Optimal. Leaf size=43 \[ -\frac{\log (x) (b c-a d)}{a^2}+\frac{(b c-a d) \log (a+b x)}{a^2}-\frac{c}{a x} \]
[Out]
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Rubi [A] time = 0.0740463, antiderivative size = 43, 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 \[ -\frac{\log (x) (b c-a d)}{a^2}+\frac{(b c-a d) \log (a+b x)}{a^2}-\frac{c}{a x} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)/(x^2*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 15.3107, size = 34, normalized size = 0.79 \[ - \frac{c}{a x} + \frac{\left (a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right ) \log{\left (a + b x \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)/x**2/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0274577, size = 42, normalized size = 0.98 \[ \frac{\log (x) (a d-b c)}{a^2}+\frac{(b c-a d) \log (a+b x)}{a^2}-\frac{c}{a x} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)/(x^2*(a + b*x)),x]
[Out]
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Maple [A] time = 0.012, size = 51, normalized size = 1.2 \[ -{\frac{c}{ax}}+{\frac{\ln \left ( x \right ) d}{a}}-{\frac{b\ln \left ( x \right ) c}{{a}^{2}}}-{\frac{\ln \left ( bx+a \right ) d}{a}}+{\frac{\ln \left ( bx+a \right ) bc}{{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)/x^2/(b*x+a),x)
[Out]
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Maxima [A] time = 1.34533, size = 58, normalized size = 1.35 \[ \frac{{\left (b c - a d\right )} \log \left (b x + a\right )}{a^{2}} - \frac{{\left (b c - a d\right )} \log \left (x\right )}{a^{2}} - \frac{c}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/((b*x + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2087, size = 55, normalized size = 1.28 \[ \frac{{\left (b c - a d\right )} x \log \left (b x + a\right ) -{\left (b c - a d\right )} x \log \left (x\right ) - a c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/((b*x + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.06773, size = 95, normalized size = 2.21 \[ - \frac{c}{a x} + \frac{\left (a d - b c\right ) \log{\left (x + \frac{a^{2} d - a b c - a \left (a d - b c\right )}{2 a b d - 2 b^{2} c} \right )}}{a^{2}} - \frac{\left (a d - b c\right ) \log{\left (x + \frac{a^{2} d - a b c + a \left (a d - b c\right )}{2 a b d - 2 b^{2} c} \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)/x**2/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.307341, size = 69, normalized size = 1.6 \[ -\frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{c}{a x} + \frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/((b*x + a)*x^2),x, algorithm="giac")
[Out]