Optimal. Leaf size=61 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{a x^6+b x^2}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{a x^6+b x^2}}\right )}{\sqrt [4]{2}} \]
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Rubi [C] time = 2.11, antiderivative size = 444, normalized size of antiderivative = 7.28, number of steps used = 20, number of rules used = 10, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2056, 6715, 6728, 246, 245, 1438, 430, 429, 511, 510} \begin {gather*} -\frac {2 x \sqrt [4]{\frac {a x^4}{b}+1} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {a^2 x^4}{-a b-2 \sqrt {1-a b}+2},-\frac {a x^4}{b}\right )}{\sqrt [4]{a x^6+b x^2}}-\frac {2 x \sqrt [4]{\frac {a x^4}{b}+1} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {a^2 x^4}{-a b+2 \sqrt {1-a b}+2},-\frac {a x^4}{b}\right )}{\sqrt [4]{a x^6+b x^2}}-\frac {2 a x^3 \left (1-\sqrt {1-a b}\right ) \sqrt [4]{\frac {a x^4}{b}+1} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {a^2 x^4}{-a b-2 \sqrt {1-a b}+2},-\frac {a x^4}{b}\right )}{5 \left (-a b-2 \sqrt {1-a b}+2\right ) \sqrt [4]{a x^6+b x^2}}-\frac {2 a x^3 \left (\sqrt {1-a b}+1\right ) \sqrt [4]{\frac {a x^4}{b}+1} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {a^2 x^4}{-a b+2 \sqrt {1-a b}+2},-\frac {a x^4}{b}\right )}{5 \left (-a b+2 \sqrt {1-a b}+2\right ) \sqrt [4]{a x^6+b x^2}}+\frac {2 x \sqrt [4]{\frac {a x^4}{b}+1} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^4}{b}\right )}{\sqrt [4]{a x^6+b x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 245
Rule 246
Rule 429
Rule 430
Rule 510
Rule 511
Rule 1438
Rule 2056
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b+a x^4}{\left (b-2 x^2+a x^4\right ) \sqrt [4]{b x^2+a x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^4}\right ) \int \frac {-b+a x^4}{\sqrt {x} \sqrt [4]{b+a x^4} \left (b-2 x^2+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {-b+a x^8}{\sqrt [4]{b+a x^8} \left (b-2 x^4+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{b+a x^8}}-\frac {2 \left (b-x^4\right )}{\sqrt [4]{b+a x^8} \left (b-2 x^4+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (4 \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {b-x^4}{\sqrt [4]{b+a x^8} \left (b-2 x^4+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=-\frac {\left (4 \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-\sqrt {1-a b}}{\left (-2-2 \sqrt {1-a b}+2 a x^4\right ) \sqrt [4]{b+a x^8}}+\frac {-1+\sqrt {1-a b}}{\left (-2+2 \sqrt {1-a b}+2 a x^4\right ) \sqrt [4]{b+a x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}+\frac {\left (2 \sqrt {x} \sqrt [4]{1+\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1+\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^4}{b}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (4 \left (-1-\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2-2 \sqrt {1-a b}+2 a x^4\right ) \sqrt [4]{b+a x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (4 \left (-1+\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 \sqrt {1-a b}+2 a x^4\right ) \sqrt [4]{b+a x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1+\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^4}{b}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (4 \left (-1-\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-\sqrt {1-a b}}{2 \sqrt [4]{b+a x^8} \left (2-a b+2 \sqrt {1-a b}-a^2 x^8\right )}-\frac {a x^4}{2 \sqrt [4]{b+a x^8} \left (2-a b+2 \sqrt {1-a b}-a^2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (4 \left (-1+\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1-\sqrt {1-a b}}{2 \sqrt [4]{b+a x^8} \left (-2+a b+2 \sqrt {1-a b}+a^2 x^8\right )}+\frac {a x^4}{2 \sqrt [4]{b+a x^8} \left (-2+a b+2 \sqrt {1-a b}+a^2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1+\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^4}{b}\right )}{\sqrt [4]{b x^2+a x^6}}+\frac {\left (2 a \left (-1-\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^8} \left (2-a b+2 \sqrt {1-a b}-a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (2 \left (-1-\sqrt {1-a b}\right )^2 \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^8} \left (2-a b+2 \sqrt {1-a b}-a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (2 a \left (-1+\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^8} \left (-2+a b+2 \sqrt {1-a b}+a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (2 \left (1-\sqrt {1-a b}\right ) \left (-1+\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^8} \left (-2+a b+2 \sqrt {1-a b}+a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1+\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^4}{b}\right )}{\sqrt [4]{b x^2+a x^6}}+\frac {\left (2 a \left (-1-\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{1+\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (2-a b+2 \sqrt {1-a b}-a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (2 \left (-1-\sqrt {1-a b}\right )^2 \sqrt {x} \sqrt [4]{1+\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2-a b+2 \sqrt {1-a b}-a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (2 a \left (-1+\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{1+\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-2+a b+2 \sqrt {1-a b}+a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {\left (2 \left (1-\sqrt {1-a b}\right ) \left (-1+\sqrt {1-a b}\right ) \sqrt {x} \sqrt [4]{1+\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+a b+2 \sqrt {1-a b}+a^2 x^8\right ) \sqrt [4]{1+\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^6}}\\ &=-\frac {2 x \sqrt [4]{1+\frac {a x^4}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {a^2 x^4}{2-a b-2 \sqrt {1-a b}},-\frac {a x^4}{b}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {2 x \sqrt [4]{1+\frac {a x^4}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {a^2 x^4}{2-a b+2 \sqrt {1-a b}},-\frac {a x^4}{b}\right )}{\sqrt [4]{b x^2+a x^6}}-\frac {2 a \left (1-\sqrt {1-a b}\right ) x^3 \sqrt [4]{1+\frac {a x^4}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {a^2 x^4}{2-a b-2 \sqrt {1-a b}},-\frac {a x^4}{b}\right )}{5 \left (2-a b-2 \sqrt {1-a b}\right ) \sqrt [4]{b x^2+a x^6}}-\frac {2 a \left (1+\sqrt {1-a b}\right ) x^3 \sqrt [4]{1+\frac {a x^4}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {a^2 x^4}{2-a b+2 \sqrt {1-a b}},-\frac {a x^4}{b}\right )}{5 \left (2-a b+2 \sqrt {1-a b}\right ) \sqrt [4]{b x^2+a x^6}}+\frac {2 x \sqrt [4]{1+\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-\frac {a x^4}{b}\right )}{\sqrt [4]{b x^2+a x^6}}\\ \end {align*}
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Mathematica [F] time = 1.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+a x^4}{\left (b-2 x^2+a x^4\right ) \sqrt [4]{b x^2+a x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.39, size = 61, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{b x^2+a x^6}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{b x^2+a x^6}}\right )}{\sqrt [4]{2}} \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 {a x^{4} - b}{{\left (a x^{6} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, x^{2} + 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 \[\int \frac {a \,x^{4}-b}{\left (a \,x^{4}-2 x^{2}+b \right ) \left (a \,x^{6}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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^{4} - b}{{\left (a x^{6} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, x^{2} + 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^4}{{\left (a\,x^6+b\,x^2\right )}^{1/4}\,\left (a\,x^4-2\,x^2+b\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 {a x^{4} - b}{\sqrt [4]{x^{2} \left (a x^{4} + b\right )} \left (a x^{4} + b - 2 x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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