Optimal. Leaf size=23 \[ -\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {648} \begin {gather*} -\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2 \sqrt {x}}{c \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 30, normalized size = 1.30 \begin {gather*} -\frac {2 \sqrt {b x+c x^2}}{c \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 30, normalized size = 1.30 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} \sqrt {x}}{c^{2} x^{2} + b c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2}{\sqrt {c x + b} c} + \frac {2}{\sqrt {b} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.09 \begin {gather*} -\frac {2 \left (c x +b \right ) x^{\frac {3}{2}}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^{3/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \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^{\frac {3}{2}}}{\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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