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 = 0.02, 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} \[ -\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d x}}-\frac {4 b n}{d \sqrt {d x}} \]
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*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.65 \[ -\frac {2 x \left (a+b \log \left (c x^n\right )+2 b n\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 28, normalized size = 0.76 \[ -\frac {2 \, {\left (b n \log \relax (x) + 2 \, b n + b \log \relax (c) + a\right )} \sqrt {d x}}{d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 43, normalized size = 1.16 \[ -\frac {2 \, {\left (\frac {b n \log \left (d x\right )}{\sqrt {d x}} - \frac {b n \log \relax (d) - 2 \, b n - b \log \relax (c) - a}{\sqrt {d x}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 122, normalized size = 3.30 \[ -\frac {2 b \ln \left (x^{n}\right )}{\sqrt {d x}\, d}-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +2 b \ln \relax (c )+2 a}{\sqrt {d x}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 41, normalized size = 1.11 \[ -\frac {4 \, b n}{\sqrt {d x} d} - \frac {2 \, b \log \left (c x^{n}\right )}{\sqrt {d x} d} - \frac {2 \, a}{\sqrt {d x} d} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.86, size = 65, normalized size = 1.76 \[ - \frac {2 a}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b n \log {\relax (x )}}{d^{\frac {3}{2}} \sqrt {x}} - \frac {4 b n}{d^{\frac {3}{2}} \sqrt {x}} - \frac {2 b \log {\relax (c )}}{d^{\frac {3}{2}} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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