Optimal. Leaf size=15 \[ \frac {c (d+e x)^5}{5 e} \]
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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 = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {27, 12, 32}
\begin {gather*} \frac {c (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 (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\int c (d+e x)^4 \, dx\\ &=c \int (d+e x)^4 \, dx\\ &=\frac {c (d+e x)^5}{5 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)^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. \(47\) vs.
\(2(13)=26\).
time = 0.45, size = 48, normalized size = 3.20
method | result | size |
gosper | \(\frac {x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) c}{5}\) | \(45\) |
default | \(\frac {1}{5} e^{4} c \,x^{5}+d \,e^{3} c \,x^{4}+2 d^{2} e^{2} c \,x^{3}+2 c \,d^{3} e \,x^{2}+c \,d^{4} x\) | \(48\) |
norman | \(\frac {1}{5} e^{4} c \,x^{5}+d \,e^{3} c \,x^{4}+2 d^{2} e^{2} c \,x^{3}+2 c \,d^{3} e \,x^{2}+c \,d^{4} x\) | \(48\) |
risch | \(\frac {1}{5} e^{4} c \,x^{5}+d \,e^{3} c \,x^{4}+2 d^{2} e^{2} c \,x^{3}+2 c \,d^{3} e \,x^{2}+c \,d^{4} x\) | \(48\) |
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. 45 vs.
\(2 (13) = 26\).
time = 0.28, size = 45, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, c x^{5} e^{4} + c d x^{4} e^{3} + 2 \, c d^{2} x^{3} e^{2} + 2 \, c d^{3} x^{2} e + c d^{4} x \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. 45 vs.
\(2 (13) = 26\).
time = 3.00, size = 45, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, c x^{5} e^{4} + c d x^{4} e^{3} + 2 \, c d^{2} x^{3} e^{2} + 2 \, c d^{3} x^{2} e + c 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. 51 vs.
\(2 (10) = 20\).
time = 0.01, size = 51, normalized size = 3.40 \begin {gather*} c d^{4} x + 2 c d^{3} e x^{2} + 2 c d^{2} e^{2} x^{3} + c d e^{3} x^{4} + \frac {c 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. 45 vs.
\(2 (13) = 26\).
time = 3.08, size = 45, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, c x^{5} e^{4} + c d x^{4} e^{3} + 2 \, c d^{2} x^{3} e^{2} + 2 \, c d^{3} x^{2} e + c d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 47, normalized size = 3.13 \begin {gather*} c\,d^4\,x+2\,c\,d^3\,e\,x^2+2\,c\,d^2\,e^2\,x^3+c\,d\,e^3\,x^4+\frac {c\,e^4\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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