Optimal. Leaf size=697 \[ \frac {182 b \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{729 a c^{16/3}}-\frac {91 b \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{729 a c^{16/3}}+\frac {182 b \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{243 \sqrt {3} a c^{16/3}}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (3645 a c^5 x-2730 b c\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3} \left (3640 a b x+1944 b c^4\right )+\sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (6561 a c^7 x-2106 b c^3\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt {\sqrt {a^2 x^2-b}+a x} \left (2340 b c^2-4374 a c^6 x\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt {a^2 x^2-b} \left (6561 c^7 \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}-4374 c^6 \sqrt {\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+3645 c^5 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+3640 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}\right )}{4860 a c^5 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}} \]
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Rubi [F] time = 0.36, 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 [4]{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 [4]{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 [4]{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 [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.52, size = 697, normalized size = 1.00 \begin {gather*} \frac {\left (1944 b c^4+3640 a b x\right ) \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-2106 b c^3+6561 a c^7 x\right ) \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}+\left (2340 b c^2-4374 a c^6 x\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-2730 b c+3645 a c^5 x\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\sqrt {-b+a^2 x^2} \left (3640 b \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+6561 c^7 \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}-4374 c^6 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+3645 c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{4860 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {182 b \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 )}{243 \sqrt {3} a c^{16/3}}+\frac {182 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{729 a c^{16/3}}-\frac {91 b \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 )}{729 a c^{16/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 1036, normalized size = 1.49
result too large to display
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 \[\int \frac {1}{\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}\, dx\]
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}{{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} {\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}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}} \,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 [4]{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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