Optimal. Leaf size=41 \[ -2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{a x^6+b x^3}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{a x^6+b x^3}}\right ) \]
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Rubi [F] time = 1.71, 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+2 a x^3}{\left (b-x+a x^3\right ) \sqrt [4]{b 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+2 a x^3}{\left (b-x+a x^3\right ) \sqrt [4]{b x^3+a x^6}} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{b+a x^3}\right ) \int \frac {-b+2 a x^3}{x^{3/4} \sqrt [4]{b+a x^3} \left (b-x+a x^3\right )} \, dx}{\sqrt [4]{b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {-b+2 a x^{12}}{\sqrt [4]{b+a x^{12}} \left (b-x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt [4]{b+a x^{12}}}-\frac {3 b-2 x^4}{\sqrt [4]{b+a x^{12}} \left (b-x^4+a x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {3 b-2 x^4}{\sqrt [4]{b+a x^{12}} \left (b-x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}+\frac {\left (8 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {3 b}{\sqrt [4]{b+a x^{12}} \left (b-x^4+a x^{12}\right )}-\frac {2 x^4}{\sqrt [4]{b+a x^{12}} \left (b-x^4+a x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}+\frac {\left (8 x^{3/4} \sqrt [4]{1+\frac {a x^3}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {a x^{12}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ &=\frac {8 x \sqrt [4]{1+\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{12},\frac {1}{4};\frac {13}{12};-\frac {a x^3}{b}\right )}{\sqrt [4]{b x^3+a x^6}}+\frac {\left (8 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^{12}} \left (b-x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}-\frac {\left (12 b x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^{12}} \left (b-x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ \end {align*}
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Mathematica [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+2 a x^3}{\left (b-x+a x^3\right ) \sqrt [4]{b x^3+a x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.73, size = 41, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{a x^6+b x^3}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{a x^6+b x^3}}\right ) \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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{3} - b}{{\left (a x^{6} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{3} + b - x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 a \,x^{3}-b}{\left (a \,x^{3}+b -x \right ) \left (a \,x^{6}+b \,x^{3}\right )^{\frac {1}{4}}}\, 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 {2 \, a x^{3} - b}{{\left (a x^{6} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{3} + b - x\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-2\,a\,x^3}{{\left (a\,x^6+b\,x^3\right )}^{1/4}\,\left (a\,x^3-x+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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