Optimal. Leaf size=185 \[ \frac {\log \left (5 \sqrt [3]{x^2+1}+\sqrt [3]{10} x-2 \sqrt [3]{10}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (10^{2/3} x^2+25 \left (x^2+1\right )^{2/3}+\left (10 \sqrt [3]{10}-5 \sqrt [3]{10} x\right ) \sqrt [3]{x^2+1}-4\ 10^{2/3} x+4\ 10^{2/3}\right )}{10 \sqrt [3]{10}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}-\frac {2 \sqrt [3]{2} x}{\sqrt {3} 5^{2/3}}+\frac {4 \sqrt [3]{2}}{\sqrt {3} 5^{2/3}}}{\sqrt [3]{x^2+1}}\right )}{5 \sqrt [3]{10}} \]
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Rubi [C] time = 0.57, antiderivative size = 238, normalized size of antiderivative = 1.29, number of steps used = 18, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6742, 757, 429, 444, 55, 617, 204, 31} \begin {gather*} \frac {2}{15} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};\frac {x^2}{9},-x^2\right )+\frac {1}{5} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};4 x^2,-x^2\right )-\frac {\log \left (1-4 x^2\right )}{20 \sqrt [3]{10}}-\frac {\log \left (9-x^2\right )}{10 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-2 \sqrt [3]{x^2+1}\right )}{20 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-\sqrt [3]{x^2+1}\right )}{10 \sqrt [3]{10}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2+1}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right )}{5 \sqrt [3]{10}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2\ 2^{2/3} \sqrt [3]{x^2+1}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right )}{10 \sqrt [3]{10}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 31
Rule 55
Rule 204
Rule 429
Rule 444
Rule 617
Rule 757
Rule 6742
Rubi steps
\begin {align*} \int \frac {1+x}{(3+x) (1+2 x) \sqrt [3]{1+x^2}} \, dx &=\int \left (\frac {2}{5 (3+x) \sqrt [3]{1+x^2}}+\frac {1}{5 (1+2 x) \sqrt [3]{1+x^2}}\right ) \, dx\\ &=\frac {1}{5} \int \frac {1}{(1+2 x) \sqrt [3]{1+x^2}} \, dx+\frac {2}{5} \int \frac {1}{(3+x) \sqrt [3]{1+x^2}} \, dx\\ &=\frac {1}{5} \int \left (\frac {1}{\left (1-4 x^2\right ) \sqrt [3]{1+x^2}}+\frac {2 x}{\sqrt [3]{1+x^2} \left (-1+4 x^2\right )}\right ) \, dx+\frac {2}{5} \int \left (-\frac {3}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}}+\frac {x}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}}\right ) \, dx\\ &=\frac {1}{5} \int \frac {1}{\left (1-4 x^2\right ) \sqrt [3]{1+x^2}} \, dx+\frac {2}{5} \int \frac {x}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}} \, dx+\frac {2}{5} \int \frac {x}{\sqrt [3]{1+x^2} \left (-1+4 x^2\right )} \, dx-\frac {6}{5} \int \frac {1}{\left (-9+x^2\right ) \sqrt [3]{1+x^2}} \, dx\\ &=\frac {2}{15} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};\frac {x^2}{9},-x^2\right )+\frac {1}{5} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};4 x^2,-x^2\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{(-9+x) \sqrt [3]{1+x}} \, dx,x,x^2\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x} (-1+4 x)} \, dx,x,x^2\right )\\ &=\frac {2}{15} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};\frac {x^2}{9},-x^2\right )+\frac {1}{5} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};4 x^2,-x^2\right )-\frac {\log \left (1-4 x^2\right )}{20 \sqrt [3]{10}}-\frac {\log \left (9-x^2\right )}{10 \sqrt [3]{10}}+\frac {3}{40} \operatorname {Subst}\left (\int \frac {1}{\frac {5^{2/3}}{2 \sqrt [3]{2}}+\frac {\sqrt [3]{5} x}{2^{2/3}}+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )+\frac {3}{10} \operatorname {Subst}\left (\int \frac {1}{10^{2/3}+\sqrt [3]{10} x+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{5}}{2^{2/3}}-x} \, dx,x,\sqrt [3]{1+x^2}\right )}{20 \sqrt [3]{10}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{10}-x} \, dx,x,\sqrt [3]{1+x^2}\right )}{10 \sqrt [3]{10}}\\ &=\frac {2}{15} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};\frac {x^2}{9},-x^2\right )+\frac {1}{5} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};4 x^2,-x^2\right )-\frac {\log \left (1-4 x^2\right )}{20 \sqrt [3]{10}}-\frac {\log \left (9-x^2\right )}{10 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-2 \sqrt [3]{1+x^2}\right )}{20 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-\sqrt [3]{1+x^2}\right )}{10 \sqrt [3]{10}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2\ 2^{2/3} \sqrt [3]{1+x^2}}{\sqrt [3]{5}}\right )}{10 \sqrt [3]{10}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{1+x^2}}{\sqrt [3]{5}}\right )}{5 \sqrt [3]{10}}\\ &=\frac {2}{15} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};\frac {x^2}{9},-x^2\right )+\frac {1}{5} x F_1\left (\frac {1}{2};1,\frac {1}{3};\frac {3}{2};4 x^2,-x^2\right )+\frac {\sqrt {3} \tan ^{-1}\left (\frac {5+10^{2/3} \sqrt [3]{1+x^2}}{5 \sqrt {3}}\right )}{5 \sqrt [3]{10}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {5+2\ 10^{2/3} \sqrt [3]{1+x^2}}{5 \sqrt {3}}\right )}{10 \sqrt [3]{10}}-\frac {\log \left (1-4 x^2\right )}{20 \sqrt [3]{10}}-\frac {\log \left (9-x^2\right )}{10 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-2 \sqrt [3]{1+x^2}\right )}{20 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-\sqrt [3]{1+x^2}\right )}{10 \sqrt [3]{10}}\\ \end {align*}
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Mathematica [F] time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{(3+x) (1+2 x) \sqrt [3]{1+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.27, size = 185, normalized size = 1.00 \begin {gather*} \frac {\log \left (5 \sqrt [3]{x^2+1}+\sqrt [3]{10} x-2 \sqrt [3]{10}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (10^{2/3} x^2+25 \left (x^2+1\right )^{2/3}+\left (10 \sqrt [3]{10}-5 \sqrt [3]{10} x\right ) \sqrt [3]{x^2+1}-4\ 10^{2/3} x+4\ 10^{2/3}\right )}{10 \sqrt [3]{10}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}-\frac {2 \sqrt [3]{2} x}{\sqrt {3} 5^{2/3}}+\frac {4 \sqrt [3]{2}}{\sqrt {3} 5^{2/3}}}{\sqrt [3]{x^2+1}}\right )}{5 \sqrt [3]{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {x + 1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (2 \, x + 1\right )} {\left (x + 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 16.31, size = 1371, normalized size = 7.41 \begin {gather*} \text {Expression too large to display} \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 + 1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (2 \, x + 1\right )} {\left (x + 3\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 = -0.01 \begin {gather*} \int \frac {x+1}{\left (2\,x+1\right )\,{\left (x^2+1\right )}^{1/3}\,\left (x+3\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 + 1}{\left (x + 3\right ) \left (2 x + 1\right ) \sqrt [3]{x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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