Optimal. Leaf size=77 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^9 (-d)+3 \text {$\#$1}^6 a d-3 \text {$\#$1}^3 a^2 d+a^3 d+b^3 c\& ,\frac {\log \left (\sqrt [3]{a x^3-b x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 d} \]
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Rubi [F] time = 1.52, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-b+a x^2} \left (d+c x^6\right )} \, dx}{\sqrt [3]{-b x+a x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-b+a x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-b x+a x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-b x+a x^3}}\\ &=-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}\\ \end {align*}
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Mathematica [B] time = 3.96, size = 571, normalized size = 7.42 \begin {gather*} -\frac {x \sqrt [3]{\frac {b}{x^2}-a} \left (a^2 \text {RootSum}\left [\text {$\#$1}^3 d+3 \text {$\#$1}^2 a d+3 \text {$\#$1} a^2 d+a^3 d+b^3 c\&,\frac {-\frac {\log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {b}{x^2}-a}+\left (\frac {b}{x^2}-a\right )^{2/3}\right )}{\sqrt [3]{\text {$\#$1}}}+\frac {2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {b}{x^2}-a}\right )}{\sqrt [3]{\text {$\#$1}}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b}{x^2}-a}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\sqrt [3]{\text {$\#$1}}}}{\text {$\#$1}^2+2 \text {$\#$1} a+a^2}\&\right ]+2 a \text {RootSum}\left [\text {$\#$1}^3 d+3 \text {$\#$1}^2 a d+3 \text {$\#$1} a^2 d+a^3 d+b^3 c\&,\frac {2 \text {$\#$1}^{2/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{2/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {b}{x^2}-a}+\left (\frac {b}{x^2}-a\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b}{x^2}-a}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\text {$\#$1}^2+2 \text {$\#$1} a+a^2}\&\right ]+\text {RootSum}\left [\text {$\#$1}^3 d+3 \text {$\#$1}^2 a d+3 \text {$\#$1} a^2 d+a^3 d+b^3 c\&,\frac {2 \text {$\#$1}^{5/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{5/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {b}{x^2}-a}+\left (\frac {b}{x^2}-a\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{5/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b}{x^2}-a}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\text {$\#$1}^2+2 \text {$\#$1} a+a^2}\&\right ]\right )}{12 d \sqrt [3]{a x^3-b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.58, size = 77, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d} \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(-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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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{3}-b x \right )^{\frac {1}{3}} \left (c \,x^{6}+d \right )}\, dx\]
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 {1}{{\left (c x^{6} + d\right )} {\left (a x^{3} - b 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 {1}{{\left (a\,x^3-b\,x\right )}^{1/3}\,\left (c\,x^6+d\right )} \,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 (a x^{2} - b\right )} \left (c x^{6} + d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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