Optimal. Leaf size=41 \[ \frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}+\frac{x}{c^2} \]
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Rubi [A] time = 0.0560585, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}+\frac{x}{c^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/(a*c - b*c*x)^2,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{4 a^{2}}{b c^{2} \left (a - b x\right )} + \frac{4 a \log{\left (a - b x \right )}}{b c^{2}} + \int \frac{1}{c^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/(-b*c*x+a*c)**2,x)
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Mathematica [A] time = 0.0491715, size = 35, normalized size = 0.85 \[ \frac{\frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x}{c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/(a*c - b*c*x)^2,x]
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Maple [A] time = 0.008, size = 44, normalized size = 1.1 \[{\frac{x}{{c}^{2}}}+4\,{\frac{a\ln \left ( bx-a \right ) }{{c}^{2}b}}-4\,{\frac{{a}^{2}}{{c}^{2}b \left ( bx-a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/(-b*c*x+a*c)^2,x)
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Maxima [A] time = 1.3402, size = 62, normalized size = 1.51 \[ -\frac{4 \, a^{2}}{b^{2} c^{2} x - a b c^{2}} + \frac{x}{c^{2}} + \frac{4 \, a \log \left (b x - a\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(b*c*x - a*c)^2,x, algorithm="maxima")
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Fricas [A] time = 0.198504, size = 77, normalized size = 1.88 \[ \frac{b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \,{\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} c^{2} x - a b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(b*c*x - a*c)^2,x, algorithm="fricas")
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Sympy [A] time = 1.38417, size = 39, normalized size = 0.95 \[ - \frac{4 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac{4 a \log{\left (- a + b x \right )}}{b c^{2}} + \frac{x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/(-b*c*x+a*c)**2,x)
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GIAC/XCAS [A] time = 0.206212, size = 107, normalized size = 2.61 \[ -\frac{4 \, a^{2}}{{\left (b c x - a c\right )} b c} - \frac{4 \, a{\rm ln}\left (\frac{{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2}{\left | b \right |}{\left | c \right |}}\right )}{b c^{2}} + \frac{b c x - a c}{b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(b*c*x - a*c)^2,x, algorithm="giac")
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