Optimal. Leaf size=17 \[ -\frac{1}{2 c^2 e (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0043401, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {27, 12, 32} \[ -\frac{1}{2 c^2 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac{1}{c^2 (d+e x)^3} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^3} \, dx}{c^2}\\ &=-\frac{1}{2 c^2 e (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0020644, size = 17, normalized size = 1. \[ -\frac{1}{2 c^2 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{c}^{2}e \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.08769, size = 41, normalized size = 2.41 \begin{align*} -\frac{1}{2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.03139, size = 65, normalized size = 3.82 \begin{align*} -\frac{1}{2 \,{\left (c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.621539, size = 36, normalized size = 2.12 \begin{align*} - \frac{1}{2 c^{2} d^{2} e + 4 c^{2} d e^{2} x + 2 c^{2} e^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]