Optimal. Leaf size=265 \[ \frac {4 c^{4/3} \log \left (\sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}-\sqrt [3]{c}\right )}{3 a}-\frac {2 c^{4/3} \log \left (\sqrt [3]{c} \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}+\left (\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c\right )^{2/3}+c^{2/3}\right )}{3 a}-\frac {4 c^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a}+\frac {4 c \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}}{a}+\frac {\left (\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c\right )^{4/3}}{a} \]
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Rubi [F] time = 0.40, 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 {\left (c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )^{4/3}}{\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 {\left (c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )^{4/3}}{\sqrt {-b+a^2 x^2}} \, dx &=\int \frac {\left (c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )^{4/3}}{\sqrt {-b+a^2 x^2}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.36, size = 199, normalized size = 0.75 \begin {gather*} \frac {-2 c^2 \left (\frac {c \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt {a^2 x^2-b}+a x}{\sqrt {a^2 x^2-b}+a x}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {c}{\left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}\right )+6 c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}+5 c^2}{a \left (\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.96, size = 288, normalized size = 1.09 \begin {gather*} \frac {5 c \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}}{a}+\frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}}{a}-\frac {4 c^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} a}+\frac {4 c^{4/3} \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}\right )}{3 a}-\frac {2 c^{4/3} \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}+\left (c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )^{2/3}\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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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 \[\int \frac {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {3}{4}}\right )^{\frac {4}{3}}}{\sqrt {a^{2} x^{2}-b}}\, dx\]
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 {{\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}\right )}^{\frac {4}{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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{3/4}\right )}^{4/3}}{\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 {\left (c + \left (a x + \sqrt {a^{2} x^{2} - b}\right )^{\frac {3}{4}}\right )^{\frac {4}{3}}}{\sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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