Optimal. Leaf size=61 \[ \frac{a}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0146494, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {640, 607} \[ \frac{a}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 607
Rubi steps
\begin{align*} \int \frac{x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac{1}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a \int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx}{b}\\ &=-\frac{1}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0096303, size = 33, normalized size = 0.54 \[ \frac{-a-2 b x}{2 b^2 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.176, size = 26, normalized size = 0.4 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 2\,bx+a \right ) }{2\,{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.83118, size = 59, normalized size = 0.97 \begin{align*} -\frac{1}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{a}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58122, size = 68, normalized size = 1.11 \begin{align*} -\frac{2 \, b x + a}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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