Optimal. Leaf size=17 \[ -\frac {1}{3 c^3 e (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32}
\begin {gather*} -\frac {1}{3 c^3 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\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)^4} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^4} \, dx}{c^3}\\ &=-\frac {1}{3 c^3 e (d+e x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{3 c^3 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.68, size = 16, normalized size = 0.94
method | result | size |
gosper | \(-\frac {1}{3 c^{3} e \left (e x +d \right )^{3}}\) | \(16\) |
default | \(-\frac {1}{3 c^{3} e \left (e x +d \right )^{3}}\) | \(16\) |
risch | \(-\frac {1}{3 e \left (e x +d \right ) \left (e^{2} x^{2}+2 d x e +d^{2}\right ) c^{3}}\) | \(34\) |
norman | \(\frac {-\frac {d^{2}}{3 e c}-\frac {e \,x^{2}}{3 c}-\frac {2 x d}{3 c}}{c^{2} \left (e x +d \right )^{5}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (15) = 30\).
time = 0.31, size = 45, normalized size = 2.65 \begin {gather*} -\frac {1}{3 \, {\left (c^{3} x^{3} e^{4} + 3 \, c^{3} d x^{2} e^{3} + 3 \, c^{3} d^{2} x e^{2} + c^{3} d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (15) = 30\).
time = 2.22, size = 45, normalized size = 2.65 \begin {gather*} -\frac {1}{3 \, {\left (c^{3} x^{3} e^{4} + 3 \, c^{3} d x^{2} e^{3} + 3 \, c^{3} d^{2} x e^{2} + c^{3} d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (15) = 30\).
time = 0.12, size = 51, normalized size = 3.00 \begin {gather*} - \frac {1}{3 c^{3} d^{3} e + 9 c^{3} d^{2} e^{2} x + 9 c^{3} d e^{3} x^{2} + 3 c^{3} e^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.14, size = 15, normalized size = 0.88 \begin {gather*} -\frac {e^{\left (-1\right )}}{3 \, {\left (x e + d\right )}^{3} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.41, size = 49, normalized size = 2.88 \begin {gather*} -\frac {1}{3\,c^3\,d^3\,e+9\,c^3\,d^2\,e^2\,x+9\,c^3\,d\,e^3\,x^2+3\,c^3\,e^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________