Optimal. Leaf size=37 \[ \frac {2}{3} x^{3/2} (A c+b B)+2 A b \sqrt {x}+\frac {2}{5} B c x^{5/2} \]
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Rubi [A] time = 0.01, antiderivative size = 37, 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*} \frac {2}{3} x^{3/2} (A c+b B)+2 A b \sqrt {x}+\frac {2}{5} B c x^{5/2} \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^{3/2}} \, dx &=\int \left (\frac {A b}{\sqrt {x}}+(b B+A c) \sqrt {x}+B c x^{3/2}\right ) \, dx\\ &=2 A b \sqrt {x}+\frac {2}{3} (b B+A c) x^{3/2}+\frac {2}{5} B c x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.84 \begin {gather*} \frac {2}{15} \sqrt {x} (5 A (3 b+c x)+B x (5 b+3 c x)) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 41, normalized size = 1.11 \begin {gather*} \frac {2}{15} \left (15 A b \sqrt {x}+5 A c x^{3/2}+5 b B x^{3/2}+3 B c x^{5/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{15} \, {\left (3 \, B c x^{2} + 15 \, A b + 5 \, {\left (B b + A c\right )} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{5} \, B c x^{\frac {5}{2}} + \frac {2}{3} \, B b x^{\frac {3}{2}} + \frac {2}{3} \, A c x^{\frac {3}{2}} + 2 \, A b \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.76 \begin {gather*} \frac {2 \left (3 B c \,x^{2}+5 A c x +5 B b x +15 A b \right ) \sqrt {x}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{5} \, B c x^{\frac {5}{2}} + 2 \, A b \sqrt {x} + \frac {2}{3} \, {\left (B b + A c\right )} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 27, normalized size = 0.73 \begin {gather*} \frac {2\,\sqrt {x}\,\left (15\,A\,b+5\,A\,c\,x+5\,B\,b\,x+3\,B\,c\,x^2\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 44, normalized size = 1.19 \begin {gather*} 2 A b \sqrt {x} + \frac {2 A c x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {3}{2}}}{3} + \frac {2 B c x^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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