Optimal. Leaf size=72 \[ \frac {\sqrt [4]{x^3+x^4} \left (7315-4180 x+3040 x^2-2432 x^3+2048 x^4\right )}{10240}+\frac {4389 \text {ArcTan}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096}-\frac {4389 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(168\) vs. \(2(72)=144\).
time = 0.09, antiderivative size = 168, normalized size of antiderivative = 2.33, number of steps
used = 11, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2081, 52, 65,
338, 304, 209, 212} \begin {gather*} \frac {4389 \sqrt [4]{x^4+x^3} \text {ArcTan}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4096 \sqrt [4]{x+1} x^{3/4}}+\frac {1}{5} \sqrt [4]{x^4+x^3} x^4-\frac {19}{80} \sqrt [4]{x^4+x^3} x^3-\frac {209}{512} \sqrt [4]{x^4+x^3} x+\frac {1463 \sqrt [4]{x^4+x^3}}{2048}-\frac {4389 \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4096 \sqrt [4]{x+1} x^{3/4}}+\frac {19}{64} \sqrt [4]{x^4+x^3} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 212
Rule 304
Rule 338
Rule 2081
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{19/4}}{(1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (19 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{15/4}}{(1+x)^{3/4}} \, dx}{20 x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {\left (57 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{11/4}}{(1+x)^{3/4}} \, dx}{64 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (209 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{7/4}}{(1+x)^{3/4}} \, dx}{256 x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {\left (1463 \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{2048 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{8192 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2048 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}-\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4389 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1463 \sqrt [4]{x^3+x^4}}{2048}-\frac {209}{512} x \sqrt [4]{x^3+x^4}+\frac {19}{64} x^2 \sqrt [4]{x^3+x^4}-\frac {19}{80} x^3 \sqrt [4]{x^3+x^4}+\frac {1}{5} x^4 \sqrt [4]{x^3+x^4}+\frac {4389 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}-\frac {4389 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4096 x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 95, normalized size = 1.32 \begin {gather*} \frac {x^{9/4} \left (2 x^{3/4} \left (7315+3135 x-1140 x^2+608 x^3-384 x^4+2048 x^5\right )-21945 (1+x)^{3/4} \text {ArcTan}\left (\frac {1}{\sqrt [4]{\frac {x}{1+x}}}\right )-21945 (1+x)^{3/4} \tanh ^{-1}\left (\frac {1}{\sqrt [4]{\frac {x}{1+x}}}\right )\right )}{20480 \left (x^3 (1+x)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.66, size = 15, normalized size = 0.21
method | result | size |
meijerg | \(\frac {4 x^{\frac {23}{4}} \hypergeom \left (\left [\frac {3}{4}, \frac {23}{4}\right ], \left [\frac {27}{4}\right ], -x \right )}{23}\) | \(15\) |
trager | \(\left (\frac {1}{5} x^{4}-\frac {19}{80} x^{3}+\frac {19}{64} x^{2}-\frac {209}{512} x +\frac {1463}{2048}\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}+\frac {4389 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}+2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}}{x^{2}}\right )}{8192}+\frac {4389 \ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}+x^{3}}\, x +2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}-2 x^{3}-x^{2}}{x^{2}}\right )}{8192}\) | \(167\) |
risch | \(\frac {\left (2048 x^{4}-2432 x^{3}+3040 x^{2}-4180 x +7315\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{10240}+\frac {\left (\frac {4389 \ln \left (\frac {2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, x +2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-2 x^{3}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}-4 x -1}{\left (1+x \right )^{2}}\right )}{8192}+\frac {4389 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-4 x \RootOf \left (\textit {\_Z}^{2}+1\right )-4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}}{\left (1+x \right )^{2}}\right )}{8192}\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (\left (1+x \right )^{3} x \right )^{\frac {1}{4}}}{x \left (1+x \right )}\) | \(390\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 87, normalized size = 1.21 \begin {gather*} \frac {1}{10240} \, {\left (2048 \, x^{4} - 2432 \, x^{3} + 3040 \, x^{2} - 4180 \, x + 7315\right )} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} - \frac {4389}{4096} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {4389}{8192} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {4389}{8192} \, \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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \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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Giac [A]
time = 0.41, size = 87, normalized size = 1.21 \begin {gather*} \frac {1}{10240} \, {\left (7315 \, {\left (\frac {1}{x} + 1\right )}^{\frac {17}{4}} - 33440 \, {\left (\frac {1}{x} + 1\right )}^{\frac {13}{4}} + 59470 \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} - 50312 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 19015 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} - \frac {4389}{4096} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {4389}{8192} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4389}{8192} \, \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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,{\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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