Optimal. Leaf size=138 \[ \frac {1135}{64} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-8\ 2^{3/4} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4-x^3}}\right )-\frac {1135}{64} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+8\ 2^{3/4} \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {1}{96} \sqrt [4]{x^4-x^3} \left (32 x^2-100 x+401\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 258, normalized size of antiderivative = 1.87, number of steps used = 27, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2042, 101, 157, 50, 63, 240, 212, 206, 203, 105, 93, 298} \begin {gather*} -\frac {25}{24} \sqrt [4]{x^4-x^3} x+\frac {401}{96} \sqrt [4]{x^4-x^3}-\frac {1135 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{x-1} x^{3/4}}-\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {1135 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{x-1} x^{3/4}}+\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x^3} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 93
Rule 101
Rule 105
Rule 157
Rule 203
Rule 206
Rule 212
Rule 240
Rule 298
Rule 2042
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{11/4}}{2+x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\sqrt [4]{-x^3+x^4} \int \frac {x^{7/4} \left (-\frac {11}{2}+\frac {25 x}{4}\right )}{(-1+x)^{3/4} (2+x)} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (25 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{7/4}}{(-1+x)^{3/4}} \, dx}{12 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (6 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{7/4}}{(-1+x)^{3/4} (2+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x)^{3/4}} \, dx}{96 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (6 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x)^{3/4}} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (12 \sqrt [4]{-x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x)^{3/4} (2+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{128 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (12 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (24 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} (2+x)} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (18 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (48 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (96 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{2-3 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (18 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (48 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (16 \sqrt {3} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (16 \sqrt {3} \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {3} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (175 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (9 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (24 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (24 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {401}{96} \sqrt [4]{-x^3+x^4}-\frac {25}{24} x \sqrt [4]{-x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{-x^3+x^4}-\frac {1135 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}-\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {1135 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{64 \sqrt [4]{-1+x} x^{3/4}}+\frac {8\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 124, normalized size = 0.90 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (\sqrt [4]{x} \, _2F_1\left (-\frac {11}{4},\frac {1}{4};\frac {5}{4};1-x\right )-3 \sqrt [4]{x} \, _2F_1\left (-\frac {7}{4},\frac {1}{4};\frac {5}{4};1-x\right )+6 \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )-12 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )+8 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {2 (x-1)}{3 x}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.65, size = 138, normalized size = 1.00 \begin {gather*} \frac {1135}{64} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-8\ 2^{3/4} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4-x^3}}\right )-\frac {1135}{64} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+8\ 2^{3/4} \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {1}{96} \sqrt [4]{x^4-x^3} \left (32 x^2-100 x+401\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 207, normalized size = 1.50 \begin {gather*} \frac {1}{96} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (32 \, x^{2} - 100 \, x + 401\right )} - 16 \cdot 24^{\frac {1}{4}} \arctan \left (\frac {24^{\frac {3}{4}} \sqrt {2} x \sqrt {\frac {\sqrt {6} x^{2} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \cdot 24^{\frac {3}{4}} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{24 \, x}\right ) + 4 \cdot 24^{\frac {1}{4}} \log \left (\frac {24^{\frac {1}{4}} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \cdot 24^{\frac {1}{4}} \log \left (-\frac {24^{\frac {1}{4}} x - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1135}{64} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1135}{128} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1135}{128} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 149, normalized size = 1.08 \begin {gather*} -\frac {1}{96} \, {\left (401 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 702 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 333 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{3} - 8 \cdot 24^{\frac {1}{4}} \arctan \left (\frac {2}{3} \, \left (\frac {3}{2}\right )^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 4 \cdot 24^{\frac {1}{4}} \log \left (\left (\frac {3}{2}\right )^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 4 \cdot 24^{\frac {1}{4}} \log \left ({\left | -\left (\frac {3}{2}\right )^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {1135}{64} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1135}{128} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1135}{128} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.45, size = 1030, normalized size = 7.46
result too large to display
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 (x^{4} - x^{3}\right )}^{\frac {1}{4}} x^{2}}{x + 2}\,{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 = -0.01 \begin {gather*} \int \frac {x^2\,{\left (x^4-x^3\right )}^{1/4}}{x+2} \,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 {x^{2} \sqrt [4]{x^{3} \left (x - 1\right )}}{x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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