Optimal. Leaf size=55 \[ -\frac {a e^2-b d e+c d^2}{e^3 (d+e x)}-\frac {(2 c d-b e) \log (d+e x)}{e^3}+\frac {c x}{e^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {698} \begin {gather*} -\frac {a e^2-b d e+c d^2}{e^3 (d+e x)}-\frac {(2 c d-b e) \log (d+e x)}{e^3}+\frac {c x}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx &=\int \left (\frac {c}{e^2}+\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^2}+\frac {-2 c d+b e}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {c x}{e^2}-\frac {c d^2-b d e+a e^2}{e^3 (d+e x)}-\frac {(2 c d-b e) \log (d+e x)}{e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 49, normalized size = 0.89 \begin {gather*} \frac {-\frac {a e^2-b d e+c d^2}{d+e x}+(b e-2 c d) \log (d+e x)+c e x}{e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 78, normalized size = 1.42 \begin {gather*} \frac {c e^{2} x^{2} + c d e x - c d^{2} + b d e - a e^{2} - {\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 106, normalized size = 1.93 \begin {gather*} -{\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 )} b e^{\left (-1\right )} + {\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 - \frac {a e^{\left (-1\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 74, normalized size = 1.35 \begin {gather*} -\frac {a}{\left (e x +d \right ) e}+\frac {b d}{\left (e x +d \right ) e^{2}}+\frac {b \ln \left (e x +d \right )}{e^{2}}-\frac {c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 c d \ln \left (e x +d \right )}{e^{3}}+\frac {c x}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.06, size = 58, normalized size = 1.05 \begin {gather*} -\frac {c d^{2} - b d e + a e^{2}}{e^{4} x + d e^{3}} + \frac {c x}{e^{2}} - \frac {{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.66, size = 59, normalized size = 1.07 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (b\,e-2\,c\,d\right )}{e^3}-\frac {c\,d^2-b\,d\,e+a\,e^2}{e\,\left (x\,e^3+d\,e^2\right )}+\frac {c\,x}{e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.33, size = 49, normalized size = 0.89 \begin {gather*} \frac {c x}{e^{2}} + \frac {- a e^{2} + b d e - c d^{2}}{d e^{3} + e^{4} x} + \frac {\left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________