Optimal. Leaf size=23 \[ \frac {2}{3} \sqrt {x^4+x} \left (2 x^4+2 x+3\right ) \]
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Rubi [A] time = 0.36, antiderivative size = 42, normalized size of antiderivative = 1.83, number of steps used = 18, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2053, 2011, 329, 225, 2029, 206, 2024} \begin {gather*} \frac {4}{3} \sqrt {x^4+x} x^4+\frac {4}{3} \sqrt {x^4+x} x+2 \sqrt {x^4+x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 225
Rule 329
Rule 2011
Rule 2024
Rule 2029
Rule 2053
Rubi steps
\begin {align*} \int \frac {\left (1+4 x^3\right ) \left (1+2 x+2 x^4\right )}{\sqrt {x+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {x+x^4}}+\frac {2 x}{\sqrt {x+x^4}}+\frac {4 x^3}{\sqrt {x+x^4}}+\frac {10 x^4}{\sqrt {x+x^4}}+\frac {8 x^7}{\sqrt {x+x^4}}\right ) \, dx\\ &=2 \int \frac {x}{\sqrt {x+x^4}} \, dx+4 \int \frac {x^3}{\sqrt {x+x^4}} \, dx+8 \int \frac {x^7}{\sqrt {x+x^4}} \, dx+10 \int \frac {x^4}{\sqrt {x+x^4}} \, dx+\int \frac {1}{\sqrt {x+x^4}} \, dx\\ &=2 \sqrt {x+x^4}+\frac {10}{3} x \sqrt {x+x^4}+\frac {4}{3} x^4 \sqrt {x+x^4}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )-5 \int \frac {x}{\sqrt {x+x^4}} \, dx-6 \int \frac {x^4}{\sqrt {x+x^4}} \, dx+\frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}-\int \frac {1}{\sqrt {x+x^4}} \, dx\\ &=2 \sqrt {x+x^4}+\frac {4}{3} x \sqrt {x+x^4}+\frac {4}{3} x^4 \sqrt {x+x^4}+\frac {4}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )+3 \int \frac {x}{\sqrt {x+x^4}} \, dx-\frac {10}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )-\frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}\\ &=2 \sqrt {x+x^4}+\frac {4}{3} x \sqrt {x+x^4}+\frac {4}{3} x^4 \sqrt {x+x^4}-2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )+\frac {x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}+2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )-\frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}\\ &=2 \sqrt {x+x^4}+\frac {4}{3} x \sqrt {x+x^4}+\frac {4}{3} x^4 \sqrt {x+x^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 23, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {x^4+x} \left (2 x^4+2 x+3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 23, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {x^4+x} \left (2 x^4+2 x+3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 19, normalized size = 0.83 \begin {gather*} \frac {2}{3} \, {\left (2 \, x^{4} + 2 \, x + 3\right )} \sqrt {x^{4} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 19, normalized size = 0.83 \begin {gather*} \frac {4}{3} \, {\left (x^{4} + x\right )}^{\frac {3}{2}} + 2 \, \sqrt {x^{4} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 1.39 \begin {gather*} \frac {2 x \left (1+x \right ) \left (x^{2}-x +1\right ) \left (2 x^{4}+2 x +3\right )}{3 \sqrt {x^{4}+x}} \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 {{\left (2 \, x^{4} + 2 \, x + 1\right )} {\left (4 \, x^{3} + 1\right )}}{\sqrt {x^{4} + x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 19, normalized size = 0.83 \begin {gather*} \frac {2\,\sqrt {x^4+x}\,\left (2\,x^4+2\,x+3\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 37, normalized size = 1.61 \begin {gather*} \frac {4 x^{4} \sqrt {x^{4} + x}}{3} + \frac {4 x \sqrt {x^{4} + x}}{3} + 2 \sqrt {x^{4} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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