Optimal. Leaf size=61 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^9 b-3 \text {$\#$1}^6 b+3 \text {$\#$1}^3 b+a-b\& ,\frac {\log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 b} \]
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Rubi [F] time = 0.74, 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}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (b+a x^6\right )} \, dx}{\sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (b+a x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-\sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+\sqrt [9]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{2/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+\sqrt [3]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{4/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+(-1)^{5/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{2/3} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+(-1)^{7/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{8/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-\sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+\sqrt [9]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{2/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+\sqrt [3]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{4/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+(-1)^{5/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{2/3} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+(-1)^{7/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{8/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 74, normalized size = 1.21 \begin {gather*} -\frac {\sqrt [3]{\frac {1}{x^2}+1} x \text {RootSum}\left [\text {$\#$1}^9 b-3 \text {$\#$1}^6 b+3 \text {$\#$1}^3 b+a-b\&,\frac {\log \left (\sqrt [3]{\frac {1}{x^2}+1}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 61, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b} \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 {1}{{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{3}+x \right )^{\frac {1}{3}} \left (a \,x^{6}+b \right )}\, dx\]
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*} -\frac {3 \, {\left (9 \, x^{7} + 3 \, x^{5} - x^{3} + 5 \, x\right )}}{80 \, {\left (a x^{\frac {19}{3}} + b x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} + \int \frac {9 \, {\left (9 \, b x^{6} + 3 \, b x^{4} - b x^{2} + 5 \, b\right )}}{40 \, {\left (a^{2} x^{\frac {37}{3}} + 2 \, a b x^{\frac {19}{3}} + b^{2} x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{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 = -0.02 \begin {gather*} \int \frac {1}{\left (a\,x^6+b\right )\,{\left (x^3+x\right )}^{1/3}} \,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 {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{6} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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