Optimal. Leaf size=515 \[ \frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{128 a c^{7/2}}-\frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{a}+\frac {\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (2560 a^3 c^4 x^3+2048 a^2 c^6 x^2-1575 a b^2 x+1920 a b c^4 x-840 b^2 c^2+1024 b c^6\right )+\sqrt {a^2 x^2+b} \left (\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (2560 a^2 c^4 x^2+2048 a c^6 x-1575 b^2+640 b c^4\right )+\left (35840 a^3 c^3 x^3-3072 a^2 c^5 x^2+179200 a b c^3 x-4096 a c^7 x+1050 b^2 c-768 b c^5\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}\right )+\left (35840 a^4 c^3 x^4-3072 a^3 c^5 x^3+197120 a^2 b c^3 x^2-4096 a^2 c^7 x^2+1050 a b^2 c x-2304 a b c^5 x+80080 b^2 c^3-2048 b c^7\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{40320 a c^3 \left (2 a^2 x^2+b\right )+80640 a^2 c^3 x \sqrt {a^2 x^2+b}} \]
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Rubi [F] time = 0.31, 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 {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 {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx &=\int \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 [A] time = 1.26, size = 392, normalized size = 0.76 \begin {gather*} \frac {\left (\left (\sqrt {a^2 x^2+b}+a x\right )^2+b\right ) \left (\sqrt {c} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (-105 b^2 \left (8 c^2 \sqrt {\sqrt {a^2 x^2+b}+a x}-10 c \left (\sqrt {a^2 x^2+b}+a x\right )+15 \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}+48 c^3\right )+80640 b c^3 \left (\sqrt {a^2 x^2+b}+a x\right )^2-128 c^3 \left (\sqrt {a^2 x^2+b}+a x\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b}+a x}+c\right ) \left (-24 c^2 \sqrt {\sqrt {a^2 x^2+b}+a x}+30 c \left (\sqrt {a^2 x^2+b}+a x\right )-35 \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}+16 c^3\right )\right )+315 b \left (5 b-256 c^4\right ) \left (\sqrt {a^2 x^2+b}+a x\right )^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )\right )}{80640 a c^{7/2} \sqrt {a^2 x^2+b} \left (\sqrt {a^2 x^2+b}+a x\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.36, size = 515, normalized size = 1.00 \begin {gather*} \frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{128 a c^{7/2}}-\frac {2 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{a}+\frac {\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (2560 a^3 c^4 x^3+2048 a^2 c^6 x^2-1575 a b^2 x+1920 a b c^4 x-840 b^2 c^2+1024 b c^6\right )+\sqrt {a^2 x^2+b} \left (\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (2560 a^2 c^4 x^2+2048 a c^6 x-1575 b^2+640 b c^4\right )+\left (35840 a^3 c^3 x^3-3072 a^2 c^5 x^2+179200 a b c^3 x-4096 a c^7 x+1050 b^2 c-768 b c^5\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}\right )+\left (35840 a^4 c^3 x^4-3072 a^3 c^5 x^3+197120 a^2 b c^3 x^2-4096 a^2 c^7 x^2+1050 a b^2 c x-2304 a b c^5 x+80080 b^2 c^3-2048 b c^7\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{40320 a c^3 \left (2 a^2 x^2+b\right )+80640 a^2 c^3 x \sqrt {a^2 x^2+b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 554, normalized size = 1.08 \begin {gather*} \left [\frac {315 \, {\left (256 \, b c^{4} - 5 \, b^{2}\right )} \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 (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 80080 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} - 9520 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} - 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} + 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{80640 \, a c^{4}}, \frac {315 \, {\left (256 \, b c^{4} - 5 \, b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) - {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 80080 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} - 9520 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} - 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} + 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{40320 \, a c^{4}}\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] time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \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^{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 {a^2\,x^2+b}\,\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^{2} x^{2} + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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