Optimal. Leaf size=32 \[ -\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643}
\begin {gather*} -\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.66 \begin {gather*} -\frac {1}{c e \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 35, normalized size = 1.09
method | result | size |
risch | \(-\frac {1}{c \sqrt {\left (e x +d \right )^{2} c}\, e}\) | \(20\) |
gosper | \(-\frac {\left (e x +d \right )^{2}}{e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(35\) |
default | \(-\frac {\left (e x +d \right )^{2}}{e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(35\) |
trager | \(\frac {x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{c^{2} d \left (e x +d \right )^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 29, normalized size = 0.91 \begin {gather*} -\frac {e^{\left (-1\right )}}{\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.47, size = 54, normalized size = 1.69 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{c^{2} x^{2} e^{3} + 2 \, c^{2} d x e^{2} + c^{2} d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.52, size = 42, normalized size = 1.31 \begin {gather*} \begin {cases} - \frac {1}{c e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {d x}{\left (c d^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.66, size = 24, normalized size = 0.75 \begin {gather*} -\frac {e^{\left (-1\right )}}{{\left (x e + d\right )} c^{\frac {3}{2}} \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 = 0.46, size = 37, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c^2\,e\,{\left (d+e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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