Optimal. Leaf size=455 \[ \frac {\left (32760 b^2 c^2-1024 b c^6+3465 a b^2 x-1920 a b c^4 x+114240 a^2 b c^2 x^2-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3+40320 a^4 c^2 x^4\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-2310 b^2 c+768 b c^5+38640 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (3465 b^2-640 b c^4+94080 a b c^2 x-2048 a c^6 x-2560 a^2 c^4 x^2+40320 a^3 c^2 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (37520 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{55440 a c^2 \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{16 a c^{5/2}} \]
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Rubi [F]
time = 0.60, 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 {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 {b+a^2 x^2} \sqrt {a x+\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 {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 [A]
time = 1.58, size = 417, normalized size = 0.92 \begin {gather*} \frac {\frac {\sqrt {c} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \left (105 b^2 \left (312 c^2-22 c \sqrt {a x+\sqrt {b+a^2 x^2}}+33 \left (a x+\sqrt {b+a^2 x^2}\right )\right )+16 b c^2 \left (-64 c^4+48 c^3 \sqrt {a x+\sqrt {b+a^2 x^2}}-40 c^2 \left (3 a x+\sqrt {b+a^2 x^2}\right )+420 a x \left (17 a x+14 \sqrt {b+a^2 x^2}\right )+35 c \sqrt {a x+\sqrt {b+a^2 x^2}} \left (69 a x+67 \sqrt {b+a^2 x^2}\right )\right )+64 c^2 \left (a x+\sqrt {b+a^2 x^2}\right ) \left (630 a^3 x^3+32 c^5 \sqrt {a x+\sqrt {b+a^2 x^2}}+8 a c^3 x \left (-4 c+3 \sqrt {a x+\sqrt {b+a^2 x^2}}\right )+5 a^2 c x^2 \left (-8 c+7 \sqrt {a x+\sqrt {b+a^2 x^2}}\right )\right )\right )}{\left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}-3465 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{55440 a c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {a^{2} x^{2}+b}\, \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 535, normalized size = 1.18 \begin {gather*} \left [\frac {3465 \, b^{2} \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} + 37520 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 385 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 1155 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} + 8400 \, a^{2} c^{3} x^{2} - 32760 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 693 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 5712 \, a c^{3} x - 693 \, 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}}}}{110880 \, a c^{3}}, \frac {3465 \, b^{2} \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} + 37520 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 385 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 1155 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} + 8400 \, a^{2} c^{3} x^{2} - 32760 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 693 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 5712 \, a c^{3} x - 693 \, 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}}}}{55440 \, a c^{3}}\right ] \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}} \sqrt {a^{2} x^{2} + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {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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