Optimal. Leaf size=75 \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 c^6}+\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c} \]
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Rubi [A] time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6097, 50, 63, 206} \[ \frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+\frac {b x^{3/2}}{9 c^3}+\frac {b \sqrt {x}}{3 c^5}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 c^6}+\frac {b x^{5/2}}{15 c} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 6097
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {1}{6} (b c) \int \frac {x^{5/2}}{1-c^2 x} \, dx\\ &=\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {x^{3/2}}{1-c^2 x} \, dx}{6 c}\\ &=\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {\sqrt {x}}{1-c^2 x} \, dx}{6 c^3}\\ &=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )} \, dx}{6 c^5}\\ &=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{3 c^5}\\ &=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 c^6}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 101, normalized size = 1.35 \[ \frac {a x^3}{3}+\frac {b \log \left (1-c \sqrt {x}\right )}{6 c^6}-\frac {b \log \left (c \sqrt {x}+1\right )}{6 c^6}+\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {1}{3} b x^3 \tanh ^{-1}\left (c \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 80, normalized size = 1.07 \[ \frac {30 \, a c^{6} x^{3} + 15 \, {\left (b c^{6} x^{3} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 2 \, {\left (3 \, b c^{5} x^{2} + 5 \, b c^{3} x + 15 \, b c\right )} \sqrt {x}}{90 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 301, normalized size = 4.01 \[ \frac {1}{3} \, a x^{3} + \frac {2}{45} \, b c {\left (\frac {\frac {45 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} - \frac {90 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {140 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} - \frac {70 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 23}{c^{7} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{5}} + \frac {15 \, {\left (\frac {3 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {10 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {3 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1}\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{c^{7} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 75, normalized size = 1.00 \[ \frac {x^{3} a}{3}+\frac {b \,x^{3} \arctanh \left (c \sqrt {x}\right )}{3}+\frac {b \,x^{\frac {5}{2}}}{15 c}+\frac {b \,x^{\frac {3}{2}}}{9 c^{3}}+\frac {b \sqrt {x}}{3 c^{5}}+\frac {b \ln \left (c \sqrt {x}-1\right )}{6 c^{6}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{6 c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 78, normalized size = 1.04 \[ \frac {1}{3} \, a x^{3} + \frac {1}{90} \, {\left (30 \, x^{3} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{\frac {5}{2}} + 5 \, c^{2} x^{\frac {3}{2}} + 15 \, \sqrt {x}\right )}}{c^{6}} - \frac {15 \, \log \left (c \sqrt {x} + 1\right )}{c^{7}} + \frac {15 \, \log \left (c \sqrt {x} - 1\right )}{c^{7}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 58, normalized size = 0.77 \[ \frac {a\,x^3}{3}+\frac {\frac {b\,c^3\,x^{3/2}}{9}-\frac {b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3}+\frac {b\,c^5\,x^{5/2}}{15}+\frac {b\,c\,\sqrt {x}}{3}}{c^6}+\frac {b\,x^3\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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