Optimal. Leaf size=18 \[ -\frac {4 \left (x^4+x^3\right )^{9/4}}{9 x^9} \]
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Rubi [B] time = 0.12, antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2052, 2016, 2014} \begin {gather*} -\frac {4 \left (x^4+x^3\right )^{5/4}}{9 x^6}-\frac {4 \left (x^4+x^3\right )^{5/4}}{9 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rule 2052
Rubi steps
\begin {align*} \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx &=\int \left (\frac {\sqrt [4]{x^3+x^4}}{x^4}+\frac {\sqrt [4]{x^3+x^4}}{x^3}\right ) \, dx\\ &=\int \frac {\sqrt [4]{x^3+x^4}}{x^4} \, dx+\int \frac {\sqrt [4]{x^3+x^4}}{x^3} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}-\frac {4 \left (x^3+x^4\right )^{5/4}}{5 x^5}-\frac {4}{9} \int \frac {\sqrt [4]{x^3+x^4}}{x^3} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {4 \left (x^3 (x+1)\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 18, normalized size = 1.00 \begin {gather*} -\frac {4 \left (x^3+x^4\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 22, normalized size = 1.22 \begin {gather*} -\frac {4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 2 \, x + 1\right )}}{9 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 9, normalized size = 0.50 \begin {gather*} -\frac {4}{9} \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 1.11
method | result | size |
gosper | \(-\frac {4 \left (1+x \right )^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(20\) |
trager | \(-\frac {4 \left (x^{2}+2 x +1\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(23\) |
meijerg | \(-\frac {4 \left (-\frac {4}{5} x^{2}+\frac {1}{5} x +1\right ) \left (1+x \right )^{\frac {1}{4}}}{9 x^{\frac {9}{4}}}-\frac {4 \left (1+x \right )^{\frac {5}{4}}}{5 x^{\frac {5}{4}}}\) | \(32\) |
risch | \(-\frac {4 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (x^{3}+3 x^{2}+3 x +1\right )}{9 \left (1+x \right ) x^{3}}\) | \(33\) |
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}} {\left (x + 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 43, normalized size = 2.39 \begin {gather*} -\frac {8\,x\,{\left (x^4+x^3\right )}^{1/4}+4\,{\left (x^4+x^3\right )}^{1/4}+4\,x^2\,{\left (x^4+x^3\right )}^{1/4}}{9\,x^3} \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 {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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