Optimal. Leaf size=17 \[ -\frac {1}{3 c e (d+e x)^3} \]
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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 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )} \, dx &=\int \frac {1}{c (d+e x)^4} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^4} \, dx}{c}\\ &=-\frac {1}{3 c e (d+e x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{3 c e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 16, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{3 c e \left (e x +d \right )^{3}}\) | \(16\) |
norman | \(-\frac {1}{3 c e \left (e x +d \right )^{3}}\) | \(16\) |
gosper | \(-\frac {1}{3 \left (e x +d \right ) e c \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\) | \(34\) |
risch | \(-\frac {1}{3 \left (e x +d \right ) e c \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (15) = 30\).
time = 0.27, size = 37, normalized size = 2.18 \begin {gather*} -\frac {1}{3 \, {\left (c x^{3} e^{4} + 3 \, c d x^{2} e^{3} + 3 \, c d^{2} x e^{2} + c d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (15) = 30\).
time = 4.05, size = 37, normalized size = 2.18 \begin {gather*} -\frac {1}{3 \, {\left (c x^{3} e^{4} + 3 \, c d x^{2} e^{3} + 3 \, c d^{2} x e^{2} + c d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (14) = 28\).
time = 0.11, size = 44, normalized size = 2.59 \begin {gather*} - \frac {1}{3 c d^{3} e + 9 c d^{2} e^{2} x + 9 c d e^{3} x^{2} + 3 c e^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 15, normalized size = 0.88 \begin {gather*} -\frac {e^{\left (-1\right )}}{3 \, {\left (x e + d\right )}^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 41, normalized size = 2.41 \begin {gather*} -\frac {1}{3\,c\,d^3\,e+9\,c\,d^2\,e^2\,x+9\,c\,d\,e^3\,x^2+3\,c\,e^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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