Optimal. Leaf size=43 \[ \frac {(d+e x)^{2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \log (d+e x)}{e} \]
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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 = {658, 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 658
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 = 0.00, size = 32, normalized size = 0.74 \begin {gather*} \frac {(d+e x)^{2 p} \left (c (d+e x)^2\right )^{-p} \log (d+e x)}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 74, normalized size = 1.72
method | result | size |
norman | \(\left (x \ln \left (e x +d \right ) {\mathrm e}^{\left (-1+2 p \right ) \ln \left (e x +d \right )}+\frac {d \ln \left (e x +d \right ) {\mathrm e}^{\left (-1+2 p \right ) \ln \left (e x +d \right )}}{e}\right ) {\mathrm e}^{-p \ln \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.35 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (x e + d\right )}{c^{p}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.12, size = 15, normalized size = 0.35 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (x e + d\right )}{c^{p}} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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