Optimal. Leaf size=37 \[ \frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {4 b n \sqrt {d x}}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2304} \[ \frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {4 b n \sqrt {d x}}{d} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx &=-\frac {4 b n \sqrt {d x}}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.65 \[ \frac {2 x \left (a+b \log \left (c x^n\right )-2 b n\right )}{\sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 25, normalized size = 0.68 \[ \frac {2 \, {\left (b n \log \relax (x) - 2 \, b n + b \log \relax (c) + a\right )} \sqrt {d x}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 41, normalized size = 1.11 \[ \frac {2 \, {\left ({\left (\sqrt {d x} \log \relax (x) - 2 \, \sqrt {d x}\right )} b n + \sqrt {d x} b \log \relax (c) + \sqrt {d x} a\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 1.14 \[ -\frac {4 \sqrt {d x}\, b n}{d}+\frac {2 \sqrt {d x}\, b \ln \left (c \,x^{n}\right )}{d}+\frac {2 \sqrt {d x}\, a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 41, normalized size = 1.11 \[ -\frac {4 \, \sqrt {d x} b n}{d} + \frac {2 \, \sqrt {d x} b \log \left (c x^{n}\right )}{d} + \frac {2 \, \sqrt {d x} a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{\sqrt {d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 63, normalized size = 1.70 \[ \frac {2 a \sqrt {x}}{\sqrt {d}} + \frac {2 b n \sqrt {x} \log {\relax (x )}}{\sqrt {d}} - \frac {4 b n \sqrt {x}}{\sqrt {d}} + \frac {2 b \sqrt {x} \log {\relax (c )}}{\sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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