Optimal. Leaf size=35 \[ 2 \sqrt {x} (A c+b B)-\frac {2 A b}{\sqrt {x}}+\frac {2}{3} B c x^{3/2} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} 2 \sqrt {x} (A c+b B)-\frac {2 A b}{\sqrt {x}}+\frac {2}{3} B c x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac {A b}{x^{3/2}}+\frac {b B+A c}{\sqrt {x}}+B c \sqrt {x}\right ) \, dx\\ &=-\frac {2 A b}{\sqrt {x}}+2 (b B+A c) \sqrt {x}+\frac {2}{3} B c x^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.83 \begin {gather*} \frac {2 (B x (3 b+c x)-3 A (b-c x))}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 30, normalized size = 0.86 \begin {gather*} \frac {2 \left (-3 A b+3 A c x+3 b B x+B c x^2\right )}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 26, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (B c x^{2} - 3 \, A b + 3 \, {\left (B b + A c\right )} x\right )}}{3 \, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.83 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, B b \sqrt {x} + 2 \, A c \sqrt {x} - \frac {2 \, A b}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.80 \begin {gather*} -\frac {2 \left (-B c \,x^{2}-3 A c x -3 B b x +3 A b \right )}{3 \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 27, normalized size = 0.77 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} - \frac {2 \, A b}{\sqrt {x}} + 2 \, {\left (B b + A c\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 27, normalized size = 0.77 \begin {gather*} \frac {6\,A\,c\,x-6\,A\,b+6\,B\,b\,x+2\,B\,c\,x^2}{3\,\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 41, normalized size = 1.17 \begin {gather*} - \frac {2 A b}{\sqrt {x}} + 2 A c \sqrt {x} + 2 B b \sqrt {x} + \frac {2 B c x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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