Optimal. Leaf size=37 \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{4 b n}{d \sqrt{d x}} \]
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Rubi [A] time = 0.0159402, 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} \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{4 b n}{d \sqrt{d x}} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx &=-\frac{4 b n}{d \sqrt{d x}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}\\ \end{align*}
Mathematica [A] time = 0.0067746, size = 24, normalized size = 0.65 \[ -\frac{2 x \left (a+b \log \left (c x^n\right )+2 b n\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.083, size = 122, normalized size = 3.3 \begin{align*} -2\,{\frac{b\ln \left ({x}^{n} \right ) }{d\sqrt{dx}}}-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +4\,bn+2\,a}{d}{\frac{1}{\sqrt{dx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17758, size = 55, normalized size = 1.49 \begin{align*} -\frac{4 \, b n}{\sqrt{d x} d} - \frac{2 \, b \log \left (c x^{n}\right )}{\sqrt{d x} d} - \frac{2 \, a}{\sqrt{d x} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.864841, size = 78, normalized size = 2.11 \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^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.86456, size = 65, normalized size = 1.76 \begin{align*} - \frac{2 a}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b n \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 b n}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b \log{\left (c \right )}}{d^{\frac{3}{2}} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36171, size = 58, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (\frac{b n \log \left (d x\right )}{\sqrt{d x}} - \frac{b n \log \left (d\right ) - 2 \, b n - b \log \left (c\right ) - a}{\sqrt{d x}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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