Optimal. Leaf size=41 \[ -\frac {2 a A}{\sqrt {x}}+2 a B \sqrt {x}+\frac {2}{3} A c x^{3/2}+\frac {2}{5} B c x^{5/2} \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \[ -\frac {2 a A}{\sqrt {x}}+2 a B \sqrt {x}+\frac {2}{3} A c x^{3/2}+\frac {2}{5} B c x^{5/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{x^{3/2}} \, dx &=\int \left (\frac {a A}{x^{3/2}}+\frac {a B}{\sqrt {x}}+A c \sqrt {x}+B c x^{3/2}\right ) \, dx\\ &=-\frac {2 a A}{\sqrt {x}}+2 a B \sqrt {x}+\frac {2}{3} 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 = 32, normalized size = 0.78 \[ \frac {2 \left (c x^2 (5 A+3 B x)-15 a (A-B x)\right )}{15 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 29, normalized size = 0.71 \[ \frac {2 \, {\left (3 \, B c x^{3} + 5 \, A c x^{2} + 15 \, B a x - 15 \, A a\right )}}{15 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.71 \[ \frac {2}{5} \, B c x^{\frac {5}{2}} + \frac {2}{3} \, A c x^{\frac {3}{2}} + 2 \, B a \sqrt {x} - \frac {2 \, A a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.73 \[ -\frac {2 \left (-3 B c \,x^{3}-5 A c \,x^{2}-15 B a x +15 a A \right )}{15 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 29, normalized size = 0.71 \[ \frac {2}{5} \, B c x^{\frac {5}{2}} + \frac {2}{3} \, A c x^{\frac {3}{2}} + 2 \, B a \sqrt {x} - \frac {2 \, A a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 29, normalized size = 0.71 \[ 2\,B\,a\,\sqrt {x}-\frac {2\,A\,a}{\sqrt {x}}+\frac {2\,A\,c\,x^{3/2}}{3}+\frac {2\,B\,c\,x^{5/2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 42, normalized size = 1.02 \[ - \frac {2 A a}{\sqrt {x}} + \frac {2 A c x^{\frac {3}{2}}}{3} + 2 B a \sqrt {x} + \frac {2 B c x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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