Optimal. Leaf size=42 \[ -\frac{d \log (b+c x)}{b^2}+\frac{d \log (x)}{b^2}+\frac{c d-b e}{b c (b+c x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0308703, 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, Rules used = {765} \[ -\frac{d \log (b+c x)}{b^2}+\frac{d \log (x)}{b^2}+\frac{c d-b e}{b c (b+c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin{align*} \int \frac{x (d+e x)}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{d}{b^2 x}+\frac{-c d+b e}{b (b+c x)^2}-\frac{c d}{b^2 (b+c x)}\right ) \, dx\\ &=\frac{c d-b e}{b c (b+c x)}+\frac{d \log (x)}{b^2}-\frac{d \log (b+c x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.026959, size = 38, normalized size = 0.9 \[ \frac{\frac{b (c d-b e)}{c (b+c x)}-d \log (b+c x)+d \log (x)}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 46, normalized size = 1.1 \begin{align*}{\frac{d\ln \left ( x \right ) }{{b}^{2}}}-{\frac{e}{c \left ( cx+b \right ) }}+{\frac{d}{b \left ( cx+b \right ) }}-{\frac{d\ln \left ( cx+b \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12003, size = 58, normalized size = 1.38 \begin{align*} \frac{c d - b e}{b c^{2} x + b^{2} c} - \frac{d \log \left (c x + b\right )}{b^{2}} + \frac{d \log \left (x\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9282, size = 131, normalized size = 3.12 \begin{align*} \frac{b c d - b^{2} e -{\left (c^{2} d x + b c d\right )} \log \left (c x + b\right ) +{\left (c^{2} d x + b c d\right )} \log \left (x\right )}{b^{2} c^{2} x + b^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.580045, size = 32, normalized size = 0.76 \begin{align*} - \frac{b e - c d}{b^{2} c + b c^{2} x} + \frac{d \left (\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15894, size = 65, normalized size = 1.55 \begin{align*} -\frac{d \log \left ({\left | c x + b \right |}\right )}{b^{2}} + \frac{d \log \left ({\left | x \right |}\right )}{b^{2}} + \frac{b c d - b^{2} e}{{\left (c x + b\right )} b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]