Optimal. Leaf size=34 \[ a \log (x)-\frac {1}{3} b \text {Li}_2\left (-c x^{3/2}\right )+\frac {1}{3} b \text {Li}_2\left (c x^{3/2}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6095, 5912} \[ -\frac {1}{3} b \text {PolyLog}\left (2,-c x^{3/2}\right )+\frac {1}{3} b \text {PolyLog}\left (2,c x^{3/2}\right )+a \log (x) \]
Antiderivative was successfully verified.
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Rule 5912
Rule 6095
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x} \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^{3/2}\right )\\ &=a \log (x)-\frac {1}{3} b \text {Li}_2\left (-c x^{3/2}\right )+\frac {1}{3} b \text {Li}_2\left (c x^{3/2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.94 \[ a \log (x)+\frac {1}{3} b \left (\text {Li}_2\left (c x^{3/2}\right )-\text {Li}_2\left (-c x^{3/2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 63, normalized size = 1.85 \[ \frac {2 a \ln \left (c \,x^{\frac {3}{2}}\right )}{3}+\frac {2 b \ln \left (c \,x^{\frac {3}{2}}\right ) \arctanh \left (c \,x^{\frac {3}{2}}\right )}{3}-\frac {b \dilog \left (c \,x^{\frac {3}{2}}\right )}{3}-\frac {b \dilog \left (c \,x^{\frac {3}{2}}+1\right )}{3}-\frac {b \ln \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 62, normalized size = 1.82 \[ -\frac {1}{3} \, {\left (\log \left (c x^{\frac {3}{2}}\right ) \log \left (-c x^{\frac {3}{2}} + 1\right ) + {\rm Li}_2\left (-c x^{\frac {3}{2}} + 1\right )\right )} b + \frac {1}{3} \, {\left (\log \left (c x^{\frac {3}{2}} + 1\right ) \log \left (-c x^{\frac {3}{2}}\right ) + {\rm Li}_2\left (c x^{\frac {3}{2}} + 1\right )\right )} b + a \log \relax (x) \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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