Optimal. Leaf size=20 \[ -\frac {1}{3 c d (a e+c d x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32}
\begin {gather*} -\frac {1}{3 c d (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {1}{(a e+c d x)^4} \, dx\\ &=-\frac {1}{3 c d (a e+c d x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{3 c d (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.70, size = 19, normalized size = 0.95
method | result | size |
gosper | \(-\frac {1}{3 c d \left (c d x +a e \right )^{3}}\) | \(19\) |
default | \(-\frac {1}{3 c d \left (c d x +a e \right )^{3}}\) | \(19\) |
risch | \(-\frac {1}{3 c d \left (c d x +a e \right )^{3}}\) | \(19\) |
norman | \(\frac {-\frac {d e x}{c}-\frac {e^{2} x^{2}}{c}-\frac {d^{2}}{3 c}-\frac {e^{3} x^{3}}{3 c d}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\) | \(61\) |
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. 51 vs.
\(2 (19) = 38\).
time = 0.28, size = 51, normalized size = 2.55 \begin {gather*} -\frac {1}{3 \, {\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} x^{2} e + 3 \, a^{2} c^{2} d^{2} x e^{2} + a^{3} c d e^{3}\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. 51 vs.
\(2 (19) = 38\).
time = 4.57, size = 51, normalized size = 2.55 \begin {gather*} -\frac {1}{3 \, {\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} x^{2} e + 3 \, a^{2} c^{2} d^{2} x e^{2} + a^{3} c d e^{3}\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. 58 vs.
\(2 (17) = 34\).
time = 0.17, size = 58, normalized size = 2.90 \begin {gather*} - \frac {1}{3 a^{3} c d e^{3} + 9 a^{2} c^{2} d^{2} e^{2} x + 9 a c^{3} d^{3} e x^{2} + 3 c^{4} d^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.61, size = 19, normalized size = 0.95 \begin {gather*} -\frac {1}{3 \, {\left (c d x + a e\right )}^{3} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.57, size = 54, normalized size = 2.70 \begin {gather*} -\frac {1}{3\,a^3\,c\,d\,e^3+9\,a^2\,c^2\,d^2\,e^2\,x+9\,a\,c^3\,d^3\,e\,x^2+3\,c^4\,d^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________