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 = 0.0145035, antiderivative size = 37, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.005951, 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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Maple [A] time = 0.043, size = 42, normalized size = 1.1 \begin{align*} 2\,{\frac{\sqrt{dx}b\ln \left ( c{x}^{n} \right ) }{d}}-4\,{\frac{bn\sqrt{dx}}{d}}+2\,{\frac{\sqrt{dx}a}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17155, size = 55, normalized size = 1.49 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.869924, size = 69, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt{d x}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.6799, size = 63, normalized size = 1.7 \begin{align*} \frac{2 a \sqrt{x}}{\sqrt{d}} + \frac{2 b n \sqrt{x} \log{\left (x \right )}}{\sqrt{d}} - \frac{4 b n \sqrt{x}}{\sqrt{d}} + \frac{2 b \sqrt{x} \log{\left (c \right )}}{\sqrt{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37406, size = 55, normalized size = 1.49 \begin{align*} \frac{2 \,{\left ({\left (\sqrt{d x} \log \left (x\right ) - 2 \, \sqrt{d x}\right )} b n + \sqrt{d x} b \log \left (c\right ) + \sqrt{d x} a\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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