Optimal. Leaf size=15 \[ \frac {c (d+e x)^4}{4 e} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 12, 32}
\begin {gather*} \frac {c (d+e x)^4}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\int c (d+e x)^3 \, dx\\ &=c \int (d+e x)^3 \, dx\\ &=\frac {c (d+e x)^4}{4 e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {c (d+e x)^4}{4 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. \(30\) vs.
\(2(13)=26\).
time = 0.11, size = 31, normalized size = 2.07
method | result | size |
default | \(\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{2}}{4 c e}\) | \(31\) |
gosper | \(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) c}{4}\) | \(34\) |
norman | \(c \,d^{3} x +c d \,e^{2} x^{3}+\frac {1}{4} c \,x^{4} e^{3}+\frac {3}{2} c \,d^{2} e \,x^{2}\) | \(36\) |
risch | \(\frac {c \,x^{4} e^{3}}{4}+c d \,e^{2} x^{3}+\frac {3 c \,d^{2} e \,x^{2}}{2}+c \,d^{3} x +\frac {c \,d^{4}}{4 e}\) | \(45\) |
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. 29 vs.
\(2 (13) = 26\).
time = 0.27, size = 29, normalized size = 1.93 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{2} e^{\left (-1\right )}}{4 \, c} \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. 34 vs.
\(2 (13) = 26\).
time = 2.44, size = 34, normalized size = 2.27 \begin {gather*} \frac {1}{4} \, c x^{4} e^{3} + c d x^{3} e^{2} + \frac {3}{2} \, c d^{2} x^{2} e + c d^{3} 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. 39 vs.
\(2 (10) = 20\).
time = 0.01, size = 39, normalized size = 2.60 \begin {gather*} c d^{3} x + \frac {3 c d^{2} e x^{2}}{2} + c d e^{2} x^{3} + \frac {c e^{3} x^{4}}{4} \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. 36 vs.
\(2 (13) = 26\).
time = 5.45, size = 36, normalized size = 2.40 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} c d^{2} + \frac {1}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} c e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 35, normalized size = 2.33 \begin {gather*} c\,d^3\,x+\frac {3\,c\,d^2\,e\,x^2}{2}+c\,d\,e^2\,x^3+\frac {c\,e^3\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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