Optimal. Leaf size=114 \[ -\frac {155}{64} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )+\frac {155}{64} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )+\frac {1}{96} \sqrt [4]{x^4+x^3} \left (32 x^2+52 x+101\right ) \]
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Rubi [B] time = 0.21, antiderivative size = 230, normalized size of antiderivative = 2.02, number of steps used = 27, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2042, 101, 157, 50, 63, 331, 298, 203, 206, 105, 93} \begin {gather*} \frac {13}{24} \sqrt [4]{x^4+x^3} x+\frac {101}{96} \sqrt [4]{x^4+x^3}-\frac {155 \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{64 \sqrt [4]{x+1} x^{3/4}}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt [4]{x+1} x^{3/4}}+\frac {155 \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{64 \sqrt [4]{x+1} x^{3/4}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{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 298
Rule 331
Rule 2042
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4} \sqrt [4]{1+x}}{-1+x} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}-\frac {\sqrt [4]{x^3+x^4} \int \frac {\left (-\frac {11}{4}-\frac {13 x}{4}\right ) x^{7/4}}{(-1+x) (1+x)^{3/4}} \, dx}{3 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {\left (13 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(1+x)^{3/4}} \, dx}{12 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(-1+x) (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}-\frac {\left (91 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{96 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(-1+x) (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {\left (91 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{128 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{(-1+x) \sqrt [4]{x} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {\left (91 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (6 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {\left (91 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{32 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (6 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (8 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt {2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt {2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (91 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (91 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {101}{96} \sqrt [4]{x^3+x^4}+\frac {13}{24} x \sqrt [4]{x^3+x^4}+\frac {1}{3} x^2 \sqrt [4]{x^3+x^4}-\frac {155 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}}+\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {155 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{64 x^{3/4} \sqrt [4]{1+x}}-\frac {2 \sqrt [4]{2} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 124, normalized size = 1.09 \begin {gather*} \frac {4 \sqrt [4]{x^3 (x+1)} \left (21 (x+1) x^2 \, _2F_1\left (-\frac {1}{4},\frac {11}{4};\frac {15}{4};-x\right )+33 (x+1) x \, _2F_1\left (-\frac {1}{4},\frac {7}{4};\frac {11}{4};-x\right )+77 (x+1) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-x\right )+77 (x+1) \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-154 \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 x}{x+1}\right )\right )}{231 (x+1)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 114, normalized size = 1.00 \begin {gather*} -\frac {155}{64} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )+\frac {155}{64} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )+\frac {1}{96} \sqrt [4]{x^4+x^3} \left (32 x^2+52 x+101\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 183, normalized size = 1.61 \begin {gather*} \frac {1}{96} \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (32 \, x^{2} + 52 \, x + 101\right )} + 4 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} + x^{3}}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{2 \, x}\right ) - 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {155}{64} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {155}{128} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {155}{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 = 0.53, size = 123, normalized size = 1.08 \begin {gather*} \frac {1}{96} \, {\left (101 \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} - 150 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 81 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{3} - 2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {155}{64} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {155}{128} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {155}{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 = 1.88, size = 878, normalized size = 7.70
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 - 1}\,{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-1} \,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 - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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