Optimal. Leaf size=86 \[ \frac{(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac{(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac{(e f-d g)^2}{2 d e^3 (d+e x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0855682, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {848, 88} \[ \frac{(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac{(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac{(e f-d g)^2}{2 d e^3 (d+e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 848
Rule 88
Rubi steps
\begin{align*} \int \frac{(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )} \, dx &=\int \frac{(f+g x)^2}{(d-e x) (d+e x)^2} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{4 d^2 e^2 (d-e x)}+\frac{(-e f+d g)^2}{2 d e^2 (d+e x)^2}+\frac{(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)}\right ) \, dx\\ &=-\frac{(e f-d g)^2}{2 d e^3 (d+e x)}-\frac{(e f+d g)^2 \log (d-e x)}{4 d^2 e^3}+\frac{(e f-d g) (e f+3 d g) \log (d+e x)}{4 d^2 e^3}\\ \end{align*}
Mathematica [A] time = 0.0486066, size = 82, normalized size = 0.95 \[ \frac{(e f-d g) ((d+e x) (3 d g+e f) \log (d+e x)+2 d (d g-e f))-(d+e x) (d g+e f)^2 \log (d-e x)}{4 d^2 e^3 (d+e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.054, size = 149, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{4\,{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{2\,d{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{4\,{d}^{2}e}}-{\frac{3\,\ln \left ( ex+d \right ){g}^{2}}{4\,{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) fg}{2\,d{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{4\,{d}^{2}e}}-{\frac{{g}^{2}d}{2\,{e}^{3} \left ( ex+d \right ) }}+{\frac{fg}{{e}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{2\,de \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.985877, size = 153, normalized size = 1.78 \begin{align*} -\frac{e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}}{2 \,{\left (d e^{4} x + d^{2} e^{3}\right )}} + \frac{{\left (e^{2} f^{2} + 2 \, d e f g - 3 \, d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{2} e^{3}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{2} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.83723, size = 343, normalized size = 3.99 \begin{align*} -\frac{2 \, d e^{2} f^{2} - 4 \, d^{2} e f g + 2 \, d^{3} g^{2} -{\left (d e^{2} f^{2} + 2 \, d^{2} e f g - 3 \, d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x + d\right ) +{\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g + d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{4 \,{\left (d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.14778, size = 182, normalized size = 2.12 \begin{align*} - \frac{d^{2} g^{2} - 2 d e f g + e^{2} f^{2}}{2 d^{2} e^{3} + 2 d e^{4} x} - \frac{\left (d g - e f\right ) \left (3 d g + e f\right ) \log{\left (x + \frac{- 2 d^{3} g^{2} + d \left (d g - e f\right ) \left (3 d g + e f\right )}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} - \frac{\left (d g + e f\right )^{2} \log{\left (x + \frac{- 2 d^{3} g^{2} + d \left (d g + e f\right )^{2}}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]