Optimal. Leaf size=38 \[ -\frac {a^3}{x}-\frac {6 a^2 b}{\sqrt {x}}+3 a b^2 \log (x)+2 b^3 \sqrt {x} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {6 a^2 b}{\sqrt {x}}-\frac {a^3}{x}+3 a b^2 \log (x)+2 b^3 \sqrt {x} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^3}{x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (b^3+\frac {a^3}{x^3}+\frac {3 a^2 b}{x^2}+\frac {3 a b^2}{x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^3}{x}-\frac {6 a^2 b}{\sqrt {x}}+2 b^3 \sqrt {x}+3 a b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.00 \[ -\frac {a^3}{x}-\frac {6 a^2 b}{\sqrt {x}}+3 a b^2 \log (x)+2 b^3 \sqrt {x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 38, normalized size = 1.00 \[ \frac {6 \, a b^{2} x \log \left (\sqrt {x}\right ) - a^{3} + 2 \, {\left (b^{3} x - 3 \, a^{2} b\right )} \sqrt {x}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 36, normalized size = 0.95 \[ 3 \, a b^{2} \log \left ({\left | x \right |}\right ) + 2 \, b^{3} \sqrt {x} - \frac {6 \, a^{2} b \sqrt {x} + a^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.92 \[ 3 a \,b^{2} \ln \relax (x )+2 b^{3} \sqrt {x}-\frac {6 a^{2} b}{\sqrt {x}}-\frac {a^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 35, normalized size = 0.92 \[ 3 \, a b^{2} \log \relax (x) + 2 \, b^{3} \sqrt {x} - \frac {6 \, a^{2} b \sqrt {x} + a^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 37, normalized size = 0.97 \[ 2\,b^3\,\sqrt {x}-\frac {a^3+6\,a^2\,b\,\sqrt {x}}{x}+6\,a\,b^2\,\ln \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 36, normalized size = 0.95 \[ - \frac {a^{3}}{x} - \frac {6 a^{2} b}{\sqrt {x}} + 3 a b^{2} \log {\relax (x )} + 2 b^{3} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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