Optimal. Leaf size=38 \[ -\frac {1}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, 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, 621}
\begin {gather*} -\frac {1}{2 e (d+e x) \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 621
Rule 656
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=c \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.68 \begin {gather*} -\frac {c (d+e x)}{2 e \left (c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 35, normalized size = 0.92
method | result | size |
risch | \(-\frac {1}{2 \left (e x +d \right ) \sqrt {\left (e x +d \right )^{2} c}\, e}\) | \(24\) |
gosper | \(-\frac {1}{2 e \left (e x +d \right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\) | \(35\) |
default | \(-\frac {1}{2 e \left (e x +d \right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\) | \(35\) |
trager | \(\frac {\left (e x +2 d \right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{2 d^{2} c \left (e x +d \right )^{3}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 32, normalized size = 0.84 \begin {gather*} -\frac {1}{2 \, {\left (\sqrt {c} x^{2} e^{3} + 2 \, \sqrt {c} d x e^{2} + \sqrt {c} d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.84, size = 59, normalized size = 1.55 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{2 \, {\left (c x^{3} e^{4} + 3 \, c d x^{2} e^{3} + 3 \, c d^{2} x e^{2} + c d^{3} e\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 {1}{\sqrt {c \left (d + e x\right )^{2}} \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 = 1.82, size = 24, normalized size = 0.63 \begin {gather*} -\frac {e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{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 [B]
time = 0.45, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,c\,e\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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