Optimal. Leaf size=41 \[ \frac{(d+e x)^{-2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e} \]
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Rubi [A] time = 0.0147421, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, 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 = 0.0070629, size = 30, normalized size = 0.73 \[ \frac{(d+e x)^{-2 p} \log (d+e x) \left (c (d+e x)^2\right )^p}{e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.079, size = 95, normalized size = 2.3 \begin{align*} x\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1-2\,p \right ) \ln \left ( ex+d \right ) }}{{\rm e}^{p\ln \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) }}+{\frac{d\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1-2\,p \right ) \ln \left ( ex+d \right ) }}{{\rm e}^{p\ln \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) }}}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34949, size = 18, normalized size = 0.44 \begin{align*} \frac{c^{p} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40496, size = 27, normalized size = 0.66 \begin{align*} \frac{c^{p} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \left (d + e x\right )^{2}\right )^{p} \left (d + e x\right )^{- 2 p - 1}\, dx \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{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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