Optimal. Leaf size=38 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x) (b d-a e)} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {767} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x) (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 767
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(b d-a e) (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.74 \begin {gather*} -\frac {a+b x}{e \sqrt {(a+b x)^2} (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 13, normalized size = 0.34 \begin {gather*} -\frac {1}{e^{2} x + d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 18, normalized size = 0.47 \begin {gather*} -\frac {e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right )}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.71 \begin {gather*} -\frac {b x +a}{\left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}
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: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.10, size = 28, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}}{e\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 10, normalized size = 0.26 \begin {gather*} - \frac {1}{d e + e^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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