Optimal. Leaf size=34 \[ -\frac {1}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {607} \begin {gather*} -\frac {1}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 607
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.68 \begin {gather*} -\frac {a+b x}{2 b \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.47, size = 125, normalized size = 3.68 \begin {gather*} \frac {\sqrt {b^2} (a-b x) \sqrt {a^2+2 a b x+b^2 x^2}+a^2 b+b^3 x^2}{b \sqrt {b^2} x^2 \left (2 a^2 b^2+4 a b^3 x+2 b^4 x^2\right )+b x^2 \left (-2 a b^3-2 b^4 x\right ) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 24, normalized size = 0.71 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.59 \begin {gather*} -\frac {b x +a}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 14, normalized size = 0.41 \begin {gather*} -\frac {1}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 30, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b\,{\left (a+b\,x\right )}^3} \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}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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