Optimal. Leaf size=32 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {643, 629} \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]
Antiderivative was successfully verified.
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Rule 629
Rule 643
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{d+e x} \, dx &=c \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p} \, dx\\ &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.66 \[ \frac {\left (c (d+e x)^2\right )^p}{2 e p} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 30, normalized size = 0.94 \[ \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p} \]
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 \[ \int \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.97 \[ \frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{2 e p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 20, normalized size = 0.62 \[ \frac {{\left (e x + d\right )}^{2 \, p} c^{p}}{2 \, e p} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 30, normalized size = 0.94 \[ \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p}{2\,e\,p} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 48, normalized size = 1.50 \[ \begin {cases} \frac {x}{d} & \text {for}\: e = 0 \wedge p = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{e} & \text {for}\: p = 0 \\\frac {x \left (c d^{2}\right )^{p}}{d} & \text {for}\: e = 0 \\\frac {\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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