Optimal. Leaf size=448 \[ \frac {\left (\sqrt [4]{-1}-1\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}{(-1)^{3/4} x^2 \sqrt [4]{a^2+b}+\sqrt {a x^4-b}}\right )}{8 \left (a^2+b\right )^{5/8}}+\frac {i \left (\sqrt {2 \left (3+2 \sqrt {2}\right )}+i \sqrt {2}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \left (\sqrt {2}-2\right ) x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} x^2 \sqrt [4]{a^2+b}+\sqrt {2-\sqrt {2}} \sqrt {a x^4-b}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (\sqrt {2}-i \sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2+b}-\sqrt [8]{-1} \sqrt {a x^4-b}}{\sqrt {2-\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (\sqrt [4]{-1}-1\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2+b}-\sqrt [8]{-1} \sqrt {a x^4-b}}{\sqrt {2+\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}\right )}{8 \left (a^2+b\right )^{5/8}} \]
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Rubi [A] time = 0.71, antiderivative size = 409, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1528, 377, 212, 208, 205} \begin {gather*} -\frac {\sqrt [4]{a-\sqrt {a^2+b}} \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} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \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} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \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} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \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} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 377
Rule 1528
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx &=\int \left (\frac {1-\frac {a}{\sqrt {a^2+b}}}{\left (2 a-2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {1+\frac {a}{\sqrt {a^2+b}}}{\left (2 a+2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=\left (1-\frac {a}{\sqrt {a^2+b}}\right ) \int \frac {1}{\left (2 a-2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx+\left (1+\frac {a}{\sqrt {a^2+b}}\right ) \int \frac {1}{\left (2 a+2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\\ &=\left (1-\frac {a}{\sqrt {a^2+b}}\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 )+\left (1+\frac {a}{\sqrt {a^2+b}}\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 )\\ &=-\frac {\sqrt {a-\sqrt {a^2+b}} \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}}-\frac {\sqrt {a-\sqrt {a^2+b}} \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}}+\frac {\sqrt {a+\sqrt {a^2+b}} \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}}+\frac {\sqrt {a+\sqrt {a^2+b}} \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}}\\ &=-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \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} \sqrt [4]{a^2+b-a \sqrt {a^2+b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+b}} \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} \sqrt [4]{a^2+b+a \sqrt {a^2+b}}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \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} \sqrt [4]{a^2+b-a \sqrt {a^2+b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+b}} \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} \sqrt [4]{a^2+b+a \sqrt {a^2+b}}}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 376, normalized size = 0.84 \begin {gather*} \frac {-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \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 )}{\sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \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 )}{\sqrt [4]{a \sqrt {a^2+b}+a^2+b}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \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 )}{\sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \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 )}{\sqrt [4]{a \sqrt {a^2+b}+a^2+b}}}{4 \sqrt {a^2+b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 39.94, size = 510, normalized size = 1.14 \begin {gather*} -\frac {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 \left (a^2+b\right )^{5/8}}+\frac {\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 \left (a^2+b\right )^{5/8}}+\frac {\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 \left (a^2+b\right )^{5/8}}+\frac {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 \left (a^2+b\right )^{5/8}} \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 {x^{4}}{{\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 \[\int \frac {x^{4}}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (x^{8}+2 a \,x^{4}-b \right )}\, dx\]
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 {x^{4}}{{\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 {x^4}{{\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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