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 = 0.01, 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 = {621}
\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 621
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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Maple [A]
time = 0.59, size = 20, normalized size = 0.59
method | result | size |
gosper | \(-\frac {b x +a}{2 b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(20\) |
default | \(-\frac {b x +a}{2 b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(20\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}}{2 \left (b x +a \right )^{3} b}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, 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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Fricas [A]
time = 2.79, 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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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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Giac [A]
time = 1.20, size = 20, normalized size = 0.59 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, 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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