Optimal. Leaf size=319 \[ \frac {8 \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{9 a c^{7/3}}-\frac {4 \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{9 a c^{7/3}}+\frac {8 \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} a c^{7/3}}+\frac {8 \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}-6 c \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}}{3 a c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}} \]
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Rubi [F] time = 1.26, 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 {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{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 {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.31, size = 74, normalized size = 0.23 \begin {gather*} -\frac {6 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3} \, _2F_1\left (\frac {2}{3},3;\frac {5}{3};\frac {c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{c}\right )}{a c^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.97, size = 319, normalized size = 1.00 \begin {gather*} \frac {-6 c \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+8 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}}{3 a c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}}+\frac {8 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} a c^{7/3}}+\frac {8 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{9 a c^{7/3}}-\frac {4 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{9 a c^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 770, normalized size = 2.41 \begin {gather*} \left [\frac {2 \, {\left (6 \, \sqrt {\frac {1}{3}} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (6 \, \sqrt {\frac {1}{3}} {\left (a c^{\frac {2}{3}} x - \sqrt {a^{2} x^{2} - b} c^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 3 \, {\left (a c^{\frac {2}{3}} x + \sqrt {\frac {1}{3}} {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - \sqrt {a^{2} x^{2} - b} c^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + 3 \, {\left (a c x - \sqrt {\frac {1}{3}} {\left (a c^{\frac {4}{3}} x - \sqrt {a^{2} x^{2} - b} c^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 2 \, b\right ) - 2 \, b c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c^{\frac {1}{3}} + c^{\frac {2}{3}}\right ) + 4 \, b c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} - c^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} - 3 \, {\left (a c^{2} x - \sqrt {a^{2} x^{2} - b} c^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{9 \, a b c^{3}}, \frac {2 \, {\left (12 \, \sqrt {\frac {1}{3}} b c^{\frac {2}{3}} \arctan \left (\sqrt {\frac {1}{3}} + \frac {2 \, \sqrt {\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) - 2 \, b c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c^{\frac {1}{3}} + c^{\frac {2}{3}}\right ) + 4 \, b c^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} - c^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} - 3 \, {\left (a c^{2} x - \sqrt {a^{2} x^{2} - b} c^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{9 \, a b c^{3}}\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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a^{2} x^{2}-b}\, \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}\, dx\]
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 {1}{\sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}\,{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 {1}{\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}\,\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 {1}{\sqrt [3]{c + \sqrt [4]{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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