Optimal. Leaf size=38 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {609} \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin {align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.66 \[ \frac {(d+e x) \left (c (d+e x)^2\right )^p}{2 e p+e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 36, normalized size = 0.95 \[ \frac {{\left (e x + d\right )} {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p + e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 62, normalized size = 1.63 \[ \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x e + {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d}{2 \, p e + e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 1.03 \[ \frac {\left (e x +d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{\left (2 p +1\right ) e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 32, normalized size = 0.84 \[ \frac {{\left (c^{p} e x + c^{p} d\right )} {\left (e x + d\right )}^{2 \, p}}{e {\left (2 \, p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 45, normalized size = 1.18 \[ \left (\frac {x}{2\,p+1}+\frac {d}{e\,\left (2\,p+1\right )}\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p \]
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 \[ \begin {cases} \frac {x}{\sqrt {c d^{2}}} & \text {for}\: e = 0 \wedge p = - \frac {1}{2} \\x \left (c d^{2}\right )^{p} & \text {for}\: e = 0 \\\int \frac {1}{\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {d \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + e} + \frac {e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + e} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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