Optimal. Leaf size=51 \[ -\frac{c \log (x) (b c-2 a d)}{a^2}+\frac{(b c-a d)^2 \log (a+b x)}{a^2 b}-\frac{c^2}{a x} \]
[Out]
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Rubi [A] time = 0.0925061, antiderivative size = 51, 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 \[ -\frac{c \log (x) (b c-2 a d)}{a^2}+\frac{(b c-a d)^2 \log (a+b x)}{a^2 b}-\frac{c^2}{a x} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^2/(x^2*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 21.3072, size = 42, normalized size = 0.82 \[ - \frac{c^{2}}{a x} + \frac{c \left (2 a d - b c\right ) \log{\left (x \right )}}{a^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**2/x**2/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0383928, size = 51, normalized size = 1. \[ \frac{-a b c^2+b c x \log (x) (2 a d-b c)+x (b c-a d)^2 \log (a+b x)}{a^2 b x} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^2/(x^2*(a + b*x)),x]
[Out]
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Maple [A] time = 0.011, size = 73, normalized size = 1.4 \[ -{\frac{{c}^{2}}{ax}}+2\,{\frac{c\ln \left ( x \right ) d}{a}}-{\frac{{c}^{2}\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{2}}{b}}-2\,{\frac{\ln \left ( bx+a \right ) cd}{a}}+{\frac{b\ln \left ( bx+a \right ){c}^{2}}{{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^2/x^2/(b*x+a),x)
[Out]
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Maxima [A] time = 1.3496, size = 86, normalized size = 1.69 \[ -\frac{c^{2}}{a x} - \frac{{\left (b c^{2} - 2 \, a c d\right )} \log \left (x\right )}{a^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209106, size = 89, normalized size = 1.75 \[ -\frac{a b c^{2} -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \log \left (b x + a\right ) +{\left (b^{2} c^{2} - 2 \, a b c d\right )} x \log \left (x\right )}{a^{2} b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.55078, size = 141, normalized size = 2.76 \[ - \frac{c^{2}}{a x} + \frac{c \left (2 a d - b c\right ) \log{\left (x + \frac{- 2 a^{2} c d + a b c^{2} + a c \left (2 a d - b c\right )}{a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}} \right )}}{a^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (x + \frac{- 2 a^{2} c d + a b c^{2} + \frac{a \left (a d - b c\right )^{2}}{b}}{a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}} \right )}}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**2/x**2/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.263427, size = 89, normalized size = 1.75 \[ -\frac{c^{2}}{a x} - \frac{{\left (b c^{2} - 2 \, a c d\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)*x^2),x, algorithm="giac")
[Out]