Optimal. Leaf size=19 \[ \frac {2 x}{b \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {636} \begin {gather*} \frac {2 x}{b \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin {align*} \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac {2 x}{b \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 x}{b \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 25, normalized size = 1.32 \begin {gather*} \frac {2 \sqrt {b x+c x^2}}{b (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 23, normalized size = 1.21 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x}}{b c x + b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 32, normalized size = 1.68 \begin {gather*} \frac {2}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 1.32 \begin {gather*} \frac {2 \left (c x +b \right ) x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 \, x}{\sqrt {c x^{2} + b x} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 15, normalized size = 0.79 \begin {gather*} \frac {2\,x}{b\,\sqrt {x\,\left (b+c\,x\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 {x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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