Optimal. Leaf size=50 \[ -\frac{b x (b d-a e)}{e^2}+\frac{(b d-a e)^2 \log (d+e x)}{e^3}+\frac{(a+b x)^2}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.021483, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac{b x (b d-a e)}{e^2}+\frac{(b d-a e)^2 \log (d+e x)}{e^3}+\frac{(a+b x)^2}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{d+e x} \, dx &=\int \frac{(a+b x)^2}{d+e x} \, dx\\ &=\int \left (-\frac{b (b d-a e)}{e^2}+\frac{b (a+b x)}{e}+\frac{(-b d+a e)^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{b (b d-a e) x}{e^2}+\frac{(a+b x)^2}{2 e}+\frac{(b d-a e)^2 \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0182524, size = 43, normalized size = 0.86 \[ \frac{b e x (4 a e-2 b d+b e x)+2 (b d-a e)^2 \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 74, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,e}}+2\,{\frac{abx}{e}}-{\frac{{b}^{2}dx}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){a}^{2}}{e}}-2\,{\frac{\ln \left ( ex+d \right ) abd}{{e}^{2}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ){b}^{2}}{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.19115, size = 81, normalized size = 1.62 \begin{align*} \frac{b^{2} e x^{2} - 2 \,{\left (b^{2} d - 2 \, a b e\right )} x}{2 \, e^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64159, size = 135, normalized size = 2.7 \begin{align*} \frac{b^{2} e^{2} x^{2} - 2 \,{\left (b^{2} d e - 2 \, a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.401215, size = 44, normalized size = 0.88 \begin{align*} \frac{b^{2} x^{2}}{2 e} + \frac{x \left (2 a b e - b^{2} d\right )}{e^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14471, size = 82, normalized size = 1.64 \begin{align*}{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{2} x^{2} e - 2 \, b^{2} d x + 4 \, a b x e\right )} e^{\left (-2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]