Optimal. Leaf size=26 \[ \frac{b x}{d}-\frac{(b c-a d) \log (c+d x)}{d^2} \]
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Rubi [A] time = 0.0605462, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{b x}{d}-\frac{(b c-a d) \log (c+d x)}{d^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/(a*c + (b*c + a*d)*x + b*d*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int b\, dx}{d} + \frac{\left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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Mathematica [A] time = 0.0133209, size = 25, normalized size = 0.96 \[ \frac{(a d-b c) \log (c+d x)}{d^2}+\frac{b x}{d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/(a*c + (b*c + a*d)*x + b*d*x^2),x]
[Out]
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Maple [A] time = 0.004, size = 32, normalized size = 1.2 \[{\frac{bx}{d}}+{\frac{\ln \left ( dx+c \right ) a}{d}}-{\frac{\ln \left ( dx+c \right ) bc}{{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/(a*c+(a*d+b*c)*x+x^2*b*d),x)
[Out]
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Maxima [A] time = 0.748402, size = 35, normalized size = 1.35 \[ \frac{b x}{d} - \frac{{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203786, size = 34, normalized size = 1.31 \[ \frac{b d x -{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.25226, size = 20, normalized size = 0.77 \[ \frac{b x}{d} + \frac{\left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.212336, size = 36, normalized size = 1.38 \[ \frac{b x}{d} - \frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="giac")
[Out]