Optimal. Leaf size=28 \[ \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, 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 = {657, 643}
\begin {gather*} \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rule 657
Rubi steps
\begin {align*} \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx &=c \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.75 \begin {gather*} \frac {c x (d+e x)}{\sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 32, normalized size = 1.14
method | result | size |
risch | \(\frac {\sqrt {\left (e x +d \right )^{2} c}\, x}{e x +d}\) | \(21\) |
default | \(\frac {x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{e x +d}\) | \(32\) |
trager | \(\frac {x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{e x +d}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.06, size = 32, normalized size = 1.14 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} x}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.77, size = 37, normalized size = 1.32 \begin {gather*} \begin {cases} \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {x \sqrt {c d^{2}}}{d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.00, size = 25, normalized size = 0.89 \begin {gather*} {\left (d e^{\left (-1\right )} \mathrm {sgn}\left (x e + d\right ) + x \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 15, normalized size = 0.54 \begin {gather*} \frac {\sqrt {c\,{\left (d+e\,x\right )}^2}}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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