Optimal. Leaf size=33 \[ -\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {665}
\begin {gather*} -\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^3} \, dx &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 40, normalized size = 1.21 \begin {gather*} \frac {(-a+b x) \sqrt {a^2-b^2 x^2}}{3 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 46, normalized size = 1.39
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3 \left (b x +a \right )^{2} b a}\) | \(36\) |
trager | \(-\frac {\left (-b x +a \right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3 \left (b x +a \right )^{2} b a}\) | \(36\) |
default | \(-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{3 b^{4} a \left (x +\frac {a}{b}\right )^{3}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (29) = 58\).
time = 0.29, size = 69, normalized size = 2.09 \begin {gather*} -\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{3 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{3 \, {\left (a b^{2} x + a^{2} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (29) = 58\).
time = 3.16, size = 66, normalized size = 2.00 \begin {gather*} -\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - \sqrt {-b^{2} x^{2} + a^{2}} {\left (b x - a\right )}}{3 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} 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 {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (29) = 58\).
time = 1.34, size = 74, normalized size = 2.24 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + 1\right )}}{3 \, a {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 35, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {a^2-b^2\,x^2}\,\left (a-b\,x\right )}{3\,a\,b\,{\left (a+b\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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