Optimal. Leaf size=13 \[ \frac {\log (d+e x)}{c^3 e} \]
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Rubi [A]
time = 0.00, antiderivative size = 13, 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, 31}
\begin {gather*} \frac {\log (d+e x)}{c^3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3 (d+e x)} \, dx\\ &=\frac {\int \frac {1}{d+e x} \, dx}{c^3}\\ &=\frac {\log (d+e x)}{c^3 e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {\log (d+e x)}{c^3 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 14, normalized size = 1.08
method | result | size |
default | \(\frac {\ln \left (e x +d \right )}{c^{3} e}\) | \(14\) |
norman | \(\frac {\ln \left (e x +d \right )}{c^{3} e}\) | \(14\) |
risch | \(\frac {\ln \left (e x +d \right )}{c^{3} e}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (x e + d\right )}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.83, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (x e + d\right )}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 17, normalized size = 1.31 \begin {gather*} \frac {\log {\left (c^{3} d + c^{3} e x \right )}}{c^{3} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 14, normalized size = 1.08 \begin {gather*} \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 13, normalized size = 1.00 \begin {gather*} \frac {\ln \left (d+e\,x\right )}{c^3\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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