Optimal. Leaf size=17 \[ -\frac{1}{4 c^3 e (d+e x)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0044241, 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} \[ -\frac{1}{4 c^3 e (d+e x)^4} \]
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 )^3} \, dx &=\int \frac{1}{c^3 (d+e x)^5} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^5} \, dx}{c^3}\\ &=-\frac{1}{4 c^3 e (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.0026144, size = 17, normalized size = 1. \[ -\frac{1}{4 c^3 e (d+e x)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{c}^{3}e \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.97872, size = 41, normalized size = 2.41 \begin{align*} -\frac{1}{4 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2} c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.02453, size = 119, normalized size = 7. \begin{align*} -\frac{1}{4 \,{\left (c^{3} e^{5} x^{4} + 4 \, c^{3} d e^{4} x^{3} + 6 \, c^{3} d^{2} e^{3} x^{2} + 4 \, c^{3} d^{3} e^{2} x + c^{3} d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.772306, size = 66, normalized size = 3.88 \begin{align*} - \frac{1}{4 c^{3} d^{4} e + 16 c^{3} d^{3} e^{2} x + 24 c^{3} d^{2} e^{3} x^{2} + 16 c^{3} d e^{4} x^{3} + 4 c^{3} e^{5} x^{4}} \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]