Optimal. Leaf size=36 \[ \frac {\log (d+e x)}{e f-d g}-\frac {\log (f+g x)}{e f-d g} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {616, 31} \[ \frac {\log (d+e x)}{e f-d g}-\frac {\log (f+g x)}{e f-d g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 616
Rubi steps
\begin {align*} \int \frac {1}{d f+(e f+d g) x+e g x^2} \, dx &=-\frac {(e g) \int \frac {1}{e f+e g x} \, dx}{e f-d g}+\frac {(e g) \int \frac {1}{d g+e g x} \, dx}{e f-d g}\\ &=\frac {\log (d+e x)}{e f-d g}-\frac {\log (f+g x)}{e f-d g}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 26, normalized size = 0.72 \[ \frac {\log (d+e x)-\log (f+g x)}{e f-d g} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 26, normalized size = 0.72 \[ \frac {\log \left (e x + d\right ) - \log \left (g x + f\right )}{e f - d g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 49, normalized size = 1.36 \[ \frac {g \log \left ({\left | g x + f \right |}\right )}{d g^{2} - f g e} - \frac {e \log \left ({\left | x e + d \right |}\right )}{d g e - f e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 37, normalized size = 1.03 \[ -\frac {\ln \left (e x +d \right )}{d g -e f}+\frac {\ln \left (g x +f \right )}{d g -e f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.10, size = 36, normalized size = 1.00 \[ \frac {\log \left (e x + d\right )}{e f - d g} - \frac {\log \left (g x + f\right )}{e f - d g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 40, normalized size = 1.11 \[ \frac {\mathrm {atan}\left (\frac {e\,f\,2{}\mathrm {i}+e\,g\,x\,2{}\mathrm {i}}{d\,g-e\,f}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d\,g-e\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.31, size = 128, normalized size = 3.56 \[ \frac {\log {\left (x + \frac {- \frac {d^{2} g^{2}}{d g - e f} + \frac {2 d e f g}{d g - e f} + d g - \frac {e^{2} f^{2}}{d g - e f} + e f}{2 e g} \right )}}{d g - e f} - \frac {\log {\left (x + \frac {\frac {d^{2} g^{2}}{d g - e f} - \frac {2 d e f g}{d g - e f} + d g + \frac {e^{2} f^{2}}{d g - e f} + e f}{2 e g} \right )}}{d g - e f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________