Optimal. Leaf size=288 \[ \frac {\sqrt [3]{b} \log \left (-\sqrt [3]{b} \sqrt [3]{x^3+x} x \sqrt [3]{a-b}+x^2 (a-b)^{2/3}+b^{2/3} \left (x^3+x\right )^{2/3}\right )}{4 a \sqrt [3]{a-b}}-\frac {\sqrt [3]{b} \log \left (x \sqrt [3]{a-b}+\sqrt [3]{b} \sqrt [3]{x^3+x}\right )}{2 a \sqrt [3]{a-b}}+\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{a-b}}{x \sqrt [3]{a-b}-2 \sqrt [3]{b} \sqrt [3]{x^3+x}}\right )}{2 a \sqrt [3]{a-b}}-\frac {\log \left (\sqrt [3]{x^3+x}-x\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )}{2 a}+\frac {\log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )}{4 a} \]
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Rubi [A] time = 0.58, antiderivative size = 445, normalized size of antiderivative = 1.55, number of steps used = 17, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2042, 466, 465, 494, 481, 200, 31, 634, 618, 204, 628, 617} \begin {gather*} \frac {\sqrt [3]{b} \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 a \sqrt [3]{x^3+x} \sqrt [3]{a-b}}-\frac {\sqrt [3]{b} \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 a \sqrt [3]{x^3+x} \sqrt [3]{a-b}}+\frac {\sqrt {3} \sqrt [3]{b} \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 a \sqrt [3]{x^3+x} \sqrt [3]{a-b}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )}{2 a \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )}{4 a \sqrt [3]{x^3+x}}+\frac {\sqrt {3} \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 a \sqrt [3]{x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 465
Rule 466
Rule 481
Rule 494
Rule 617
Rule 618
Rule 628
Rule 634
Rule 2042
Rubi steps
\begin {align*} \int \frac {x^2}{\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 {x^{5/3}}{\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^7}{\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 {x^3}{\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 {x^3}{\left (1-x^3\right ) \left (b-(-a+b) x^3\right )} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\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}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 a \sqrt [3]{x+x^3}}-\frac {\left (3 b \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 a \sqrt [3]{x+x^3}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 a \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2+x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 a \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{b} \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 a \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{b} \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 a \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 a \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{b} \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 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\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 )}{4 a \sqrt [3]{x+x^3}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 a \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{b} \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 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}-\frac {\left (3 b^{2/3} \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 a \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 a \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 a \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{b} \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 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{b} \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 a \sqrt [3]{a-b} \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 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 a \sqrt [3]{x+x^3}}-\frac {\left (3 \sqrt [3]{b} \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 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}\\ &=\frac {\sqrt {3} \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 a \sqrt [3]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{b} \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 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 a \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 a \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{b} \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 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{b} \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 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 53, normalized size = 0.18 \begin {gather*} \frac {3 x^3 \sqrt [3]{x^2+1} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-x^2,-\frac {a x^2}{b}\right )}{8 b \sqrt [3]{x^3+x}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.80, size = 291, normalized size = 1.01 \begin {gather*} \frac {\sqrt [3]{b} \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 a \sqrt [3]{a-b}}-\frac {\sqrt [3]{b} \log \left (x \sqrt [3]{a-b}+\sqrt [3]{b} \sqrt [3]{x^3+x}\right )}{2 a \sqrt [3]{a-b}}+\frac {\sqrt {3} \sqrt [3]{b} \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 a \sqrt [3]{a-b}}-\frac {\log \left (a \sqrt [3]{x^3+x}-a x\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )}{2 a}+\frac {\log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )}{4 a} \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 = 0.41, size = 256, normalized size = 0.89 \begin {gather*} -\frac {b \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^{2} - a b\right )}} - \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^{2} b - \sqrt {3} a b^{2}\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^{2} b - a b^{2}\right )}} - \frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right )}{2 \, a} + \frac {\log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}{4 \, a} - \frac {\log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a \,x^{2}+b \right ) \left (x^{3}+x \right )^{\frac {1}{3}}}\, dx \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^{2}}{{\left (a x^{2} + 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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{\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 {x^{2}}{\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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