Optimal. Leaf size=482 \[ \frac {3 \left (1+\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4+b}}{(-1)^{3/4} x^2 \sqrt [4]{a^2+b}+\sqrt {a x^4+b}}\right )}{8 \sqrt [8]{a^2+b}}-\frac {3 i \left (\sqrt {2 \left (3-2 \sqrt {2}\right )}-i \sqrt {2}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \left (\sqrt {2}-2\right ) x \sqrt [8]{a^2+b} \sqrt [4]{a x^4+b}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} x^2 \sqrt [4]{a^2+b}+\sqrt {2-\sqrt {2}} \sqrt {a x^4+b}}\right )}{16 \sqrt [8]{a^2+b}}+\frac {3 \left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2+b}-\sqrt [8]{-1} \sqrt {a x^4+b}}{\sqrt {2-\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4+b}}\right )}{16 \sqrt [8]{a^2+b}}+\frac {3 \left (1+\sqrt [4]{-1}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2+b}-\sqrt [8]{-1} \sqrt {a x^4+b}}{\sqrt {2+\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4+b}}\right )}{8 \sqrt [8]{a^2+b}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \]
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Rubi [A] time = 1.05, antiderivative size = 451, normalized size of antiderivative = 0.94, number of steps used = 25, number of rules used = 10, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6728, 240, 212, 206, 203, 1428, 408, 377, 208, 205} \begin {gather*} \frac {3 \left (-a \sqrt {a^2+b}+a^2+b\right )^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (a-\sqrt {a^2+b}\right )^{3/4}}-\frac {3 \left (a \sqrt {a^2+b}+a^2+b\right )^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (\sqrt {a^2+b}+a\right )^{3/4}}+\frac {3 \left (-a \sqrt {a^2+b}+a^2+b\right )^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (a-\sqrt {a^2+b}\right )^{3/4}}-\frac {3 \left (a \sqrt {a^2+b}+a^2+b\right )^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \sqrt {a^2+b} \left (\sqrt {a^2+b}+a\right )^{3/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 377
Rule 408
Rule 1428
Rule 6728
Rubi steps
\begin {align*} \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{b+a x^4}}+\frac {3 \left (b+a x^4\right )^{3/4}}{-b-2 a x^4+x^8}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+3 \int \frac {\left (b+a x^4\right )^{3/4}}{-b-2 a x^4+x^8} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {3 \int \frac {\left (b+a x^4\right )^{3/4}}{-2 a-2 \sqrt {a^2+b}+2 x^4} \, dx}{\sqrt {a^2+b}}-\frac {3 \int \frac {\left (b+a x^4\right )^{3/4}}{-2 a+2 \sqrt {a^2+b}+2 x^4} \, dx}{\sqrt {a^2+b}}\\ &=-\frac {\left (3 \left (a^2+b-a \sqrt {a^2+b}\right )\right ) \int \frac {1}{\left (-2 a+2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+b}}+\frac {\left (3 \left (b+a \left (a+\sqrt {a^2+b}\right )\right )\right ) \int \frac {1}{\left (-2 a-2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2+b}}+\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\left (3 \left (a^2+b-a \sqrt {a^2+b}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 a+2 \sqrt {a^2+b}-\left (-2 b+a \left (-2 a+2 \sqrt {a^2+b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+b}}+\frac {\left (3 \left (b+a \left (a+\sqrt {a^2+b}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 a-2 \sqrt {a^2+b}-\left (-2 b+a \left (-2 a-2 \sqrt {a^2+b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2+b}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (3 \left (a^2+b-a \sqrt {a^2+b}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+b}}-\sqrt {a^2+b-a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt {a-\sqrt {a^2+b}}}+\frac {\left (3 \left (a^2+b-a \sqrt {a^2+b}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+b}}+\sqrt {a^2+b-a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt {a-\sqrt {a^2+b}}}-\frac {\left (3 \left (b+a \left (a+\sqrt {a^2+b}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+b}}-\sqrt {a^2+b+a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt {a+\sqrt {a^2+b}}}-\frac {\left (3 \left (b+a \left (a+\sqrt {a^2+b}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+b}}+\sqrt {a^2+b+a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt {a+\sqrt {a^2+b}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (a^2+b-a \sqrt {a^2+b}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+b}} x}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \left (a-\sqrt {a^2+b}\right )^{3/4}}-\frac {3 \left (a^2+b+a \sqrt {a^2+b}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+b}} x}{\sqrt [4]{a+\sqrt {a^2+b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \left (a+\sqrt {a^2+b}\right )^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (a^2+b-a \sqrt {a^2+b}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+b}} x}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \left (a-\sqrt {a^2+b}\right )^{3/4}}-\frac {3 \left (a^2+b+a \sqrt {a^2+b}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+b}} x}{\sqrt [4]{a+\sqrt {a^2+b}} \sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a^2+b} \left (a+\sqrt {a^2+b}\right )^{3/4}}\\ \end {align*}
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Mathematica [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b-a x^4+2 x^8}{\sqrt [4]{b+a x^4} \left (-b-2 a x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 133.15, size = 536, normalized size = 1.11 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 i \left (i \sqrt {2}+\sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\left ((-1+i)-(1+i) (-1)^{3/4}\right ) \sqrt [4]{a^2+b} x^2+(1+i) \sqrt {b+a x^4}+(1+i) (-1)^{3/4} \sqrt {b+a x^4}}{2 \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}\right )}{16 \sqrt [8]{a^2+b}}+\frac {3 \left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {2 \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}{\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{a^2+b} x^2-(1+i) \sqrt {b+a x^4}-\sqrt {2} \sqrt {b+a x^4}}\right )}{16 \sqrt [8]{a^2+b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (i+(-1)^{3/4}\right ) \tanh ^{-1}\left (\frac {\left ((-2+2 i)-(2+2 i) (-1)^{3/4}\right ) \sqrt [4]{a^2+b} x^2-(2+2 i) \sqrt {b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt {b+a x^4}}{4 \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}\right )}{8 \sqrt [8]{a^2+b}}-\frac {3 i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{a^2+b} x^2+(1+i) \sqrt {b+a x^4}+\sqrt {2} \sqrt {b+a x^4}}{2 \sqrt [8]{a^2+b} x \sqrt [4]{b+a x^4}}\right )}{16 \sqrt [8]{a^2+b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 567, normalized size = 1.18 \begin {gather*} -\frac {3 \, \sqrt {2} \arctan \left (\frac {\frac {\sqrt {2} x \sqrt {\frac {{\left (a^{2} + b\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a^{2} + b\right )}^{\frac {1}{8}} x + \sqrt {a x^{4} + b}}{x^{2}}}}{{\left (a^{2} + b\right )}^{\frac {1}{8}}} - x - \frac {\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{{\left (a^{2} + b\right )}^{\frac {1}{8}}}}{x}\right )}{4 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {3 \, \sqrt {2} \arctan \left (\frac {\frac {\sqrt {2} x \sqrt {\frac {{\left (a^{2} + b\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a^{2} + b\right )}^{\frac {1}{8}} x + \sqrt {a x^{4} + b}}{x^{2}}}}{{\left (a^{2} + b\right )}^{\frac {1}{8}}} + x - \frac {\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{{\left (a^{2} + b\right )}^{\frac {1}{8}}}}{x}\right )}{4 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {3 \, \sqrt {2} \log \left (\frac {{\left (a^{2} + b\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a^{2} + b\right )}^{\frac {1}{8}} x + \sqrt {a x^{4} + b}}{x^{2}}\right )}{16 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {3 \, \sqrt {2} \log \left (\frac {{\left (a^{2} + b\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a^{2} + b\right )}^{\frac {1}{8}} x + \sqrt {a x^{4} + b}}{x^{2}}\right )}{16 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {3 \, \arctan \left (\frac {\frac {x \sqrt {\frac {{\left (a^{2} + b\right )}^{\frac {1}{4}} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}}}{{\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{{\left (a^{2} + b\right )}^{\frac {1}{8}}}}{x}\right )}{2 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} - \frac {3 \, \log \left (\frac {{\left (a^{2} + b\right )}^{\frac {1}{8}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{8 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {3 \, \log \left (-\frac {{\left (a^{2} + b\right )}^{\frac {1}{8}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{8 \, {\left (a^{2} + b\right )}^{\frac {1}{8}}} + \frac {2 \, \arctan \left (\frac {\frac {x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}}{x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \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 \, x^{8} - a x^{4} + b}{{\left (x^{8} - 2 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-a \,x^{4}+b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (x^{8}-2 a \,x^{4}-b \right )}\, dx\]
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 {2 \, x^{8} - a x^{4} + b}{{\left (x^{8} - 2 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}\,{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 = -0.00 \begin {gather*} \int -\frac {2\,x^8-a\,x^4+b}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+2\,a\,x^4+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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