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 = 0.02, antiderivative size = 41, 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} \[ -\frac {2 \left (a+b \log \left (c x^n\right )\right )}{3 d (d x)^{3/2}}-\frac {4 b n}{9 d (d x)^{3/2}} \]
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.48, size = 32, normalized size = 0.78 \[ -\frac {2 \, {\left (3 \, b n \log \relax (x) + 2 \, b n + 3 \, b \log \relax (c) + 3 \, a\right )} \sqrt {d x}}{9 \, d^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 67, normalized size = 1.63 \[ -\frac {2 \, {\left (\frac {3 \, b d n \log \left (d x\right )}{\sqrt {d x} x} - \frac {3 \, b d^{2} n \log \relax (d) - 2 \, b d^{2} n - 3 \, b d^{2} \log \relax (c) - 3 \, a d^{2}}{\sqrt {d x} d x}\right )}}{9 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 128, normalized size = 3.12 \[ -\frac {2 b \ln \left (x^{n}\right )}{3 \sqrt {d x}\, d^{2} x}-\frac {-3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +6 b \ln \relax (c )+6 a}{9 \sqrt {d x}\, d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 41, normalized size = 1.00 \[ -\frac {4 \, b n}{9 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, a}{3 \, \left (d x\right )^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.21, size = 71, normalized size = 1.73 \[ - \frac {2 a}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b n \log {\relax (x )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 b n}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b \log {\relax (c )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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