Optimal. Leaf size=71 \[ \frac {\text {RootSum}\left [\text {$\#$1}^9 (-b)+3 \text {$\#$1}^6 b-3 \text {$\#$1}^3 b+a+b\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x^3-x}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^3-1}\& \right ]}{6 b} \]
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Rubi [B] time = 3.36, antiderivative size = 192, normalized size of antiderivative = 2.70, number of steps used = 55, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2056, 6725, 959, 466, 465, 511, 510} \begin {gather*} \frac {x \sqrt [3]{x^3-x} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {\sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{x^3-x} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,-\frac {\sqrt [3]{-1} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{x^3-x} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {(-1)^{2/3} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 465
Rule 466
Rule 510
Rule 511
Rule 959
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{b+a x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^3\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^3\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt {b}-\sqrt {-a} x^3} \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt {b}+\sqrt {-a} x^3} \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}-\sqrt [6]{-a} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x\right )}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}-\sqrt [6]{-a} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x\right )}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}-\sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}-\sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}-\sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}-\sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}-\sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}-\sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}\\ &=\frac {x \sqrt [3]{-x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {\sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{-x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,-\frac {\sqrt [3]{-1} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{-x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {(-1)^{2/3} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}\\ \end {align*}
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Mathematica [F] time = 1.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 71, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [a+b-3 b \text {$\#$1}^3+3 b \text {$\#$1}^6-b \text {$\#$1}^9\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\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 {{\left (x^{3} - x\right )}^{\frac {1}{3}}}{a x^{6} + b}\,{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 \[\int \frac {\left (x^{3}-x \right )^{\frac {1}{3}}}{a \,x^{6}+b}\, dx\]
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*} \int \frac {{\left (x^{3} - x\right )}^{\frac {1}{3}}}{a x^{6} + b}\,{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 (x^3-x\right )}^{1/3}}{a\,x^6+b} \,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 - 1\right ) \left (x + 1\right )}}{a x^{6} + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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