b}^{2}}}+{\frac{{a}^{2}Bx}{{b}^{3}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) A}{{b}^{3}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) B}{{b}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x+A)/(b*x+a),x)
[Out]
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Maxima [A] time = 1.35289, size = 95, normalized size = 1.44 \[ \frac{2 \, B b^{2} x^{3} - 3 \,{\left (B a b - A b^{2}\right )} x^{2} + 6 \,{\left (B a^{2} - A a b\right )} x}{6 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201202, size = 96, normalized size = 1.45 \[ \frac{2 \, B b^{3} x^{3} - 3 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 6 \,{\left (B a^{2} b - A a b^{2}\right )} x - 6 \,{\left (B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.27296, size = 58, normalized size = 0.88 \[ \frac{B x^{3}}{3 b} - \frac{a^{2} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{4}} - \frac{x^{2} \left (- A b + B a\right )}{2 b^{2}} + \frac{x \left (- A a b + B a^{2}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x+A)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.31739, size = 96, normalized size = 1.45 \[ \frac{2 \, B b^{2} x^{3} - 3 \, B a b x^{2} + 3 \, A b^{2} x^{2} + 6 \, B a^{2} x - 6 \, A a b x}{6 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^2/(b*x + a),x, algorithm="giac")
[Out]