Optimal. Leaf size=191 \[ \frac {\log \left (x \sqrt [3]{a-b}+\sqrt [3]{b} \sqrt [3]{x^3+x}\right )}{2 b^{2/3} \sqrt [3]{a-b}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{a-b}}{x \sqrt [3]{a-b}-2 \sqrt [3]{b} \sqrt [3]{x^3+x}}\right )}{2 b^{2/3} \sqrt [3]{a-b}}-\frac {\log \left (-\sqrt [3]{b} \sqrt [3]{x^3+x} x \sqrt [3]{a-b}+x^2 (a-b)^{2/3}+b^{2/3} \left (x^3+x\right )^{2/3}\right )}{4 b^{2/3} \sqrt [3]{a-b}} \]
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Rubi [A] time = 0.33, antiderivative size = 264, normalized size of antiderivative = 1.38, number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {2056, 466, 465, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {x^{2/3} \sqrt [3]{a-b}}{\sqrt [3]{x^2+1}}+\sqrt [3]{b}\right )}{2 b^{2/3} \sqrt [3]{x^3+x} \sqrt [3]{a-b}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {x^{4/3} (a-b)^{2/3}}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{b} x^{2/3} \sqrt [3]{a-b}}{\sqrt [3]{x^2+1}}+b^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x^3+x} \sqrt [3]{a-b}}-\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [3]{b}-\frac {2 x^{2/3} \sqrt [3]{a-b}}{\sqrt [3]{x^2+1}}}{\sqrt {3} \sqrt [3]{b}}\right )}{2 b^{2/3} \sqrt [3]{x^3+x} \sqrt [3]{a-b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 377
Rule 465
Rule 466
Rule 617
Rule 628
Rule 634
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, 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^2\right )} \, dx}{\sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^6} \left (b+a x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\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^3\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 \frac {1}{b-(-a+b) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a-b} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 b^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a-b} x}{b^{2/3}-\sqrt [3]{a-b} \sqrt [3]{b} x+(a-b)^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 b^{2/3} \sqrt [3]{x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{a-b} \sqrt [3]{b}+2 (a-b)^{2/3} x}{b^{2/3}-\sqrt [3]{a-b} \sqrt [3]{b} x+(a-b)^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{a-b} \sqrt [3]{b} x+(a-b)^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{b} \sqrt [3]{x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (b^{2/3}+\frac {(a-b)^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{a-b} \sqrt [3]{b} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{b} \sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{b} \sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (b^{2/3}+\frac {(a-b)^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{a-b} \sqrt [3]{b} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \sqrt [3]{a-b} b^{2/3} \sqrt [3]{x+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 68, normalized size = 0.36 \begin {gather*} \frac {3 x \sqrt [3]{x^2+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {(a-b) x^2}{a x^2+b}\right )}{2 b \sqrt [3]{x^3+x} \sqrt [3]{\frac {a x^2}{b}+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.49, size = 191, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a-b} x}{\sqrt [3]{a-b} x-2 \sqrt [3]{b} \sqrt [3]{x+x^3}}\right )}{2 \sqrt [3]{a-b} b^{2/3}}+\frac {\log \left (\sqrt [3]{a-b} x+\sqrt [3]{b} \sqrt [3]{x+x^3}\right )}{2 \sqrt [3]{a-b} b^{2/3}}-\frac {\log \left ((a-b)^{2/3} x^2-\sqrt [3]{a-b} \sqrt [3]{b} x \sqrt [3]{x+x^3}+b^{2/3} \left (x+x^3\right )^{2/3}\right )}{4 \sqrt [3]{a-b} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 187, normalized size = 0.98 \begin {gather*} \frac {3 \, {\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a - b}{b}\right )^{\frac {1}{3}}}\right )}{2 \, {\left (\sqrt {3} a b^{2} - \sqrt {3} b^{3}\right )}} - \frac {{\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \log \left (\left (-\frac {a - b}{b}\right )^{\frac {2}{3}} + \left (-\frac {a - b}{b}\right )^{\frac {1}{3}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )}{4 \, {\left (a b^{2} - b^{3}\right )}} + \frac {\left (-\frac {a - b}{b}\right )^{\frac {2}{3}} \log \left ({\left | -\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right )}{2 \, {\left (a - b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{2}+b \right ) \left (x^{3}+x \right )^{\frac {1}{3}}}\, dx\]
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 (x^{3} + x\right )}}{4 \, {\left (a x^{\frac {7}{3}} + b x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} + \int \frac {3 \, {\left (b x^{2} + b\right )}}{2 \, {\left (a^{2} x^{\frac {13}{3}} + 2 \, a b x^{\frac {7}{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 = -0.01 \begin {gather*} \int \frac {1}{\left (a\,x^2+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^{2} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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