Optimal. Leaf size=699 \[ -\frac {70 b \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{243 a c^{13/3}}+\frac {35 b \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{243 a c^{13/3}}-\frac {70 b \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{81 \sqrt {3} a c^{13/3}}+\frac {\sqrt {\sqrt {a^2 x^2-b}+a x} \left (11550 b c-10935 a c^5 x\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3} \left (9720 a c^4 x-15400 b\right )+\left (8910 b c^3-19683 a c^7 x\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (13122 a c^6 x-9900 b c^2\right ) \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\sqrt {a^2 x^2-b} \left (-19683 c^7 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+13122 c^6 \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}-10935 c^5 \sqrt {\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+9720 c^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}\right )}{17820 a c^4 \sqrt {a^2 x^2-b}+17820 a^2 c^4 x} \]
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Rubi [F] time = 0.05, 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 [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 [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.76, size = 575, normalized size = 0.82 \begin {gather*} \frac {6 \left (-\frac {35 b \log \left (1-\frac {\sqrt [3]{c}}{\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}\right )}{729 c^{13/3}}+\frac {35 b \log \left (\frac {c^{2/3}}{\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}}+\frac {\sqrt [3]{c}}{\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}+1\right )}{1458 c^{13/3}}+\frac {35 b \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{c}}{\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}+1}{\sqrt {3}}\right )}{243 \sqrt {3} c^{13/3}}+\frac {b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{11/3}}{12 c^4 \left (\sqrt {a^2 x^2-b}+a x\right )}-\frac {37 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{8/3}}{108 c^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}+\frac {44 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{5/3}}{81 c^4 \sqrt {\sqrt {a^2 x^2-b}+a x}}-\frac {104 b \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}}{243 c^4 \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}-\frac {1}{2} c^3 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+\frac {3}{5} c^2 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{5/3}+\frac {1}{11} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{11/3}-\frac {3}{8} c \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{8/3}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.16, size = 699, normalized size = 1.00 \begin {gather*} \frac {\left (8910 b c^3-19683 a c^7 x\right ) \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+\left (-9900 b c^2+13122 a c^6 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 (11550 b c-10935 a c^5 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 (-15400 b+9720 a c^4 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 (-19683 c^7 \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+13122 c^6 \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}-10935 c^5 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}+9720 c^4 \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 )}{17820 a^2 c^4 x+17820 a c^4 \sqrt {-b+a^2 x^2}}-\frac {70 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 )}{81 \sqrt {3} a c^{13/3}}-\frac {70 b \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{243 a c^{13/3}}+\frac {35 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 )}{243 a c^{13/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 1039, 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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}\, dx\]
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 (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 (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}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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