Optimal. Leaf size=129 \[ \frac {1}{6} (3 b-a) \log \left (\sqrt [3]{x^3+x}-x\right )+\frac {1}{6} \left (3 \sqrt {3} b-\sqrt {3} a\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )+\frac {\sqrt [3]{x^3+x} \left (a x^2+3 b\right )}{2 x}+\frac {1}{12} (a-3 b) \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.21, antiderivative size = 215, normalized size of antiderivative = 1.67, number of steps used = 12, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2038, 2004, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{2} x \sqrt [3]{x^3+x} (a-3 b)-\frac {x^{2/3} \left (x^2+1\right )^{2/3} (a-3 b) \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )}{6 \left (x^3+x\right )^{2/3}}+\frac {x^{2/3} \left (x^2+1\right )^{2/3} (a-3 b) \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )}{12 \left (x^3+x\right )^{2/3}}-\frac {x^{2/3} \left (x^2+1\right )^{2/3} (a-3 b) \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x^3+x\right )^{2/3}}+\frac {3 b \left (x^3+x\right )^{4/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 2004
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^2\right ) \sqrt [3]{x+x^3}}{x^2} \, dx &=\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-(-a+3 b) \int \sqrt [3]{x+x^3} \, dx\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {1}{3} (-a+3 b) \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{3 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (x+x^3\right )^{2/3}}+\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (x+x^3\right )^{2/3}}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \left (x+x^3\right )^{2/3}}+\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (x+x^3\right )^{2/3}}+\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \left (x+x^3\right )^{2/3}}-\frac {\left ((-a+3 b) x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=\frac {1}{2} (a-3 b) x \sqrt [3]{x+x^3}+\frac {3 b \left (x+x^3\right )^{4/3}}{2 x^2}-\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x+x^3\right )^{2/3}}-\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \left (x+x^3\right )^{2/3}}+\frac {(a-3 b) x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \left (x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 62, normalized size = 0.48 \begin {gather*} \frac {3 \sqrt [3]{x^3+x} \left (x^2 (a-3 b) \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^2\right )+2 b \left (x^2+1\right )^{4/3}\right )}{4 x \sqrt [3]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 129, normalized size = 1.00 \begin {gather*} \frac {1}{6} (3 b-a) \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 {\sqrt [3]{x^3+x} \left (a x^2+3 b\right )}{2 x}+\frac {1}{12} (a-3 b) \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 69.45, size = 114, normalized size = 0.88 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (a - 3 \, b\right )} x \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 ) + {\left (a - 3 \, b\right )} x \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) - 6 \, {\left (a x^{2} + 3 \, b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{12 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 93, normalized size = 0.72 \begin {gather*} \frac {1}{2} \, a x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{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 ) + \frac {3}{2} \, b {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.94, size = 512, normalized size = 3.97 \begin {gather*} \frac {\left (a \,x^{2}+3 b \right ) \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{2 x}+\frac {\left (a -3 b \right ) \left (\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+20 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-100 x^{4}+30 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+14 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+36 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-48 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-140 x^{2}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-40}{x^{2}+1}\right )-2 \ln \left (-\frac {-5 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-38 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+16 x^{4}+30 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-70 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-96 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+18 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+5 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+28 x^{2}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-32 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+12}{x^{2}+1}\right )\right ) \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )^{2}\right )^{\frac {1}{3}}}{12 x \left (x^{2}+1\right )} \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 {{\left (a x^{2} - b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{x^{2}}\,{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 {\left (b-a\,x^2\right )\,{\left (x^3+x\right )}^{1/3}}{x^2} \,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 {\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{2} - b\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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