Optimal. Leaf size=52 \[ \frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {4 b \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {656, 648} \begin {gather*} \frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {4 b \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx &=\frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {(2 b) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{3 c}\\ &=-\frac {4 b \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \sqrt {b x+c x^2}}{3 c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.58 \begin {gather*} \frac {2 (c x-2 b) \sqrt {x (b+c x)}}{3 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 32, normalized size = 0.62 \begin {gather*} \frac {2 (c x-2 b) \sqrt {b x+c x^2}}{3 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 26, normalized size = 0.50 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (c x - 2 \, b\right )}}{3 \, c^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 34, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, c^{2}} - \frac {2 \, \sqrt {c x + b} b}{c^{2}} + \frac {4 \, b^{\frac {3}{2}}}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 33, normalized size = 0.63 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-c x +2 b \right ) \sqrt {x}}{3 \sqrt {c \,x^{2}+b x}\, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 30, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )}}{3 \, \sqrt {c x + b} c^{2}} \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 = -0.02 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {c\,x^2+b\,x}} \,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}}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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