Optimal. Leaf size=17 \[ -\frac{c^2}{3 e (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0047521, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ -\frac{c^2}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac{c^2}{(d+e x)^4} \, dx\\ &=c^2 \int \frac{1}{(d+e x)^4} \, dx\\ &=-\frac{c^2}{3 e (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0019206, size = 17, normalized size = 1. \[ -\frac{c^2}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 16, normalized size = 0.9 \begin{align*} -{\frac{{c}^{2}}{3\,e \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.19668, size = 51, normalized size = 3. \begin{align*} -\frac{c^{2}}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.93112, size = 76, normalized size = 4.47 \begin{align*} -\frac{c^{2}}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.66286, size = 39, normalized size = 2.29 \begin{align*} - \frac{c^{2}}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26211, size = 89, normalized size = 5.24 \begin{align*} -\frac{{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]