Optimal. Leaf size=42 \[ -\frac{(b c-a d)^2 \log (a+b x)}{a b^2}+\frac{c^2 \log (x)}{a}+\frac{d^2 x}{b} \]
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Rubi [A] time = 0.0717783, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{(b c-a d)^2 \log (a+b x)}{a b^2}+\frac{c^2 \log (x)}{a}+\frac{d^2 x}{b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^2/(x*(a + b*x)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ d^{2} \int \frac{1}{b}\, dx + \frac{c^{2} \log{\left (x \right )}}{a} - \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**2/x/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0300781, size = 42, normalized size = 1. \[ \frac{-(b c-a d)^2 \log (a+b x)+a b d^2 x+b^2 c^2 \log (x)}{a b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^2/(x*(a + b*x)),x]
[Out]
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Maple [A] time = 0.009, size = 61, normalized size = 1.5 \[{\frac{{d}^{2}x}{b}}+{\frac{{c}^{2}\ln \left ( x \right ) }{a}}-{\frac{a\ln \left ( bx+a \right ){d}^{2}}{{b}^{2}}}+2\,{\frac{\ln \left ( bx+a \right ) cd}{b}}-{\frac{\ln \left ( bx+a \right ){c}^{2}}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^2/x/(b*x+a),x)
[Out]
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Maxima [A] time = 1.36945, size = 72, normalized size = 1.71 \[ \frac{d^{2} x}{b} + \frac{c^{2} \log \left (x\right )}{a} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211574, size = 72, normalized size = 1.71 \[ \frac{a b d^{2} x + b^{2} c^{2} \log \left (x\right ) -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)*x),x, algorithm="fricas")
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Sympy [A] time = 5.57297, size = 73, normalized size = 1.74 \[ \frac{d^{2} x}{b} + \frac{c^{2} \log{\left (x \right )}}{a} - \frac{\left (a d - b c\right )^{2} \log{\left (x + \frac{a b c^{2} + \frac{a \left (a d - b c\right )^{2}}{b}}{a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}} \right )}}{a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**2/x/(b*x+a),x)
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GIAC/XCAS [A] time = 0.267028, size = 74, normalized size = 1.76 \[ \frac{d^{2} x}{b} + \frac{c^{2}{\rm ln}\left ({\left | x \right |}\right )}{a} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)*x),x, algorithm="giac")
[Out]