Optimal. Leaf size=301 \[ \frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}+\frac {b \sqrt {d} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} c^{3/4}}+\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{3 c^{3/4}}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}} \]
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Rubi [A] time = 0.25, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6097, 16, 329, 301, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}+\frac {b \sqrt {d} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} c^{3/4}}+\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{3 c^{3/4}}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 204
Rule 205
Rule 208
Rule 297
Rule 298
Rule 301
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6097
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(4 b c) \int \frac {x (d x)^{3/2}}{1-c^2 x^4} \, dx}{3 d}\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(4 b c) \int \frac {(d x)^{5/2}}{1-c^2 x^4} \, dx}{3 d^2}\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(8 b c) \operatorname {Subst}\left (\int \frac {x^6}{1-\frac {c^2 x^8}{d^4}} \, dx,x,\sqrt {d x}\right )}{3 d^3}\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {1}{3} (4 b d) \operatorname {Subst}\left (\int \frac {x^2}{d^2-c x^4} \, dx,x,\sqrt {d x}\right )+\frac {1}{3} (4 b d) \operatorname {Subst}\left (\int \frac {x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(2 b d) \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}+\frac {(2 b d) \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}-\frac {(2 b d) \operatorname {Subst}\left (\int \frac {d-\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}+\frac {(2 b d) \operatorname {Subst}\left (\int \frac {d+\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}\\ &=\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\left (b \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}+2 x}{-\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}+\frac {\left (b \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}-2 x}{-\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{3 c}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{3 c}\\ &=\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}+\frac {\left (\sqrt {2} b \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\left (\sqrt {2} b \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}\\ &=\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 227, normalized size = 0.75 \[ \frac {\sqrt {d x} \left (4 a c^{3/4} x^{3/2}+4 b c^{3/4} x^{3/2} \tanh ^{-1}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \log \left (\sqrt [4]{c} \sqrt {x}+1\right )+\sqrt {2} b \log \left (\sqrt {c} x-\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )-\sqrt {2} b \log \left (\sqrt {c} x+\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )-2 \sqrt {2} b \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+2 \sqrt {2} b \tan ^{-1}\left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )+4 b \tan ^{-1}\left (\sqrt [4]{c} \sqrt {x}\right )\right )}{6 c^{3/4} \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 34, normalized size = 0.11 \[ \frac {1}{3} \, {\left (b x \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a x\right )} \sqrt {d x} \]
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 \[ \int \sqrt {d x} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 280, normalized size = 0.93 \[ \frac {2 \left (d x \right )^{\frac {3}{2}} a}{3 d}+\frac {2 b \left (d x \right )^{\frac {3}{2}} \arctanh \left (c \,x^{2}\right )}{3 d}+\frac {2 d b \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {d b \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {d b \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{6 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {d b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {d b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 301, normalized size = 1.00 \[ \frac {4 \, \left (d x\right )^{\frac {3}{2}} a + {\left (4 \, \left (d x\right )^{\frac {3}{2}} \operatorname {artanh}\left (c x^{2}\right ) + \frac {{\left (\frac {d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )}}{c} + \frac {2 \, d^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}}\right )}}{c}\right )} c}{d^{2}}\right )} b}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {d\,x}\,\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right ) \,d x \]
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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