Optimal. Leaf size=30 \[ \frac {4 \left (7 x^3+1\right ) \left (x^4+x\right )^{3/4}}{9 x^3 \left (x^3+1\right )} \]
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Rubi [A] time = 0.15, antiderivative size = 31, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2056, 453, 264} \begin {gather*} \frac {28 x}{9 \sqrt [4]{x^4+x}}+\frac {4}{9 \sqrt [4]{x^4+x} x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rule 2056
Rubi steps
\begin {align*} \int \frac {-1+x^3}{x^3 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^3}\right ) \int \frac {-1+x^3}{x^{13/4} \left (1+x^3\right )^{5/4}} \, dx}{\sqrt [4]{x+x^4}}\\ &=\frac {4}{9 x^2 \sqrt [4]{x+x^4}}+\frac {\left (7 \sqrt [4]{x} \sqrt [4]{1+x^3}\right ) \int \frac {1}{\sqrt [4]{x} \left (1+x^3\right )^{5/4}} \, dx}{3 \sqrt [4]{x+x^4}}\\ &=\frac {4}{9 x^2 \sqrt [4]{x+x^4}}+\frac {28 x}{9 \sqrt [4]{x+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.77 \begin {gather*} \frac {28 x^3+4}{9 x^2 \sqrt [4]{x^4+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 30, normalized size = 1.00 \begin {gather*} \frac {4 \left (7 x^3+1\right ) \left (x^4+x\right )^{3/4}}{9 x^3 \left (x^3+1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 25, normalized size = 0.83 \begin {gather*} \frac {4 \, {\left (x^{4} + x\right )}^{\frac {3}{4}} {\left (7 \, x^{3} + 1\right )}}{9 \, {\left (x^{6} + x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 19, normalized size = 0.63 \begin {gather*} \frac {4}{9} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} + \frac {8}{3 \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.67 \begin {gather*} \frac {\frac {28 x^{3}}{9}+\frac {4}{9}}{\left (x^{4}+x \right )^{\frac {1}{4}} x^{2}} \end {gather*}
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 {x^{3} - 1}{{\left (x^{4} + x\right )}^{\frac {1}{4}} {\left (x^{3} + 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 26, normalized size = 0.87 \begin {gather*} \frac {4\,\left (7\,x^3+1\right )\,{\left (x^4+x\right )}^{3/4}}{9\,x^3\,\left (x^3+1\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 {\left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{3} \sqrt [4]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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