Optimal. Leaf size=39 \[ \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c e} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {656, 623}
\begin {gather*} \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 656
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac {\int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx}{c}\\ &=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 0.77 \begin {gather*} \frac {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 = 0.59, size = 38, normalized size = 0.97
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right ) \left (e x +d \right )}{2 \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\) | \(38\) |
default | \(\frac {x \left (e x +2 d \right ) \left (e x +d \right )}{2 \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\) | \(38\) |
trager | \(\frac {x \left (e x +2 d \right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{2 c \left (e x +d \right )}\) | \(43\) |
risch | \(\frac {\left (e x +d \right ) e \,x^{2}}{2 \sqrt {\left (e x +d \right )^{2} c}}+\frac {\left (e x +d \right ) d x}{\sqrt {\left (e x +d \right )^{2} c}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 48, normalized size = 1.23 \begin {gather*} \frac {x^{2} e}{2 \, \sqrt {c}} - \frac {d x}{\sqrt {c}} + \frac {2 \, \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} d e^{\left (-1\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.50, size = 46, normalized size = 1.18 \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 (c x e + c 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 \frac {\left (d + e x\right )^{2}}{\sqrt {c \left (d + e x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 25, normalized size = 0.64 \begin {gather*} \frac {x^{2} e + 2 \, d x}{2 \, \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 [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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