Optimal. Leaf size=26 \[ \left (a-\frac {c d^2}{e^2}\right ) \log (d+e x)+\frac {c d x}{e} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 43} \begin {gather*} \left (a-\frac {c d^2}{e^2}\right ) \log (d+e x)+\frac {c d x}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 24
Rule 43
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{d+e x} \, dx}{e^2}\\ &=\frac {\int \left (c d e+\frac {-c d^2 e+a e^3}{d+e x}\right ) \, dx}{e^2}\\ &=\frac {c d x}{e}+\left (a-\frac {c d^2}{e^2}\right ) \log (d+e x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 30, normalized size = 1.15 \begin {gather*} \frac {\left (a e^2-c d^2\right ) \log (d+e x)}{e^2}+\frac {c d x}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 30, normalized size = 1.15 \begin {gather*} \frac {c d e x - {\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 117, normalized size = 4.50 \begin {gather*} {\left (2 \, d e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (x e + d\right )} e^{\left (-3\right )} - \frac {d^{2} e^{\left (-3\right )}}{x e + d}\right )} c d e - {\left (c d^{2} + a e^{2}\right )} {\left (e^{\left (-1\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {d e^{\left (-1\right )}}{x e + d}\right )} e^{\left (-1\right )} - \frac {a d}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 32, normalized size = 1.23 \begin {gather*} a \ln \left (e x +d \right )-\frac {c \,d^{2} \ln \left (e x +d \right )}{e^{2}}+\frac {c d x}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 31, normalized size = 1.19 \begin {gather*} \frac {c d x}{e} - \frac {{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 30, normalized size = 1.15 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (a\,e^2-c\,d^2\right )}{e^2}+\frac {c\,d\,x}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.18, size = 26, normalized size = 1.00 \begin {gather*} \frac {c d x}{e} + \frac {\left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________