Optimal. Leaf size=110 \[ -\frac{c^2}{b^2 (b+c x) (c d-b e)}+\frac{c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac{\log (x) (b e+2 c d)}{b^3 d^2}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.118041, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{c^2}{b^2 (b+c x) (c d-b e)}+\frac{c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac{\log (x) (b e+2 c d)}{b^3 d^2}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{1}{b^2 d x^2}+\frac{-2 c d-b e}{b^3 d^2 x}-\frac{c^3}{b^2 (-c d+b e) (b+c x)^2}-\frac{c^3 (-2 c d+3 b e)}{b^3 (-c d+b e)^2 (b+c x)}+\frac{e^4}{d^2 (c d-b e)^2 (d+e x)}\right ) \, dx\\ &=-\frac{1}{b^2 d x}-\frac{c^2}{b^2 (c d-b e) (b+c x)}-\frac{(2 c d+b e) \log (x)}{b^3 d^2}+\frac{c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2}\\ \end{align*}
Mathematica [A] time = 0.0960025, size = 111, normalized size = 1.01 \[ \frac{c^2}{b^2 (b+c x) (b e-c d)}+\frac{\left (2 c^3 d-3 b c^2 e\right ) \log (b+c x)}{b^3 (b e-c d)^2}+\frac{\log (x) (-b e-2 c d)}{b^3 d^2}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.105, size = 132, normalized size = 1.2 \begin{align*} -{\frac{1}{{b}^{2}dx}}-{\frac{\ln \left ( x \right ) e}{{b}^{2}{d}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{d{b}^{3}}}+{\frac{{c}^{2}}{ \left ( be-cd \right ){b}^{2} \left ( cx+b \right ) }}-3\,{\frac{{c}^{2}\ln \left ( cx+b \right ) e}{ \left ( be-cd \right ) ^{2}{b}^{2}}}+2\,{\frac{{c}^{3}\ln \left ( cx+b \right ) d}{ \left ( be-cd \right ) ^{2}{b}^{3}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) }{{d}^{2} \left ( be-cd \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17669, size = 239, normalized size = 2.17 \begin{align*} \frac{e^{3} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} + \frac{{\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}} - \frac{b c d - b^{2} e +{\left (2 \, c^{2} d - b c e\right )} x}{{\left (b^{2} c^{2} d^{2} - b^{3} c d e\right )} x^{2} +{\left (b^{3} c d^{2} - b^{4} d e\right )} x} - \frac{{\left (2 \, c d + b e\right )} \log \left (x\right )}{b^{3} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 24.7447, size = 574, normalized size = 5.22 \begin{align*} -\frac{b^{2} c^{2} d^{3} - 2 \, b^{3} c d^{2} e + b^{4} d e^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{3} c d e^{2}\right )} x -{\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e\right )} x^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e\right )} x\right )} \log \left (c x + b\right ) -{\left (b^{3} c e^{3} x^{2} + b^{4} e^{3} x\right )} \log \left (e x + d\right ) +{\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{3} c e^{3}\right )} x^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{4} e^{3}\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e + b^{5} c d^{2} e^{2}\right )} x^{2} +{\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29849, size = 273, normalized size = 2.48 \begin{align*} \frac{{\left (2 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3} d^{2} - 2 \, b^{4} c^{2} d e + b^{5} c e^{2}} + \frac{e^{4} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}} - \frac{{\left (2 \, c d + b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{2}} - \frac{b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} +{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} x}{{\left (c d - b e\right )}^{2}{\left (c x + b\right )} b^{2} d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]