Optimal. Leaf size=49 \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \tanh ^{-1}\left (c x^{3/2}\right )}{3 c^2}+\frac {b x^{3/2}}{3 c} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6097, 321, 329, 275, 206} \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \tanh ^{-1}\left (c x^{3/2}\right )}{3 c^2}+\frac {b x^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 275
Rule 321
Rule 329
Rule 6097
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {1}{2} (b c) \int \frac {x^{7/2}}{1-c^2 x^3} \, dx\\ &=\frac {b x^{3/2}}{3 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \int \frac {\sqrt {x}}{1-c^2 x^3} \, dx}{2 c}\\ &=\frac {b x^{3/2}}{3 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {x^2}{1-c^2 x^6} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {b x^{3/2}}{3 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,x^{3/2}\right )}{3 c}\\ &=\frac {b x^{3/2}}{3 c}-\frac {b \tanh ^{-1}\left (c x^{3/2}\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 1.53 \[ \frac {a x^3}{3}+\frac {b \log \left (1-c x^{3/2}\right )}{6 c^2}-\frac {b \log \left (c x^{3/2}+1\right )}{6 c^2}+\frac {b x^{3/2}}{3 c}+\frac {1}{3} b x^3 \tanh ^{-1}\left (c x^{3/2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 64, normalized size = 1.31 \[ \frac {2 \, a c^{2} x^{3} + 2 \, b c x^{\frac {3}{2}} + {\left (b c^{2} x^{3} - b\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )}{6 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 97, normalized size = 1.98 \[ \frac {1}{3} \, a x^{3} + \frac {2}{3} \, b c {\left (\frac {1}{c^{3} {\left (\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1} - 1\right )}} + \frac {{\left (c x^{\frac {3}{2}} + 1\right )} \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right )}{{\left (c x^{\frac {3}{2}} - 1\right )} c^{3} {\left (\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 57, normalized size = 1.16 \[ \frac {x^{3} a}{3}+\frac {b \,x^{3} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{3}+\frac {b \,x^{\frac {3}{2}}}{3 c}+\frac {b \ln \left (c \,x^{\frac {3}{2}}-1\right )}{6 c^{2}}-\frac {b \ln \left (c \,x^{\frac {3}{2}}+1\right )}{6 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 58, normalized size = 1.18 \[ \frac {1}{3} \, a x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + c {\left (\frac {2 \, x^{\frac {3}{2}}}{c^{2}} - \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{c^{3}} + \frac {\log \left (c x^{\frac {3}{2}} - 1\right )}{c^{3}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.76, size = 110, normalized size = 2.24 \[ \frac {a\,x^3}{3}+\frac {b\,x^{3/2}}{3\,c}+\frac {b\,\ln \left (\frac {c\,x^{3/2}-1}{c\,x^{3/2}+1}\right )}{6\,c^2}+\frac {b\,x^3\,\ln \left (c\,x^{3/2}+1\right )}{6}+\frac {b\,x^3\,\ln \left (1-c\,x^{3/2}\right )}{3\,\left (2\,c^2\,x^3-2\right )}-\frac {b\,c^2\,x^6\,\ln \left (1-c\,x^{3/2}\right )}{3\,\left (2\,c^2\,x^3-2\right )} \]
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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