Optimal. Leaf size=24 \[ -\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, 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*} -\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x) \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 )^2}{x^2} \, dx &=2 \text {Subst}\left (\int \frac {(a+b x)^2}{x^3} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.96 \begin {gather*} -\frac {a \left (a+4 b \sqrt {x}\right )}{x}+b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 23, normalized size = 0.96
method | result | size |
derivativedivides | \(-\frac {a^{2}}{x}+b^{2} \ln \left (x \right )-\frac {4 a b}{\sqrt {x}}\) | \(23\) |
default | \(-\frac {a^{2}}{x}+b^{2} \ln \left (x \right )-\frac {4 a b}{\sqrt {x}}\) | \(23\) |
trager | \(\frac {a^{2} \left (x -1\right )}{x}-\frac {4 a b}{\sqrt {x}}-b^{2} \ln \left (\frac {1}{x}\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 23, normalized size = 0.96 \begin {gather*} b^{2} \log \left (x\right ) - \frac {4 \, a b \sqrt {x} + a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 27, normalized size = 1.12 \begin {gather*} \frac {2 \, b^{2} x \log \left (\sqrt {x}\right ) - 4 \, a b \sqrt {x} - a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 20, normalized size = 0.83 \begin {gather*} - \frac {a^{2}}{x} - \frac {4 a b}{\sqrt {x}} + b^{2} \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 24, normalized size = 1.00 \begin {gather*} b^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a b \sqrt {x} + a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 26, normalized size = 1.08 \begin {gather*} 2\,b^2\,\ln \left (\sqrt {x}\right )-\frac {a^2+4\,a\,b\,\sqrt {x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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