Optimal. Leaf size=106 \[ \frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}+\frac {4 b \sqrt {d x}}{3 c} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5916, 321, 329, 212, 208, 205} \[ \frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}+\frac {4 b \sqrt {d x}}{3 c} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 321
Rule 329
Rule 5916
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(2 b c) \int \frac {(d x)^{3/2}}{1-c^2 x^2} \, dx}{3 d}\\ &=\frac {4 b \sqrt {d x}}{3 c}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(2 b d) \int \frac {1}{\sqrt {d x} \left (1-c^2 x^2\right )} \, dx}{3 c}\\ &=\frac {4 b \sqrt {d x}}{3 c}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{1-\frac {c^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{3 c}\\ &=\frac {4 b \sqrt {d x}}{3 c}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(2 b d) \operatorname {Subst}\left (\int \frac {1}{d-c x^2} \, dx,x,\sqrt {d x}\right )}{3 c}-\frac {(2 b d) \operatorname {Subst}\left (\int \frac {1}{d+c x^2} \, dx,x,\sqrt {d x}\right )}{3 c}\\ &=\frac {4 b \sqrt {d x}}{3 c}-\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 114, normalized size = 1.08 \[ \frac {\sqrt {d x} \left (2 a c^{3/2} x^{3/2}+2 b c^{3/2} x^{3/2} \tanh ^{-1}(c x)+4 b \sqrt {c} \sqrt {x}+b \log \left (1-\sqrt {c} \sqrt {x}\right )-b \log \left (\sqrt {c} \sqrt {x}+1\right )-2 b \tan ^{-1}\left (\sqrt {c} \sqrt {x}\right )\right )}{3 c^{3/2} \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 223, normalized size = 2.10 \[ \left [-\frac {2 \, b \sqrt {\frac {d}{c}} \arctan \left (\frac {\sqrt {d x} c \sqrt {\frac {d}{c}}}{d}\right ) - b \sqrt {\frac {d}{c}} \log \left (\frac {c d x - 2 \, \sqrt {d x} c \sqrt {\frac {d}{c}} + d}{c x - 1}\right ) - {\left (b c x \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a c x + 4 \, b\right )} \sqrt {d x}}{3 \, c}, \frac {2 \, b \sqrt {-\frac {d}{c}} \arctan \left (\frac {\sqrt {d x} c \sqrt {-\frac {d}{c}}}{d}\right ) + b \sqrt {-\frac {d}{c}} \log \left (\frac {c d x - 2 \, \sqrt {d x} c \sqrt {-\frac {d}{c}} - d}{c x + 1}\right ) + {\left (b c x \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a c x + 4 \, b\right )} \sqrt {d x}}{3 \, c}\right ] \]
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 \[ \int \sqrt {d x} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 89, normalized size = 0.84 \[ \frac {2 \left (d x \right )^{\frac {3}{2}} a}{3 d}+\frac {2 b \left (d x \right )^{\frac {3}{2}} \arctanh \left (c x \right )}{3 d}+\frac {4 b \sqrt {d x}}{3 c}-\frac {2 d b \arctan \left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{3 c \sqrt {c d}}-\frac {2 d b \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{3 c \sqrt {c d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 119, normalized size = 1.12 \[ \frac {2 \, \left (d x\right )^{\frac {3}{2}} a + {\left (2 \, \left (d x\right )^{\frac {3}{2}} \operatorname {artanh}\left (c x\right ) - \frac {{\left (\frac {2 \, d^{3} \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} c^{2}} - \frac {d^{3} \log \left (\frac {\sqrt {d x} c - \sqrt {c d}}{\sqrt {d x} c + \sqrt {c d}}\right )}{\sqrt {c d} c^{2}} - \frac {4 \, \sqrt {d x} d^{2}}{c^{2}}\right )} c}{d}\right )} b}{3 \, d} \]
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 \[ \int \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\sqrt {d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.73, size = 685, normalized size = 6.46 \[ \frac {2 a \left (d x\right )^{\frac {3}{2}}}{3 d} + \frac {2 b \left (\begin {cases} \frac {4 c^{2} \sqrt {d} \left (d x\right )^{\frac {3}{2}} \sqrt {\frac {1}{c}} \operatorname {atanh}{\left (c x \right )}}{12 c^{2} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {4 i c^{2} \sqrt {d} \left (d x\right )^{\frac {3}{2}} \sqrt {\frac {1}{c}} \operatorname {atanh}{\left (c x \right )}}{12 c^{2} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {2 i c^{2} d^{2} \log {\left (i \sqrt {d} \sqrt {\frac {1}{c}} + \sqrt {d x} \right )}}{12 c^{4} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{4} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {8 c d^{\frac {3}{2}} \sqrt {d x} \sqrt {\frac {1}{c}}}{12 c^{2} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {8 i c d^{\frac {3}{2}} \sqrt {d x} \sqrt {\frac {1}{c}}}{12 c^{2} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {4 i c d^{2} \log {\left (- \sqrt {d} \sqrt {\frac {1}{c}} + \sqrt {d x} \right )}}{12 c^{3} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{3} \sqrt {d} \sqrt {\frac {1}{c}}} - \frac {6 i c d^{2} \log {\left (i \sqrt {d} \sqrt {\frac {1}{c}} + \sqrt {d x} \right )}}{12 c^{3} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{3} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {4 i c d^{2} \operatorname {atanh}{\left (c x \right )}}{12 c^{3} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{3} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {4 d^{2} \log {\left (- \sqrt {d} \sqrt {\frac {1}{c}} + \sqrt {d x} \right )}}{12 c^{2} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {d} \sqrt {\frac {1}{c}}} - \frac {4 d^{2} \log {\left (- i \sqrt {d} \sqrt {\frac {1}{c}} + \sqrt {d x} \right )}}{12 c^{2} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {d} \sqrt {\frac {1}{c}}} + \frac {4 d^{2} \operatorname {atanh}{\left (c x \right )}}{12 c^{2} \sqrt {d} \sqrt {\frac {1}{c}} + 12 i c^{2} \sqrt {d} \sqrt {\frac {1}{c}}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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