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 = {651} \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 651
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 = 0.03, size = 39, normalized size = 1.18 \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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IntegrateAlgebraic [A] time = 0.38, size = 40, normalized size = 1.21 \begin {gather*} \frac {(b x-a) \sqrt {a^2-b^2 x^2}}{3 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, 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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giac [B] time = 0.23, 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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maple [A] time = 0.04, size = 36, normalized size = 1.09 \begin {gather*} -\frac {\left (-b x +a \right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3 \left (b x +a \right )^{2} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, 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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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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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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