Optimal. Leaf size=41 \[ \frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d}-\frac {4 b n (d x)^{3/2}}{9 d} \]
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Rubi [A] time = 0.01, 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} \[ \frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d}-\frac {4 b n (d x)^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {4 b n (d x)^{3/2}}{9 d}+\frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.71 \[ \frac {2}{9} x \sqrt {d x} \left (3 a+3 b \log \left (c x^n\right )-2 b n\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 32, normalized size = 0.78 \[ \frac {2}{9} \, {\left (3 \, b n x \log \relax (x) + 3 \, b x \log \relax (c) - {\left (2 \, b n - 3 \, a\right )} x\right )} \sqrt {d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.54, size = 105, normalized size = 2.56 \[ \left (\frac {1}{3} i + \frac {1}{3}\right ) \, \sqrt {2} b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (x) - \left (\frac {1}{3} i - \frac {1}{3}\right ) \, \sqrt {2} b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {2}{9} i + \frac {2}{9}\right ) \, \sqrt {2} b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (\frac {2}{9} i - \frac {2}{9}\right ) \, \sqrt {2} b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \frac {2}{3} \, b \sqrt {d} x^{\frac {3}{2}} \log \relax (c) + \frac {2}{3} \, a \sqrt {d} x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 124, normalized size = 3.02 \[ \frac {2 b d \,x^{2} \ln \left (x^{n}\right )}{3 \sqrt {d x}}+\frac {\left (-3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b n +6 b \ln \relax (c )+6 a \right ) d \,x^{2}}{9 \sqrt {d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 41, normalized size = 1.00 \[ -\frac {4 \, \left (d x\right )^{\frac {3}{2}} b n}{9 \, d} + \frac {2 \, \left (d x\right )^{\frac {3}{2}} b \log \left (c x^{n}\right )}{3 \, d} + \frac {2 \, \left (d x\right )^{\frac {3}{2}} a}{3 \, d} \]
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 \[ \int \sqrt {d\,x}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 70, normalized size = 1.71 \[ \frac {2 a \sqrt {d} x^{\frac {3}{2}}}{3} + \frac {2 b \sqrt {d} n x^{\frac {3}{2}} \log {\relax (x )}}{3} - \frac {4 b \sqrt {d} n x^{\frac {3}{2}}}{9} + \frac {2 b \sqrt {d} x^{\frac {3}{2}} \log {\relax (c )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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