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 = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 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 = 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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fricas [A] time = 1.25, size = 13, normalized size = 0.32 \[ \frac {c^{p} \log \left (e x + d\right )}{e} \]
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 \[ \int {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 95, normalized size = 2.32 \[ x \,{\mathrm e}^{p \ln \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )} {\mathrm e}^{\left (-2 p -1\right ) \ln \left (e x +d \right )} \ln \left (e x +d \right )+\frac {d \,{\mathrm e}^{p \ln \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )} {\mathrm e}^{\left (-2 p -1\right ) \ln \left (e x +d \right )} \ln \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 13, normalized size = 0.32 \[ \frac {c^{p} \log \left (e x + d\right )}{e} \]
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 \[ \int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^{2\,p+1}} \,d x \]
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 \[ \int \left (c \left (d + e x\right )^{2}\right )^{p} \left (d + e x\right )^{- 2 p - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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