Optimal. Leaf size=66 \[ \frac{a^2 (b c-a d) \log (a+b x)}{b^4}-\frac{a x (b c-a d)}{b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^3}{3 b} \]
[Out]
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Rubi [A] time = 0.114618, 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 \[ \frac{a^2 (b c-a d) \log (a+b x)}{b^4}-\frac{a x (b c-a d)}{b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^3}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(c + d*x))/(a + b*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{d x^{3}}{3 b} - \frac{\left (a d - b c\right ) \int x\, dx}{b^{2}} + \frac{\left (a d - b c\right ) \int a\, dx}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(d*x+c)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0348711, size = 61, normalized size = 0.92 \[ \frac{b x \left (6 a^2 d-3 a b (2 c+d x)+b^2 x (3 c+2 d x)\right )+6 a^2 (b c-a d) \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(c + d*x))/(a + b*x),x]
[Out]
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Maple [A] time = 0.004, size = 76, normalized size = 1.2 \[{\frac{d{x}^{3}}{3\,b}}-{\frac{{x}^{2}ad}{2\,{b}^{2}}}+{\frac{c{x}^{2}}{2\,b}}+{\frac{{a}^{2}dx}{{b}^{3}}}-{\frac{acx}{{b}^{2}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) d}{{b}^{4}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) c}{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(d*x+c)/(b*x+a),x)
[Out]
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Maxima [A] time = 1.34616, size = 93, normalized size = 1.41 \[ \frac{2 \, b^{2} d x^{3} + 3 \,{\left (b^{2} c - a b d\right )} x^{2} - 6 \,{\left (a b c - a^{2} d\right )} x}{6 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*x^2/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203537, size = 96, normalized size = 1.45 \[ \frac{2 \, b^{3} d x^{3} + 3 \,{\left (b^{3} c - a b^{2} d\right )} x^{2} - 6 \,{\left (a b^{2} c - a^{2} b d\right )} x + 6 \,{\left (a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*x^2/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.28825, size = 58, normalized size = 0.88 \[ - \frac{a^{2} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{d x^{3}}{3 b} - \frac{x^{2} \left (a d - b c\right )}{2 b^{2}} + \frac{x \left (a^{2} d - a b c\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(d*x+c)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.245121, size = 95, normalized size = 1.44 \[ \frac{2 \, b^{2} d x^{3} + 3 \, b^{2} c x^{2} - 3 \, a b d x^{2} - 6 \, a b c x + 6 \, a^{2} d x}{6 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*x^2/(b*x + a),x, algorithm="giac")
[Out]