Optimal. Leaf size=37 \[ 6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2}+a^3 \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 37, 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 = {272, 45}
\begin {gather*} a^3 \log (x)+6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^3}{x} \, dx &=2 \text {Subst}\left (\int \frac {(a+b x)^3}{x} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (3 a^2 b+\frac {a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,\sqrt {x}\right )\\ &=6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2}+a^3 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 1.00 \begin {gather*} 6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2}+a^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 32, normalized size = 0.86
method | result | size |
derivativedivides | \(3 a \,b^{2} x +\frac {2 b^{3} x^{\frac {3}{2}}}{3}+a^{3} \ln \left (x \right )+6 a^{2} b \sqrt {x}\) | \(32\) |
default | \(3 a \,b^{2} x +\frac {2 b^{3} x^{\frac {3}{2}}}{3}+a^{3} \ln \left (x \right )+6 a^{2} b \sqrt {x}\) | \(32\) |
trager | \(3 a \,b^{2} \left (x -1\right )+\frac {2 \left (b^{2} x +9 a^{2}\right ) b \sqrt {x}}{3}-a^{3} \ln \left (\frac {1}{x}\right )\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 31, normalized size = 0.84 \begin {gather*} \frac {2}{3} \, b^{3} x^{\frac {3}{2}} + 3 \, a b^{2} x + a^{3} \log \left (x\right ) + 6 \, a^{2} b \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 34, normalized size = 0.92 \begin {gather*} 3 \, a b^{2} x + 2 \, a^{3} \log \left (\sqrt {x}\right ) + \frac {2}{3} \, {\left (b^{3} x + 9 \, a^{2} b\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 37, normalized size = 1.00 \begin {gather*} a^{3} \log {\left (x \right )} + 6 a^{2} b \sqrt {x} + 3 a b^{2} x + \frac {2 b^{3} x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 32, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, b^{3} x^{\frac {3}{2}} + 3 \, a b^{2} x + a^{3} \log \left ({\left | x \right |}\right ) + 6 \, a^{2} b \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 34, normalized size = 0.92 \begin {gather*} 2\,a^3\,\ln \left (\sqrt {x}\right )+\frac {2\,b^3\,x^{3/2}}{3}+6\,a^2\,b\,\sqrt {x}+3\,a\,b^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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