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 = {629} \[ -\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 629
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 = 0.01, size = 21, normalized size = 0.66 \[ -\frac {1}{c e \sqrt {c (d+e x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 55, normalized size = 1.72 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 41, normalized size = 1.28 \[ \frac {2 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac {e^{\left (-1\right )}}{c}}{\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.09 \[ -\frac {\left (e x +d \right )^{2}}{\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 30, normalized size = 0.94 \[ -\frac {1}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]
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 \[ -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c^2\,e\,{\left (d+e\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.23, size = 42, normalized size = 1.31 \[ \begin {cases} - \frac {1}{c e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {d x}{\left (c d^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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