Optimal. Leaf size=286 \[ \frac {\sqrt {\sqrt {a^2 x^2+b}+a x} \left (6 a c x+16 c^3\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+\sqrt {a^2 x^2+b} \left (\left (60 a x-8 c^2\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+6 c \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}\right )+\left (60 a^2 x^2-8 a c^2 x-75 b\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{105 a \sqrt {\sqrt {a^2 x^2+b}+a x}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{a \sqrt {c}} \]
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Rubi [F] time = 0.34, 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 {a x+\sqrt {b+a^2 x^2}} \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 {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx &=\int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\\ \end {align*}
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Mathematica [F] time = 9.27, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x+\sqrt {b+a^2 x^2}} \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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IntegrateAlgebraic [A] time = 0.72, size = 286, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\sqrt {a^2 x^2+b}+a x} \left (6 a c x+16 c^3\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+\sqrt {a^2 x^2+b} \left (\left (60 a x-8 c^2\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+6 c \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}\right )+\left (60 a^2 x^2-8 a c^2 x-75 b\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{105 a \sqrt {\sqrt {a^2 x^2+b}+a x}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{a \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.97, size = 359, normalized size = 1.26 \begin {gather*} \left [\frac {105 \, 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} + 6 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} - 135 \, a c x + 75 \, \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}}}}{210 \, a c}, \frac {105 \, 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} + 6 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} - 135 \, a c x + 75 \, \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}}}}{105 \, a c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \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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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \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 {\sqrt {a^2\,x^2+b}+a\,x}\,\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}}} \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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