Optimal. Leaf size=17 \[ \frac {c^2 (d+e x)^5}{5 e} \]
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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {27, 12, 32}
\begin {gather*} \frac {c^2 (d+e x)^5}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^4 \, dx\\ &=c^2 \int (d+e x)^4 \, dx\\ &=\frac {c^2 (d+e x)^5}{5 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)^5}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs.
\(2(15)=30\).
time = 0.60, size = 58, normalized size = 3.41
method | result | size |
gosper | \(\frac {x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) c^{2}}{5}\) | \(47\) |
default | \(\frac {1}{5} c^{2} x^{5} e^{4}+c^{2} d \,e^{3} x^{4}+2 c^{2} d^{2} e^{2} x^{3}+2 c^{2} d^{3} e \,x^{2}+c^{2} d^{4} x\) | \(58\) |
norman | \(\frac {1}{5} c^{2} x^{5} e^{4}+c^{2} d \,e^{3} x^{4}+2 c^{2} d^{2} e^{2} x^{3}+2 c^{2} d^{3} e \,x^{2}+c^{2} d^{4} x\) | \(58\) |
risch | \(\frac {1}{5} c^{2} x^{5} e^{4}+c^{2} d \,e^{3} x^{4}+2 c^{2} d^{2} e^{2} x^{3}+2 c^{2} d^{3} e \,x^{2}+c^{2} d^{4} x\) | \(58\) |
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. 65 vs.
\(2 (15) = 30\).
time = 0.26, size = 65, normalized size = 3.82 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} e^{4} + c^{2} d x^{4} e^{3} + \frac {4}{3} \, c^{2} d^{2} x^{3} e^{2} + c^{2} d^{4} x + \frac {2}{3} \, {\left (c x^{3} e^{2} + 3 \, c d x^{2} e\right )} c d^{2} \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. 55 vs.
\(2 (15) = 30\).
time = 2.12, size = 55, normalized size = 3.24 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} e^{4} + c^{2} d x^{4} e^{3} + 2 \, c^{2} d^{2} x^{3} e^{2} + 2 \, c^{2} d^{3} x^{2} e + c^{2} d^{4} 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. 60 vs.
\(2 (12) = 24\).
time = 0.01, size = 60, normalized size = 3.53 \begin {gather*} c^{2} d^{4} x + 2 c^{2} d^{3} e x^{2} + 2 c^{2} d^{2} e^{2} x^{3} + c^{2} d e^{3} x^{4} + \frac {c^{2} e^{4} x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (15) = 30\).
time = 3.32, size = 55, normalized size = 3.24 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} e^{4} + c^{2} d x^{4} e^{3} + 2 \, c^{2} d^{2} x^{3} e^{2} + 2 \, c^{2} d^{3} x^{2} e + c^{2} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 57, normalized size = 3.35 \begin {gather*} c^2\,d^4\,x+2\,c^2\,d^3\,e\,x^2+2\,c^2\,d^2\,e^2\,x^3+c^2\,d\,e^3\,x^4+\frac {c^2\,e^4\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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