Optimal. Leaf size=64 \[ \frac{e^2 x (3 c d-b e)}{c^2}-\frac{(c d-b e)^3 \log (b+c x)}{b c^3}+\frac{d^3 \log (x)}{b}+\frac{e^3 x^2}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0574003, antiderivative size = 64, 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} \[ \frac{e^2 x (3 c d-b e)}{c^2}-\frac{(c d-b e)^3 \log (b+c x)}{b c^3}+\frac{d^3 \log (x)}{b}+\frac{e^3 x^2}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{b x+c x^2} \, dx &=\int \left (\frac{e^2 (3 c d-b e)}{c^2}+\frac{d^3}{b x}+\frac{e^3 x}{c}+\frac{(-c d+b e)^3}{b c^2 (b+c x)}\right ) \, dx\\ &=\frac{e^2 (3 c d-b e) x}{c^2}+\frac{e^3 x^2}{2 c}+\frac{d^3 \log (x)}{b}-\frac{(c d-b e)^3 \log (b+c x)}{b c^3}\\ \end{align*}
Mathematica [A] time = 0.0275231, size = 59, normalized size = 0.92 \[ \frac{b c e^2 x (-2 b e+6 c d+c e x)-2 (c d-b e)^3 \log (b+c x)+2 c^3 d^3 \log (x)}{2 b c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 103, normalized size = 1.6 \begin{align*}{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{{e}^{3}xb}{{c}^{2}}}+3\,{\frac{d{e}^{2}x}{c}}+{\frac{{d}^{3}\ln \left ( x \right ) }{b}}+{\frac{{b}^{2}\ln \left ( cx+b \right ){e}^{3}}{{c}^{3}}}-3\,{\frac{b\ln \left ( cx+b \right ) d{e}^{2}}{{c}^{2}}}+3\,{\frac{\ln \left ( cx+b \right ){d}^{2}e}{c}}-{\frac{\ln \left ( cx+b \right ){d}^{3}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09648, size = 123, normalized size = 1.92 \begin{align*} \frac{d^{3} \log \left (x\right )}{b} + \frac{c e^{3} x^{2} + 2 \,{\left (3 \, c d e^{2} - b e^{3}\right )} x}{2 \, c^{2}} - \frac{{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.70464, size = 204, normalized size = 3.19 \begin{align*} \frac{b c^{2} e^{3} x^{2} + 2 \, c^{3} d^{3} \log \left (x\right ) + 2 \,{\left (3 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x - 2 \,{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 5.58811, size = 112, normalized size = 1.75 \begin{align*} \frac{e^{3} x^{2}}{2 c} - \frac{x \left (b e^{3} - 3 c d e^{2}\right )}{c^{2}} + \frac{d^{3} \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right )^{3} \log{\left (x + \frac{- b c^{2} d^{3} + \frac{b \left (b e - c d\right )^{3}}{c}}{b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - 2 c^{3} d^{3}} \right )}}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30506, size = 117, normalized size = 1.83 \begin{align*} \frac{d^{3} \log \left ({\left | x \right |}\right )}{b} + \frac{c x^{2} e^{3} + 6 \, c d x e^{2} - 2 \, b x e^{3}}{2 \, c^{2}} - \frac{{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]