Optimal. Leaf size=32 \[ \frac{e \log (b+c x)}{c^2}-\frac{c d-b e}{c^2 (b+c x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.029932, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{e \log (b+c x)}{c^2}-\frac{c d-b e}{c^2 (b+c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{c d-b e}{c (b+c x)^2}+\frac{e}{c (b+c x)}\right ) \, dx\\ &=-\frac{c d-b e}{c^2 (b+c x)}+\frac{e \log (b+c x)}{c^2}\\ \end{align*}
Mathematica [A] time = 0.010746, size = 31, normalized size = 0.97 \[ \frac{b e-c d}{c^2 (b+c x)}+\frac{e \log (b+c x)}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 39, normalized size = 1.2 \begin{align*}{\frac{be}{{c}^{2} \left ( cx+b \right ) }}-{\frac{d}{c \left ( cx+b \right ) }}+{\frac{e\ln \left ( cx+b \right ) }{{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12139, size = 47, normalized size = 1.47 \begin{align*} -\frac{c d - b e}{c^{3} x + b c^{2}} + \frac{e \log \left (c x + b\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74865, size = 80, normalized size = 2.5 \begin{align*} -\frac{c d - b e -{\left (c e x + b e\right )} \log \left (c x + b\right )}{c^{3} x + b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.384825, size = 27, normalized size = 0.84 \begin{align*} \frac{b e - c d}{b c^{2} + c^{3} x} + \frac{e \log{\left (b + c x \right )}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10128, size = 47, normalized size = 1.47 \begin{align*} \frac{e \log \left ({\left | c x + b \right |}\right )}{c^{2}} - \frac{c d - b e}{{\left (c x + b\right )} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]