Optimal. Leaf size=45 \[ -\frac{a (b c-a d) \log (a+b x)}{b^3}+\frac{x (b c-a d)}{b^2}+\frac{d x^2}{2 b} \]
[Out]
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Rubi [A] time = 0.0726813, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{a (b c-a d) \log (a+b x)}{b^3}+\frac{x (b c-a d)}{b^2}+\frac{d x^2}{2 b} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x))/(a + b*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} - \left (a d - b c\right ) \int \frac{1}{b^{2}}\, dx + \frac{d \int x\, dx}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x+c)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0214152, size = 41, normalized size = 0.91 \[ \frac{b x (-2 a d+2 b c+b d x)+2 a (a d-b c) \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x))/(a + b*x),x]
[Out]
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Maple [A] time = 0.004, size = 52, normalized size = 1.2 \[{\frac{d{x}^{2}}{2\,b}}-{\frac{adx}{{b}^{2}}}+{\frac{cx}{b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) d}{{b}^{3}}}-{\frac{a\ln \left ( bx+a \right ) c}{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x+c)/(b*x+a),x)
[Out]
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Maxima [A] time = 1.35166, size = 62, normalized size = 1.38 \[ \frac{b d x^{2} + 2 \,{\left (b c - a d\right )} x}{2 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*x/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205849, size = 63, normalized size = 1.4 \[ \frac{b^{2} d x^{2} + 2 \,{\left (b^{2} c - a b d\right )} x - 2 \,{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*x/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.13985, size = 37, normalized size = 0.82 \[ \frac{a \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} + \frac{d x^{2}}{2 b} - \frac{x \left (a d - b c\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x+c)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.2505, size = 62, normalized size = 1.38 \[ \frac{b d x^{2} + 2 \, b c x - 2 \, a d x}{2 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*x/(b*x + a),x, algorithm="giac")
[Out]