Optimal. Leaf size=43 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e} \]
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Rubi [A] time = 0.0146628, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {644, 31} \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e} \]
Antiderivative was successfully verified.
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Rule 644
Rule 31
Rubi steps
\begin{align*} \int (d+e x)^{-1+2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx &=\left ((d+e x)^{2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}\right ) \int \frac{1}{d+e x} \, dx\\ &=\frac{(d+e x)^{2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \log (d+e x)}{e}\\ \end{align*}
Mathematica [A] time = 0.0060687, size = 32, normalized size = 0.74 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c (d+e x)^2\right )^{-p}}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 74, normalized size = 1.7 \begin{align*}{\frac{1}{{{\rm e}^{p\ln \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) }}} \left ( x\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1+2\,p \right ) \ln \left ( ex+d \right ) }}+{\frac{d\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1+2\,p \right ) \ln \left ( ex+d \right ) }}}{e}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16358, size = 20, normalized size = 0.47 \begin{align*} \frac{\log \left (e x + d\right )}{c^{p} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36749, size = 30, normalized size = 0.7 \begin{align*} \frac{\log \left (e x + d\right )}{c^{p} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{2 \, p - 1}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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