Optimal. Leaf size=62 \[ \frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4}+\frac {b \sqrt {x}}{2 c^3}+\frac {b x^{3/2}}{6 c} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6097, 50, 63, 206} \[ \frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+\frac {b \sqrt {x}}{2 c^3}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4}+\frac {b x^{3/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 6097
Rubi steps
\begin {align*} \int x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {1}{4} (b c) \int \frac {x^{3/2}}{1-c^2 x} \, dx\\ &=\frac {b x^{3/2}}{6 c}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {\sqrt {x}}{1-c^2 x} \, dx}{4 c}\\ &=\frac {b \sqrt {x}}{2 c^3}+\frac {b x^{3/2}}{6 c}+\frac {1}{2} x^2 \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}{4 c^3}\\ &=\frac {b \sqrt {x}}{2 c^3}+\frac {b x^{3/2}}{6 c}+\frac {1}{2} x^2 \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 )}{2 c^3}\\ &=\frac {b \sqrt {x}}{2 c^3}+\frac {b x^{3/2}}{6 c}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 88, normalized size = 1.42 \[ \frac {a x^2}{2}+\frac {b \log \left (1-c \sqrt {x}\right )}{4 c^4}-\frac {b \log \left (c \sqrt {x}+1\right )}{4 c^4}+\frac {b \sqrt {x}}{2 c^3}+\frac {b x^{3/2}}{6 c}+\frac {1}{2} b x^2 \tanh ^{-1}\left (c \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 70, normalized size = 1.13 \[ \frac {6 \, a c^{4} x^{2} + 3 \, {\left (b c^{4} x^{2} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 2 \, {\left (b c^{3} x + 3 \, b c\right )} \sqrt {x}}{12 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 239, normalized size = 3.85 \[ \frac {1}{2} \, a x^{2} + \frac {2}{3} \, b c {\left (\frac {\frac {3 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} - \frac {3 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 2}{c^{5} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{3}} + \frac {3 \, {\left (\frac {{\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {c \sqrt {x} + 1}{c \sqrt {x} - 1}\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{c^{5} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 66, normalized size = 1.06 \[ \frac {a \,x^{2}}{2}+\frac {b \,x^{2} \arctanh \left (c \sqrt {x}\right )}{2}+\frac {b \,x^{\frac {3}{2}}}{6 c}+\frac {b \sqrt {x}}{2 c^{3}}+\frac {b \ln \left (c \sqrt {x}-1\right )}{4 c^{4}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{4 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 69, normalized size = 1.11 \[ \frac {1}{2} \, a x^{2} + \frac {1}{12} \, {\left (6 \, x^{2} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{\frac {3}{2}} + 3 \, \sqrt {x}\right )}}{c^{4}} - \frac {3 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 49, normalized size = 0.79 \[ \frac {\frac {b\,c^3\,x^{3/2}}{6}-\frac {b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{2}+\frac {b\,c\,\sqrt {x}}{2}}{c^4}+\frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{2} \]
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 \[ \int x \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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