Optimal. Leaf size=285 \[ 2 \text {RootSum}\left [\text {$\#$1}^3-\text {$\#$1}-1\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{3 \text {$\#$1}^2-1}\& \right ]+2 \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^4-2 \text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}+1\& ,\frac {2 \text {$\#$1}^4 \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-2 \text {$\#$1}^4 \log (x)+\text {$\#$1}^2 \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)-2 \text {$\#$1} \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )+2 \text {$\#$1} \log (x)}{6 \text {$\#$1}^5+4 \text {$\#$1}^3-6 \text {$\#$1}^2+2 \text {$\#$1}-1}\& \right ]-\log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right ) \]
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Rubi [F] time = 0.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1+x^2+x^3}{x^{2/3} \sqrt [3]{1+x} \left (-1+x^2+x^3\right )} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1+x^6+x^9}{\sqrt [3]{1+x^3} \left (-1+x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^3}}+\frac {2}{\sqrt [3]{1+x^3} \left (-1+x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1+x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{x^2+x^3}}-\frac {3 x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{x^2+x^3}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (-1+x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}\\ \end {align*}
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Mathematica [F] time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.36, size = 285, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+2 \text {RootSum}\left [-1-\text {$\#$1}+\text {$\#$1}^3\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2}\&\right ]+2 \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^6\&,\frac {2 \log (x) \text {$\#$1}-2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 \log (x) \text {$\#$1}^4+2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-1+2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+6 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 11.80, size = 868, normalized size = 3.05
result too large to display
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 {x^{3} + x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 254.25, size = 290058, normalized size = 1017.75
method | result | size |
trager | \(\text {Expression too large to display}\) | \(290058\) |
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} + x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} - 1\right )}}\,{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 = -0.00 \begin {gather*} \int \frac {x^3+x^2+1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^3+x^2-1\right )} \,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 {x^{3} + x^{2} + 1}{\sqrt [3]{x^{2} \left (x + 1\right )} \left (x^{3} + x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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