Optimal. Leaf size=48 \[ \frac {4 b \sqrt {x}}{c^2 \sqrt {b x+c x^2}}+\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 48, 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 {4 b \sqrt {x}}{c^2 \sqrt {b x+c x^2}}+\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}}-\frac {(2 b) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {4 b \sqrt {x}}{c^2 \sqrt {b x+c x^2}}+\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.58 \begin {gather*} \frac {2 \sqrt {x} (2 b+c x)}{c^2 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.47, size = 37, normalized size = 0.77 \begin {gather*} \frac {2 (2 b+c x) \sqrt {b x+c x^2}}{c^2 \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 39, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (c x + 2 \, b\right )} \sqrt {x}}{c^{3} x^{2} + b c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 38, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {c x + b}}{c} + \frac {b}{\sqrt {c x + b} c}\right )}}{c} - \frac {4 \, \sqrt {b}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 32, normalized size = 0.67 \begin {gather*} \frac {2 \left (c x +b \right ) \left (c x +2 b \right ) x^{\frac {3}{2}}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}} \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*} \frac {2 \, \sqrt {c x + b} x}{c^{2} x + b c} - \int \frac {2 \, {\left (b c x + b^{2}\right )} x}{{\left (c^{3} x^{3} + 2 \, b c^{2} x^{2} + b^{2} c x\right )} \sqrt {c x + b}}\,{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 = -0.02 \begin {gather*} \int \frac {x^{5/2}}{{\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 {5}{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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