Optimal. Leaf size=33 \[ \frac {4 \left (128 x^3-96 x^2+84 x-77\right ) \left (x^4+x^3\right )^{3/4}}{1155 x^6} \]
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Rubi [B] time = 0.10, antiderivative size = 73, normalized size of antiderivative = 2.21, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2016, 2014} \begin {gather*} \frac {512 \left (x^4+x^3\right )^{3/4}}{1155 x^3}-\frac {128 \left (x^4+x^3\right )^{3/4}}{385 x^4}-\frac {4 \left (x^4+x^3\right )^{3/4}}{15 x^6}+\frac {16 \left (x^4+x^3\right )^{3/4}}{55 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{x^3+x^4}} \, dx &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}-\frac {4}{5} \int \frac {1}{x^3 \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (x^3+x^4\right )^{3/4}}{55 x^5}+\frac {32}{55} \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (x^3+x^4\right )^{3/4}}{55 x^5}-\frac {128 \left (x^3+x^4\right )^{3/4}}{385 x^4}-\frac {128}{385} \int \frac {1}{x \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{15 x^6}+\frac {16 \left (x^3+x^4\right )^{3/4}}{55 x^5}-\frac {128 \left (x^3+x^4\right )^{3/4}}{385 x^4}+\frac {512 \left (x^3+x^4\right )^{3/4}}{1155 x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3 (x+1)\right )^{3/4} \left (128 x^3-96 x^2+84 x-77\right )}{1155 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 33, normalized size = 1.00 \begin {gather*} \frac {4 \left (-77+84 x-96 x^2+128 x^3\right ) \left (x^3+x^4\right )^{3/4}}{1155 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 29, normalized size = 0.88 \begin {gather*} \frac {4 \, {\left (x^{4} + x^{3}\right )}^{\frac {3}{4}} {\left (128 \, x^{3} - 96 \, x^{2} + 84 \, x - 77\right )}}{1155 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 37, normalized size = 1.12 \begin {gather*} -\frac {4}{15} \, {\left (\frac {1}{x} + 1\right )}^{\frac {15}{4}} + \frac {12}{11} \, {\left (\frac {1}{x} + 1\right )}^{\frac {11}{4}} - \frac {12}{7} \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{4}} + \frac {4}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 26, normalized size = 0.79
method | result | size |
meijerg | \(-\frac {4 \left (-\frac {128}{77} x^{3}+\frac {96}{77} x^{2}-\frac {12}{11} x +1\right ) \left (1+x \right )^{\frac {3}{4}}}{15 x^{\frac {15}{4}}}\) | \(26\) |
trager | \(\frac {4 \left (128 x^{3}-96 x^{2}+84 x -77\right ) \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{1155 x^{6}}\) | \(30\) |
gosper | \(\frac {4 \left (1+x \right ) \left (128 x^{3}-96 x^{2}+84 x -77\right )}{1155 x^{3} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\) | \(33\) |
risch | \(\frac {-\frac {4}{15}+\frac {4}{165} x -\frac {16}{385} x^{2}+\frac {128}{1155} x^{3}+\frac {512}{1155} x^{4}}{x^{3} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(35\) |
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 {1}{{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 57, normalized size = 1.73 \begin {gather*} \frac {512\,{\left (x^4+x^3\right )}^{3/4}}{1155\,x^3}-\frac {128\,{\left (x^4+x^3\right )}^{3/4}}{385\,x^4}+\frac {16\,{\left (x^4+x^3\right )}^{3/4}}{55\,x^5}-\frac {4\,{\left (x^4+x^3\right )}^{3/4}}{15\,x^6} \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 {1}{x^{4} \sqrt [4]{x^{3} \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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