Optimal. Leaf size=293 \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{4 a c^{3/2}}+\frac {\left (48 a^2 c x^2-16 a c^3 x-6 b c\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (8 a c^2 x-15 b\right )+\sqrt {a^2 x^2+b} \left (\left (48 a c x-16 c^3\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+8 c^2 \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}\right )}{60 a c \sqrt {a^2 x^2+b}+60 a^2 c x} \]
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Rubi [F] time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx &=\int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.59, size = 220, normalized size = 0.75 \begin {gather*} -\frac {2 \left (-\frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{8 c^{3/2}}-\frac {1}{5} \left (\sqrt {\sqrt {a^2 x^2+b}+a x}+c\right )^{5/2}+\frac {1}{3} c \left (\sqrt {\sqrt {a^2 x^2+b}+a x}+c\right )^{3/2}+\frac {b \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{8 c \sqrt {\sqrt {a^2 x^2+b}+a x}}+\frac {b \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{4 \left (\sqrt {a^2 x^2+b}+a x\right )}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 293, normalized size = 1.00 \begin {gather*} \frac {\left (-6 b c-16 a c^3 x+48 a^2 c x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-15 b+8 a c^2 x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (-16 c^3+48 a c x\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+8 c^2 \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{60 a^2 c x+60 a c \sqrt {b+a^2 x^2}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{4 a c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 359, normalized size = 1.23 \begin {gather*} \left [\frac {15 \, b \sqrt {c} \log \left (-2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) - 2 \, {\left (16 \, c^{4} - 54 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} + 15 \, a c x - 15 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{120 \, a c^{2}}, -\frac {15 \, b \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + {\left (16 \, c^{4} - 54 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} + 15 \, a c x - 15 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{60 \, a c^{2}}\right ] \end {gather*}
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 \begin {gather*} \int \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}} \,d x \end {gather*}
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 \begin {gather*} \int \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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