Optimal. Leaf size=140 \[ \frac {3}{2} \sqrt [3]{x^2+2 x+6}+\frac {1}{2} \sqrt [3]{5} \log \left (5^{2/3} \sqrt [3]{x^2+2 x+6}-5\right )-\frac {1}{4} \sqrt [3]{5} \log \left (\sqrt [3]{5} \left (x^2+2 x+6\right )^{2/3}+5^{2/3} \sqrt [3]{x^2+2 x+6}+5\right )-\frac {1}{2} \sqrt {3} \sqrt [3]{5} \tan ^{-1}\left (\frac {2 \sqrt [3]{x^2+2 x+6}}{\sqrt {3} \sqrt [3]{5}}+\frac {1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 103, normalized size of antiderivative = 0.74, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {694, 266, 50, 57, 617, 204, 31} \begin {gather*} \frac {3}{2} \sqrt [3]{(x+1)^2+5}-\frac {1}{2} \sqrt [3]{5} \log (x+1)+\frac {3}{4} \sqrt [3]{5} \log \left (\sqrt [3]{5}-\sqrt [3]{(x+1)^2+5}\right )-\frac {1}{2} \sqrt {3} \sqrt [3]{5} \tan ^{-1}\left (\frac {2 \sqrt [3]{(x+1)^2+5}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 204
Rule 266
Rule 617
Rule 694
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{6+2 x+x^2}}{1+x} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt [3]{5+x^2}}{x} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{5+x}}{x} \, dx,x,(1+x)^2\right )\\ &=\frac {3}{2} \sqrt [3]{5+(1+x)^2}+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{x (5+x)^{2/3}} \, dx,x,(1+x)^2\right )\\ &=\frac {3}{2} \sqrt [3]{5+(1+x)^2}-\frac {1}{2} \sqrt [3]{5} \log (1+x)-\frac {1}{4} \left (3 \sqrt [3]{5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{5}-x} \, dx,x,\sqrt [3]{5+(1+x)^2}\right )-\frac {1}{4} \left (3\ 5^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{5^{2/3}+\sqrt [3]{5} x+x^2} \, dx,x,\sqrt [3]{5+(1+x)^2}\right )\\ &=\frac {3}{2} \sqrt [3]{5+(1+x)^2}-\frac {1}{2} \sqrt [3]{5} \log (1+x)+\frac {3}{4} \sqrt [3]{5} \log \left (\sqrt [3]{5}-\sqrt [3]{5+(1+x)^2}\right )+\frac {1}{2} \left (3 \sqrt [3]{5}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{5+(1+x)^2}}{\sqrt [3]{5}}\right )\\ &=\frac {3}{2} \sqrt [3]{5+(1+x)^2}-\frac {1}{2} \sqrt {3} \sqrt [3]{5} \tan ^{-1}\left (\frac {5+2\ 5^{2/3} \sqrt [3]{5+(1+x)^2}}{5 \sqrt {3}}\right )-\frac {1}{2} \sqrt [3]{5} \log (1+x)+\frac {3}{4} \sqrt [3]{5} \log \left (\sqrt [3]{5}-\sqrt [3]{5+(1+x)^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 131, normalized size = 0.94 \begin {gather*} \frac {1}{4} \left (6 \sqrt [3]{x^2+2 x+6}+2 \sqrt [3]{5} \log \left (\sqrt [3]{5}-\sqrt [3]{(x+1)^2+5}\right )-\sqrt [3]{5} \log \left (\left ((x+1)^2+5\right )^{2/3}+\sqrt [3]{5} \sqrt [3]{(x+1)^2+5}+5^{2/3}\right )-2 \sqrt {3} \sqrt [3]{5} \tan ^{-1}\left (\frac {2 \sqrt [3]{(x+1)^2+5}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 140, normalized size = 1.00 \begin {gather*} \frac {3}{2} \sqrt [3]{6+2 x+x^2}-\frac {1}{2} \sqrt {3} \sqrt [3]{5} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{6+2 x+x^2}}{\sqrt {3} \sqrt [3]{5}}\right )+\frac {1}{2} \sqrt [3]{5} \log \left (-5+5^{2/3} \sqrt [3]{6+2 x+x^2}\right )-\frac {1}{4} \sqrt [3]{5} \log \left (5+5^{2/3} \sqrt [3]{6+2 x+x^2}+\sqrt [3]{5} \left (6+2 x+x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 102, normalized size = 0.73 \begin {gather*} -\frac {1}{2} \cdot 5^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {2}{15} \cdot 5^{\frac {2}{3}} \sqrt {3} {\left (x^{2} + 2 \, x + 6\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{4} \cdot 5^{\frac {1}{3}} \log \left (5^{\frac {2}{3}} + 5^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 6\right )}^{\frac {1}{3}} + {\left (x^{2} + 2 \, x + 6\right )}^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 5^{\frac {1}{3}} \log \left (-5^{\frac {1}{3}} + {\left (x^{2} + 2 \, x + 6\right )}^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (x^{2} + 2 \, x + 6\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} + 2 \, x + 6\right )}^{\frac {1}{3}}}{x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 17.38, size = 1277, normalized size = 9.12
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1277\) |
risch | \(\text {Expression too large to display}\) | \(2698\) |
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^{2} + 2 \, x + 6\right )}^{\frac {1}{3}}}{x + 1}\,{d x} \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 {{\left (x^2+2\,x+6\right )}^{1/3}}{x+1} \,d x \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 [3]{x^{2} + 2 x + 6}}{x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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