Optimal. Leaf size=104 \[ x-128 \sqrt [4]{x}+\frac {256}{3} \sqrt [3]{2} \log \left (\sqrt [4]{x}+2 \sqrt [3]{2}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (\sqrt {x}-2 \sqrt [3]{2} \sqrt [4]{x}+4\ 2^{2/3}\right )-\frac {256 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt [3]{2}-\sqrt [4]{x}}{\sqrt [3]{2} \sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {376, 459, 321, 200, 31, 634, 617, 204, 628} \[ x-128 \sqrt [4]{x}+\frac {256}{3} \sqrt [3]{2} \log \left (\sqrt [4]{x}+2 \sqrt [3]{2}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (\sqrt {x}-2 \sqrt [3]{2} \sqrt [4]{x}+4\ 2^{2/3}\right )-\frac {256 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt [3]{2}-\sqrt [4]{x}}{\sqrt [3]{2} \sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 321
Rule 376
Rule 459
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {-16+x^{3/4}}{16+x^{3/4}} \, dx &=4 \operatorname {Subst}\left (\int \frac {x^3 \left (-16+x^3\right )}{16+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=x-128 \operatorname {Subst}\left (\int \frac {x^3}{16+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+2048 \operatorname {Subst}\left (\int \frac {1}{16+x^3} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+\frac {1}{3} \left (256 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+x} \, dx,x,\sqrt [4]{x}\right )+\frac {1}{3} \left (256 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {4 \sqrt [3]{2}-x}{4\ 2^{2/3}-2 \sqrt [3]{2} x+x^2} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+\frac {256}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2}+\sqrt [4]{x}\right )-\frac {1}{3} \left (128 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{2}+2 x}{4\ 2^{2/3}-2 \sqrt [3]{2} x+x^2} \, dx,x,\sqrt [4]{x}\right )+\left (256\ 2^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{4\ 2^{2/3}-2 \sqrt [3]{2} x+x^2} \, dx,x,\sqrt [4]{x}\right )\\ &=-128 \sqrt [4]{x}+x+\frac {256}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2}+\sqrt [4]{x}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (4\ 2^{2/3}-2 \sqrt [3]{2} \sqrt [4]{x}+\sqrt {x}\right )+\left (256 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {\sqrt [4]{x}}{\sqrt [3]{2}}\right )\\ &=-128 \sqrt [4]{x}+x-\frac {256 \sqrt [3]{2} \tan ^{-1}\left (\frac {2-2^{2/3} \sqrt [4]{x}}{2 \sqrt {3}}\right )}{\sqrt {3}}+\frac {256}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2}+\sqrt [4]{x}\right )-\frac {128}{3} \sqrt [3]{2} \log \left (4\ 2^{2/3}-2 \sqrt [3]{2} \sqrt [4]{x}+\sqrt {x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 22, normalized size = 0.21 \[ x-2 x \, _2F_1\left (1,\frac {4}{3};\frac {7}{3};-\frac {x^{3/4}}{16}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 71, normalized size = 0.68 \[ \frac {256}{3} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} x^{\frac {1}{4}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {128}{3} \cdot 2^{\frac {1}{3}} \log \left (4 \cdot 2^{\frac {2}{3}} - 2 \cdot 2^{\frac {1}{3}} x^{\frac {1}{4}} + \sqrt {x}\right ) + \frac {256}{3} \cdot 2^{\frac {1}{3}} \log \left (2 \cdot 2^{\frac {1}{3}} + x^{\frac {1}{4}}\right ) + x - 128 \, x^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 71, normalized size = 0.68 \[ \frac {256}{3} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} - x^{\frac {1}{4}}\right )}\right ) - \frac {128}{3} \cdot 2^{\frac {1}{3}} \log \left (4 \cdot 2^{\frac {2}{3}} - 2 \cdot 2^{\frac {1}{3}} x^{\frac {1}{4}} + \sqrt {x}\right ) + \frac {256}{3} \cdot 2^{\frac {1}{3}} \log \left (2 \cdot 2^{\frac {1}{3}} + x^{\frac {1}{4}}\right ) + x - 128 \, x^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.63 \[ x +\frac {128 \,16^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {16^{\frac {2}{3}} x^{\frac {1}{4}}}{8}-1\right )}{3}\right )}{3}+\frac {128 \,16^{\frac {1}{3}} \ln \left (x^{\frac {1}{4}}+16^{\frac {1}{3}}\right )}{3}-\frac {64 \,16^{\frac {1}{3}} \ln \left (\sqrt {x}-16^{\frac {1}{3}} x^{\frac {1}{4}}+16^{\frac {2}{3}}\right )}{3}-128 x^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 71, normalized size = 0.68 \[ \frac {256}{3} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} - x^{\frac {1}{4}}\right )}\right ) - \frac {128}{3} \cdot 2^{\frac {1}{3}} \log \left (4 \cdot 2^{\frac {2}{3}} - 2 \cdot 2^{\frac {1}{3}} x^{\frac {1}{4}} + \sqrt {x}\right ) + \frac {256}{3} \cdot 2^{\frac {1}{3}} \log \left (2 \cdot 2^{\frac {1}{3}} + x^{\frac {1}{4}}\right ) + x - 128 \, x^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 90, normalized size = 0.87 \[ x+\frac {256\,2^{1/3}\,\ln \left (12288\,2^{1/3}+6144\,x^{1/4}\right )}{3}-128\,x^{1/4}+\frac {128\,2^{1/3}\,\ln \left (6144\,x^{1/4}+6144\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3}-\frac {128\,2^{1/3}\,\ln \left (6144\,x^{1/4}-6144\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.72, size = 102, normalized size = 0.98 \[ - 128 \sqrt [4]{x} + x + \frac {256 \sqrt [3]{2} \log {\left (\sqrt [4]{x} + 2 \sqrt [3]{2} \right )}}{3} - \frac {128 \sqrt [3]{2} \log {\left (- 2 \sqrt [3]{2} \sqrt [4]{x} + \sqrt {x} + 4 \cdot 2^{\frac {2}{3}} \right )}}{3} + \frac {256 \sqrt [3]{2} \sqrt {3} \operatorname {atan}{\left (\frac {2^{\frac {2}{3}} \sqrt {3} \sqrt [4]{x}}{6} - \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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