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 = 0.00, 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 = {623}
\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 623
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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Maple [A]
time = 0.49, size = 40, normalized size = 1.11
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{2 e x +2 d}\) | \(40\) |
default | \(\frac {x \left (e x +2 d \right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{2 e x +2 d}\) | \(40\) |
trager | \(\frac {x \left (e x +2 d \right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{2 e x +2 d}\) | \(40\) |
risch | \(\frac {\sqrt {\left (e x +d \right )^{2} c}\, e \,x^{2}}{2 e x +2 d}+\frac {\sqrt {\left (e x +d \right )^{2} c}\, d x}{e x +d}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 53, normalized size = 1.47 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} d e^{\left (-1\right )} + \frac {1}{2} \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.96, size = 43, normalized size = 1.19 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (x^{2} e + 2 \, d x\right )}}{2 \, {\left (x e + d\right )}} \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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Giac [A]
time = 1.11, size = 23, normalized size = 0.64 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} \sqrt {c} \mathrm {sgn}\left (x e + d\right ) \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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