Optimal. Leaf size=13 \[ \frac {c^2 \log (d+e x)}{e} \]
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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} \[ \frac {c^2 \log (d+e x)}{e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^5} \, dx &=\int \frac {c^2}{d+e x} \, dx\\ &=c^2 \int \frac {1}{d+e x} \, dx\\ &=\frac {c^2 \log (d+e x)}{e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ \frac {c^2 \log (d+e x)}{e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 13, normalized size = 1.00 \[ \frac {c^{2} \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 26, normalized size = 2.00 \[ -c^{2} e^{\left (-1\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 14, normalized size = 1.08 \[ \frac {c^{2} \ln \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 13, normalized size = 1.00 \[ \frac {c^{2} \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 13, normalized size = 1.00 \[ \frac {c^2\,\ln \left (d+e\,x\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 10, normalized size = 0.77 \[ \frac {c^{2} \log {\left (d + e x \right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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