Optimal. Leaf size=156 \[ -\frac {2}{3} b \text {Li}_2\left (1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {2}{3} b \text {Li}_2\left (\frac {2}{1-c x^{3/2}}-1\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {4}{3} \tanh ^{-1}\left (1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2+\frac {1}{3} b^2 \text {Li}_3\left (1-\frac {2}{1-c x^{3/2}}\right )-\frac {1}{3} b^2 \text {Li}_3\left (\frac {2}{1-c x^{3/2}}-1\right ) \]
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Rubi [A] time = 0.32, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6095, 5914, 6052, 5948, 6058, 6610} \[ -\frac {2}{3} b \text {PolyLog}\left (2,1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {2}{3} b \text {PolyLog}\left (2,\frac {2}{1-c x^{3/2}}-1\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+\frac {1}{3} b^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x^{3/2}}\right )-\frac {1}{3} b^2 \text {PolyLog}\left (3,\frac {2}{1-c x^{3/2}}-1\right )+\frac {4}{3} \tanh ^{-1}\left (1-\frac {2}{1-c x^{3/2}}\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 5914
Rule 5948
Rule 6052
Rule 6058
Rule 6095
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{x} \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx,x,x^{3/2}\right )\\ &=\frac {4}{3} \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x^{3/2}}\right )-\frac {1}{3} (8 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^{3/2}\right )\\ &=\frac {4}{3} \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x^{3/2}}\right )+\frac {1}{3} (4 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^{3/2}\right )-\frac {1}{3} (4 b c) \operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^{3/2}\right )\\ &=\frac {4}{3} \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x^{3/2}}\right )-\frac {2}{3} b \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-c x^{3/2}}\right )+\frac {2}{3} b \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-c x^{3/2}}\right )+\frac {1}{3} \left (2 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^{3/2}\right )-\frac {1}{3} \left (2 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^{3/2}\right )\\ &=\frac {4}{3} \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x^{3/2}}\right )-\frac {2}{3} b \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-c x^{3/2}}\right )+\frac {2}{3} b \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-c x^{3/2}}\right )+\frac {1}{3} b^2 \text {Li}_3\left (1-\frac {2}{1-c x^{3/2}}\right )-\frac {1}{3} b^2 \text {Li}_3\left (-1+\frac {2}{1-c x^{3/2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 167, normalized size = 1.07 \[ \frac {1}{3} \left (b \left (2 \text {Li}_2\left (\frac {c x^{3/2}+1}{1-c x^{3/2}}\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-2 \text {Li}_2\left (\frac {c x^{3/2}+1}{c x^{3/2}-1}\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )+b \left (\text {Li}_3\left (\frac {c x^{3/2}+1}{c x^{3/2}-1}\right )-\text {Li}_3\left (\frac {c x^{3/2}+1}{1-c x^{3/2}}\right )\right )\right )+4 \tanh ^{-1}\left (\frac {2}{c x^{3/2}-1}+1\right ) \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a^{2}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 785, normalized size = 5.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, b^{2} \int \frac {\log \left (c x^{\frac {3}{2}} + 1\right )^{2}}{x}\,{d x} - \frac {1}{2} \, b^{2} \int \frac {\log \left (c x^{\frac {3}{2}} + 1\right ) \log \left (-c x^{\frac {3}{2}} + 1\right )}{x}\,{d x} + \frac {1}{4} \, b^{2} \int \frac {\log \left (-c x^{\frac {3}{2}} + 1\right )^{2}}{x}\,{d x} + a b \int \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{x}\,{d x} - a b \int \frac {\log \left (-c x^{\frac {3}{2}} + 1\right )}{x}\,{d x} + a^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^{3/2}\right )\right )}^2}{x} \,d x \]
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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