Optimal. Leaf size=495 \[ -\frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {c}}\right )}{128 a c^{9/2}}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {\sqrt {a^2 x^2-b} \left (\left (-9216 a^2 c^5 x^2-12288 a c^7 x-2450 b^2 c+2304 b c^5\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (7680 a^2 c^4 x^2+6144 a c^6 x+3675 b^2-1920 b c^4\right )\right )+\left (-9216 a^3 c^5 x^3-12288 a^2 c^7 x^2-2450 a b^2 c x+6912 a b c^5 x-1680 b^2 c^3+6144 b c^7\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (7680 a^3 c^4 x^3+6144 a^2 c^6 x^2+3675 a b^2 x-5760 a b c^4 x+1960 b^2 c^2-3072 b c^6\right )}{13440 a c^4 \left (2 a^2 x^2-b\right )+26880 a^2 c^4 x \sqrt {a^2 x^2-b}} \]
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Rubi [F] time = 0.33, 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 \frac {\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 \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {\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 = 3.32, size = 536, normalized size = 1.08 \begin {gather*} \frac {\left (2 a x \left (\sqrt {a^2 x^2-b}+a x\right )-2 b\right ) \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+c\right )^{3/2} \left (\frac {\sqrt {c} \left (35 b^2 \left (56 c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}-70 c \left (\sqrt {a^2 x^2-b}+a x\right )+105 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}-48 c^3\right )+1536 a c^4 x \left (\sqrt {a^2 x^2-b}+a x\right ) \left (4 c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}+5 a x \sqrt {\sqrt {a^2 x^2-b}+a x}-6 a c x-8 c^3\right )-384 b c^4 \left (8 c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}-6 c \left (\sqrt {a^2 x^2-b}+3 a x\right )+5 \sqrt {\sqrt {a^2 x^2-b}+a x} \left (\sqrt {a^2 x^2-b}+3 a x\right )-16 c^3\right )\right )}{\left (\sqrt {\sqrt {a^2 x^2-b}+a x}+c\right )^7}-\frac {105 b \left (35 b-256 c^4\right ) \left (\sqrt {a^2 x^2-b}+a x\right )^2 \tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}\right )}{\left (\sqrt {\sqrt {a^2 x^2-b}+a x}+c\right )^{15/2}}\right )}{13440 a c^{9/2} \sqrt {4 a^2 x^2-4 b} \left (\frac {c}{\sqrt {\sqrt {a^2 x^2-b}+a x}+c}-1\right )^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.06, size = 495, normalized size = 1.00 \begin {gather*} \frac {\left (-1680 b^2 c^3+6144 b c^7-2450 a b^2 c x+6912 a b c^5 x-12288 a^2 c^7 x^2-9216 a^3 c^5 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (1960 b^2 c^2-3072 b c^6+3675 a b^2 x-5760 a b c^4 x+6144 a^2 c^6 x^2+7680 a^3 c^4 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 (-2450 b^2 c+2304 b c^5-12288 a c^7 x-9216 a^2 c^5 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (3675 b^2-1920 b c^4+6144 a c^6 x+7680 a^2 c^4 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{26880 a^2 c^4 x \sqrt {-b+a^2 x^2}+13440 a c^4 \left (-b+2 a^2 x^2\right )}-\frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{128 a c^{9/2}}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 581, normalized size = 1.17 \begin {gather*} \left [\frac {105 \, {\left (256 \, b c^{4} - 35 \, 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 (6144 \, c^{8} + 3360 \, a^{2} c^{4} x^{2} - 1680 \, b c^{4} + 2 \, {\left (1152 \, a c^{6} + 1225 \, a b c^{2}\right )} x + 2 \, {\left (1152 \, c^{6} - 1680 \, a c^{4} x - 1225 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (3072 \, c^{7} + 3920 \, a^{2} c^{3} x^{2} - 1960 \, b c^{3} + 15 \, {\left (128 \, a c^{5} + 245 \, a b c\right )} x + 5 \, {\left (384 \, c^{5} - 784 \, a c^{3} x - 735 \, 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}}}}{26880 \, a c^{5}}, -\frac {105 \, {\left (256 \, b c^{4} - 35 \, b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{c}\right ) + {\left (6144 \, c^{8} + 3360 \, a^{2} c^{4} x^{2} - 1680 \, b c^{4} + 2 \, {\left (1152 \, a c^{6} + 1225 \, a b c^{2}\right )} x + 2 \, {\left (1152 \, c^{6} - 1680 \, a c^{4} x - 1225 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (3072 \, c^{7} + 3920 \, a^{2} c^{3} x^{2} - 1960 \, b c^{3} + 15 \, {\left (128 \, a c^{5} + 245 \, a b c\right )} x + 5 \, {\left (384 \, c^{5} - 784 \, a c^{3} x - 735 \, 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}}}}{13440 \, a c^{5}}\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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}}{\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]
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*} \int \frac {\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 \frac {\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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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