Optimal. Leaf size=41 \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}-\frac{4 b n}{9 d (d x)^{3/2}} \]
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Rubi [A] time = 0.0151008, antiderivative size = 41, 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} \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}-\frac{4 b n}{9 d (d x)^{3/2}} \]
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)^{5/2}} \, dx &=-\frac{4 b n}{9 d (d x)^{3/2}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0086047, size = 29, normalized size = 0.71 \[ -\frac{2 x \left (3 a+3 b \log \left (c x^n\right )+2 b n\right )}{9 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.088, size = 128, normalized size = 3.1 \begin{align*} -{\frac{2\,b\ln \left ({x}^{n} \right ) }{3\,x{d}^{2}}{\frac{1}{\sqrt{dx}}}}-{\frac{3\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-3\,ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -3\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+3\,ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +6\,b\ln \left ( c \right ) +4\,bn+6\,a}{9\,x{d}^{2}}{\frac{1}{\sqrt{dx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04975, size = 55, normalized size = 1.34 \begin{align*} -\frac{4 \, b n}{9 \, \left (d x\right )^{\frac{3}{2}} d} - \frac{2 \, b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac{3}{2}} d} - \frac{2 \, a}{3 \, \left (d x\right )^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.866572, size = 92, normalized size = 2.24 \begin{align*} -\frac{2 \,{\left (3 \, b n \log \left (x\right ) + 2 \, b n + 3 \, b \log \left (c\right ) + 3 \, a\right )} \sqrt{d x}}{9 \, d^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 39.234, size = 71, normalized size = 1.73 \begin{align*} - \frac{2 a}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b n \log{\left (x \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{4 b n}{9 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b \log{\left (c \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40789, size = 90, normalized size = 2.2 \begin{align*} -\frac{2 \,{\left (\frac{3 \, b d n \log \left (d x\right )}{\sqrt{d x} x} - \frac{3 \, b d^{2} n \log \left (d\right ) - 2 \, b d^{2} n - 3 \, b d^{2} \log \left (c\right ) - 3 \, a d^{2}}{\sqrt{d x} d x}\right )}}{9 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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