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 = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 644
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*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.74 \begin {gather*} \frac {(d+e x)^{2 p} \log (d+e x) \left (c (d+e x)^2\right )^{-p}}{e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 32, normalized size = 0.74 \begin {gather*} \frac {(d+e x)^{2 p} \log (d+e x) \left (c (d+e x)^2\right )^{-p}}{e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 15, normalized size = 0.35 \begin {gather*} \frac {\log \left (e x + d\right )}{c^{p} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 74, normalized size = 1.72 \begin {gather*} \left (x \,{\mathrm e}^{\left (2 p -1\right ) \ln \left (e x +d \right )} \ln \left (e x +d \right )+\frac {d \,{\mathrm e}^{\left (2 p -1\right ) \ln \left (e x +d \right )} \ln \left (e x +d \right )}{e}\right ) {\mathrm e}^{-p \ln \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 15, normalized size = 0.35 \begin {gather*} \frac {\log \left (e x + d\right )}{c^{p} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{2\,p-1}}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c \left (d + e x\right )^{2}\right )^{- p} \left (d + e x\right )^{2 p - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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