Optimal. Leaf size=65 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6+4 a b\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+a x^6-b}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [F] time = 0.79, 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 {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx &=\int \left (\frac {1}{\sqrt [3]{-b+a^3 x^3+a x^6}}-\frac {2 b}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}\right ) \, dx\\ &=-\left ((2 b) \int \frac {1}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\right )+\int \frac {1}{\sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\\ &=-\left ((2 b) \int \left (\frac {1}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}+\frac {1}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}\right ) \, dx\right )+\frac {\left (\sqrt [3]{1+\frac {2 a x^3}{a^3-\sqrt {a} \sqrt {a^5+4 b}}} \sqrt [3]{1+\frac {2 a x^3}{a^3+\sqrt {a} \sqrt {a^5+4 b}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 a x^3}{a^3-\sqrt {a^6+4 a b}}} \sqrt [3]{1+\frac {2 a x^3}{a^3+\sqrt {a^6+4 a b}}}} \, dx}{\sqrt [3]{-b+a^3 x^3+a x^6}}\\ &=\frac {x \sqrt [3]{1+\frac {2 \sqrt {a} x^3}{a^{5/2}-\sqrt {a^5+4 b}}} \sqrt [3]{1+\frac {2 \sqrt {a} x^3}{a^{5/2}+\sqrt {a^5+4 b}}} F_1\left (\frac {1}{3};\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {2 a x^3}{a^3-\sqrt {a^6+4 a b}},-\frac {2 a x^3}{a^3+\sqrt {a^6+4 a b}}\right )}{\sqrt [3]{-b+a^3 x^3+a x^6}}-\sqrt {b} \int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx-\sqrt {b} \int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.75, size = 65, normalized size = 1.00 \begin {gather*} \frac {1}{6} \text {RootSum}\left [a^6+4 a b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b+a^3 x^3+a x^6}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \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 {a x^{6} - b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}-b}{\left (a \,x^{6}+b \right ) \left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}}}\, dx\]
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 {a x^{6} - b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} + b\right )}}\,{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 = -0.02 \begin {gather*} \int -\frac {b-a\,x^6}{\left (a\,x^6+b\right )\,{\left (a^3\,x^3+a\,x^6-b\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^{6} - b}{\left (a x^{6} + b\right ) \sqrt [3]{a^{3} x^{3} + a x^{6} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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