Optimal. Leaf size=101 \[ -\frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{3 c^2}+\frac {2 a b x^{3/2}}{3 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2+\frac {b^2 \log \left (1-c^2 x^3\right )}{3 c^2}+\frac {2 b^2 x^{3/2} \tanh ^{-1}\left (c x^{3/2}\right )}{3 c} \]
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Rubi [F] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx &=\int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx\\ \end {align*}
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Mathematica [A] time = 0.10, size = 122, normalized size = 1.21 \[ \frac {a^2 c^2 x^3+2 a b c x^{3/2}+b (a+b) \log \left (1-c x^{3/2}\right )-a b \log \left (c x^{3/2}+1\right )+2 b c x^{3/2} \tanh ^{-1}\left (c x^{3/2}\right ) \left (a c x^{3/2}+b\right )+b^2 \left (c^2 x^3-1\right ) \tanh ^{-1}\left (c x^{3/2}\right )^2+b^2 \log \left (c x^{3/2}+1\right )}{3 c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 179, normalized size = 1.77 \[ \frac {4 \, a^{2} c^{2} x^{3} + 8 \, a b c x^{\frac {3}{2}} + {\left (b^{2} c^{2} x^{3} - b^{2}\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )^{2} + 4 \, {\left (a b c^{2} - a b + b^{2}\right )} \log \left (c x^{\frac {3}{2}} + 1\right ) - 4 \, {\left (a b c^{2} - a b - b^{2}\right )} \log \left (c x^{\frac {3}{2}} - 1\right ) + 4 \, {\left (a b c^{2} x^{3} + b^{2} c x^{\frac {3}{2}} - a b c^{2}\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )}{12 \, c^{2}} \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 284, normalized size = 2.81 \[ \frac {x^{3} a^{2}}{3}+\frac {x^{3} b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right )^{2}}{3}+\frac {2 b^{2} x^{\frac {3}{2}} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{3 c}+\frac {b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3 c^{2}}-\frac {b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3 c^{2}}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right )^{2}}{12 c^{2}}-\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right ) \ln \left (\frac {1}{2}+\frac {c \,x^{\frac {3}{2}}}{2}\right )}{6 c^{2}}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3 c^{2}}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3 c^{2}}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}+1\right )^{2}}{12 c^{2}}+\frac {b^{2} \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c \,x^{\frac {3}{2}}}{2}\right )}{6 c^{2}}-\frac {b^{2} \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{6 c^{2}}+\frac {2 a b \,x^{3} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{3}+\frac {2 a b \,x^{\frac {3}{2}}}{3 c}+\frac {a b \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3 c^{2}}-\frac {a b \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 186, normalized size = 1.84 \[ \frac {1}{3} \, b^{2} x^{3} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )^{2} + \frac {1}{3} \, a^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + c {\left (\frac {2 \, x^{\frac {3}{2}}}{c^{2}} - \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{c^{3}} + \frac {\log \left (c x^{\frac {3}{2}} - 1\right )}{c^{3}}\right )}\right )} a b + \frac {1}{12} \, {\left (4 \, c {\left (\frac {2 \, x^{\frac {3}{2}}}{c^{2}} - \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{c^{3}} + \frac {\log \left (c x^{\frac {3}{2}} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) - \frac {2 \, {\left (\log \left (c x^{\frac {3}{2}} - 1\right ) - 2\right )} \log \left (c x^{\frac {3}{2}} + 1\right ) - \log \left (c x^{\frac {3}{2}} + 1\right )^{2} - \log \left (c x^{\frac {3}{2}} - 1\right )^{2} - 4 \, \log \left (c x^{\frac {3}{2}} - 1\right )}{c^{2}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 105, normalized size = 1.04 \[ \frac {c\,\left (\frac {2\,b^2\,x^{3/2}\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3}+\frac {2\,a\,b\,x^{3/2}}{3}\right )-\frac {b^2\,{\mathrm {atanh}\left (c\,x^{3/2}\right )}^2}{3}+\frac {b^2\,\ln \left (c^2\,x^3-1\right )}{3}-\frac {2\,a\,b\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3}}{c^2}+\frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\mathrm {atanh}\left (c\,x^{3/2}\right )}^2}{3}+\frac {2\,a\,b\,x^3\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3} \]
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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