Optimal. Leaf size=3 \[ c x \]
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Rubi [A]
time = 0.00, antiderivative size = 3, 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 = {24, 21, 8}
\begin {gather*} c x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 21
Rule 24
Rubi steps
\begin {align*} \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^2} \, dx &=\frac {\int \frac {c d e^2+c e^3 x}{d+e x} \, dx}{e^2}\\ &=c \int 1 \, dx\\ &=c x\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 3, normalized size = 1.00 \begin {gather*} c x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 4, normalized size = 1.33
method | result | size |
default | \(c x\) | \(4\) |
risch | \(c x\) | \(4\) |
norman | \(\frac {c e \,x^{2}+c d x}{e x +d}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 3, normalized size = 1.00 \begin {gather*} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.91, size = 3, normalized size = 1.00 \begin {gather*} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 2, normalized size = 0.67 \begin {gather*} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 4.07, size = 110, normalized size = 36.67 \begin {gather*} -2 \, {\left (e^{\left (-1\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {d e^{\left (-1\right )}}{x e + d}\right )} c d + {\left (2 \, d e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (x e + d\right )} e^{\left (-3\right )} - \frac {d^{2} e^{\left (-3\right )}}{x e + d}\right )} c e^{2} - \frac {c d^{2} e^{\left (-1\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.01, size = 3, normalized size = 1.00 \begin {gather*} c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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