Optimal. Leaf size=118 \[ \frac {1}{6} (a-3 b) \log \left (\sqrt [3]{x^3+x}-x\right )+\frac {1}{6} \left (3 \sqrt {3} b-\sqrt {3} a\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )+\frac {1}{12} (3 b-a) \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )+\frac {1}{2} a \left (x^3+x\right )^{2/3} \]
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Rubi [A] time = 0.16, antiderivative size = 223, normalized size of antiderivative = 1.89, number of steps used = 11, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2053, 2011, 329, 275, 239, 2024} \begin {gather*} \frac {1}{2} a \left (x^3+x\right )^{2/3}+\frac {a \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{x^3+x}}-\frac {a \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{x^3+x}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{x^3+x}}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 275
Rule 329
Rule 2011
Rule 2024
Rule 2053
Rubi steps
\begin {align*} \int \frac {b+a x^2}{\sqrt [3]{x+x^3}} \, dx &=\int \left (\frac {b}{\sqrt [3]{x+x^3}}+\frac {a x^2}{\sqrt [3]{x+x^3}}\right ) \, dx\\ &=a \int \frac {x^2}{\sqrt [3]{x+x^3}} \, dx+b \int \frac {1}{\sqrt [3]{x+x^3}} \, dx\\ &=\frac {1}{2} a \left (x+x^3\right )^{2/3}-\frac {1}{3} a \int \frac {1}{\sqrt [3]{x+x^3}} \, dx+\frac {\left (b \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x+x^3}}\\ &=\frac {1}{2} a \left (x+x^3\right )^{2/3}-\frac {\left (a \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2}} \, dx}{3 \sqrt [3]{x+x^3}}+\frac {\left (3 b \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^3}}\\ &=\frac {1}{2} a \left (x+x^3\right )^{2/3}-\frac {\left (a \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^3}}+\frac {\left (3 b \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {1}{2} a \left (x+x^3\right )^{2/3}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{x+x^3}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}}-\frac {\left (a \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {1}{2} a \left (x+x^3\right )^{2/3}-\frac {a \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{x+x^3}}+\frac {\sqrt {3} b \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt [3]{x+x^3}}+\frac {a \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}}-\frac {3 b \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 136, normalized size = 1.15 \begin {gather*} \frac {\sqrt [3]{x} \left (a x^{2/3} \left (x^2+1\right )-\frac {1}{6} \sqrt [3]{x^2+1} (a-3 b) \left (-2 \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )+\log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )\right )\right )}{2 \sqrt [3]{x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 118, normalized size = 1.00 \begin {gather*} \frac {1}{6} (a-3 b) \log \left (\sqrt [3]{x^3+x}-x\right )+\frac {1}{6} \left (3 \sqrt {3} b-\sqrt {3} a\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )+\frac {1}{12} (3 b-a) \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )+\frac {1}{2} a \left (x^3+x\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 83.61, size = 100, normalized size = 0.85 \begin {gather*} -\frac {1}{6} \, \sqrt {3} {\left (a - 3 \, b\right )} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + \frac {1}{12} \, {\left (a - 3 \, b\right )} \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) + \frac {1}{2} \, {\left (x^{3} + x\right )}^{\frac {2}{3}} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 83, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, a x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + \frac {1}{6} \, \sqrt {3} {\left (a - 3 \, b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{12} \, {\left (a - 3 \, b\right )} \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, {\left (a - 3 \, b\right )} \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 54, normalized size = 0.46 \begin {gather*} \frac {a x \left (x^{2}+1\right )}{2 \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}}}+\frac {3 b \,x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2}-\frac {a \,x^{\frac {2}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2} \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 {a x^{2} + b}{{\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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x^2+b}{{\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 {a x^{2} + b}{\sqrt [3]{x \left (x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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