Optimal. Leaf size=58 \[ -\left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \log (a+b x)+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{a b^2 (a+b x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.100585, antiderivative size = 58, 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 \[ -\left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \log (a+b x)+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{a b^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^2/(x*(a + b*x)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.6581, size = 48, normalized size = 0.83 \[ - \left (- \frac{d^{2}}{b^{2}} + \frac{c^{2}}{a^{2}}\right ) \log{\left (a + b x \right )} + \frac{\left (a d - b c\right )^{2}}{a b^{2} \left (a + b x\right )} + \frac{c^{2} \log{\left (x \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**2/x/(b*x+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0844928, size = 60, normalized size = 1.03 \[ \frac{\frac{(a d-b c) ((a+b x) (a d+b c) \log (a+b x)+a (a d-b c))}{b^2 (a+b x)}+c^2 \log (x)}{a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^2/(x*(a + b*x)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.013, size = 81, normalized size = 1.4 \[{\frac{{c}^{2}\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{2}}{{b}^{2}}}-{\frac{\ln \left ( bx+a \right ){c}^{2}}{{a}^{2}}}+{\frac{{d}^{2}a}{{b}^{2} \left ( bx+a \right ) }}-2\,{\frac{cd}{b \left ( bx+a \right ) }}+{\frac{{c}^{2}}{a \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^2/x/(b*x+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.35013, size = 105, normalized size = 1.81 \[ \frac{c^{2} \log \left (x\right )}{a^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{a b^{3} x + a^{2} b^{2}} - \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)^2*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.212731, size = 144, normalized size = 2.48 \[ \frac{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} -{\left (a b^{2} c^{2} - a^{3} d^{2} +{\left (b^{3} c^{2} - a^{2} b d^{2}\right )} x\right )} \log \left (b x + a\right ) +{\left (b^{3} c^{2} x + a b^{2} c^{2}\right )} \log \left (x\right )}{a^{2} b^{3} x + a^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)^2*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 5.53214, size = 107, normalized size = 1.84 \[ \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{a^{2} b^{2} + a b^{3} x} + \frac{c^{2} \log{\left (x \right )}}{a^{2}} + \frac{\left (a d - b c\right ) \left (a d + b c\right ) \log{\left (x + \frac{- a b c^{2} + \frac{a \left (a d - b c\right ) \left (a d + b c\right )}{b}}{a^{2} d^{2} - 2 b^{2} c^{2}} \right )}}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**2/x/(b*x+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.282531, size = 146, normalized size = 2.52 \[ -b{\left (\frac{d^{2}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{c^{2}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{2} b} - \frac{\frac{b^{3} c^{2}}{b x + a} - \frac{2 \, a b^{2} c d}{b x + a} + \frac{a^{2} b d^{2}}{b x + a}}{a b^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2/((b*x + a)^2*x),x, algorithm="giac")
[Out]