Optimal. Leaf size=36 \[ \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {609} \begin {gather*} \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin {align*} \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx &=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.86 \begin {gather*} \frac {c x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 41, normalized size = 1.14 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x^{2} + 2 \, d x\right )}}{2 \, {\left (e x + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 30, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (d e^{\left (-1\right )} + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.11 \begin {gather*} \frac {\left (e x +2 d \right ) \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, x}{2 e x +2 d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 54, normalized size = 1.50 \begin {gather*} \frac {1}{2} \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x + \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d}{2 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 33, normalized size = 0.92 \begin {gather*} \left (\frac {x}{2}+\frac {d}{2\,e}\right )\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2} \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 \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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