Optimal. Leaf size=16 \[ \frac {\log (a e+c d x)}{c d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 31}
\begin {gather*} \frac {\log (a e+c d x)}{c d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 640
Rubi steps
\begin {align*} \int \frac {d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac {1}{a e+c d x} \, dx\\ &=\frac {\log (a e+c d x)}{c d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\log (a e+c d x)}{c d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.63, size = 17, normalized size = 1.06
method | result | size |
default | \(\frac {\ln \left (c d x +a e \right )}{c d}\) | \(17\) |
norman | \(\frac {\ln \left (c d x +a e \right )}{c d}\) | \(17\) |
risch | \(\frac {\ln \left (c d x +a e \right )}{c d}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 17, normalized size = 1.06 \begin {gather*} \frac {\log \left (c d x + a e\right )}{c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.52, size = 17, normalized size = 1.06 \begin {gather*} \frac {\log \left (c d x + a e\right )}{c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {\log {\left (a e + c d x \right )}}{c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.48, size = 18, normalized size = 1.12 \begin {gather*} \frac {\log \left ({\left | c d x + a e \right |}\right )}{c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.56, size = 16, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a\,e+c\,d\,x\right )}{c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________