Optimal. Leaf size=17 \[ \frac {c^2 (d+e x)^3}{3 e} \]
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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 {c^2 (d+e x)^3}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
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)^2} \, dx &=\int c^2 (d+e x)^2 \, dx\\ &=c^2 \int (d+e x)^2 \, dx\\ &=\frac {c^2 (d+e x)^3}{3 e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (d+e x)^3}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 16, normalized size = 0.94
method | result | size |
default | \(\frac {c^{2} \left (e x +d \right )^{3}}{3 e}\) | \(16\) |
gosper | \(\frac {x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) c^{2}}{3}\) | \(25\) |
risch | \(\frac {c^{2} e^{2} x^{3}}{3}+c^{2} d e \,x^{2}+d^{2} c^{2} x +\frac {c^{2} d^{3}}{3 e}\) | \(41\) |
norman | \(\frac {c^{2} d^{3} x +\frac {1}{3} c^{2} x^{4} e^{3}+\frac {4}{3} c^{2} d \,e^{2} x^{3}+2 c^{2} d^{2} e \,x^{2}}{e x +d}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{3} \, c^{2} x^{3} e^{2} + c^{2} d x^{2} e + c^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.50, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{3} \, c^{2} x^{3} e^{2} + c^{2} d x^{2} e + c^{2} d^{2} x \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. 29 vs.
\(2 (12) = 24\).
time = 0.02, size = 29, normalized size = 1.71 \begin {gather*} c^{2} d^{2} x + c^{2} d e x^{2} + \frac {c^{2} e^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.81, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, {\left (x e + d\right )}^{3} c^{2} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 1.41 \begin {gather*} \frac {c^2\,x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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