Optimal. Leaf size=215 \[ \frac {\sqrt [4]{a x^4-b} \left (a x^4-b-5 c x^4\right )}{5 b x^5}-\frac {a \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+2 a b\& ,\frac {-\text {$\#$1}^4 c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-\text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 b \log (x)+\text {$\#$1}^4 c \log (x)+a c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+2 b c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a c \log (x)-2 b c \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{8 b} \]
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Rubi [B] time = 4.67, antiderivative size = 933, normalized size of antiderivative = 4.34, number of steps used = 43, number of rules used = 12, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6725, 264, 277, 331, 298, 203, 206, 1529, 511, 510, 1519, 494} \begin {gather*} -\frac {a \sqrt [4]{a x^4-b} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right ) x^3}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a \sqrt [4]{a x^4-b} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right ) x^3}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {\sqrt {-a} c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {\sqrt {-a} c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{a x^4-b}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {c \sqrt [4]{a x^4-b}}{b x}+\frac {\left (a x^4-b\right )^{5/4}}{5 b x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 264
Rule 277
Rule 298
Rule 331
Rule 494
Rule 510
Rule 511
Rule 1519
Rule 1529
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx &=\int \left (\frac {\sqrt [4]{-b+a x^4}}{x^6}+\frac {c \sqrt [4]{-b+a x^4}}{b x^2}-\frac {a x^2 \sqrt [4]{-b+a x^4} \left (b+2 c x^4\right )}{b \left (b+2 a x^8\right )}\right ) \, dx\\ &=-\frac {a \int \frac {x^2 \sqrt [4]{-b+a x^4} \left (b+2 c x^4\right )}{b+2 a x^8} \, dx}{b}+\frac {c \int \frac {\sqrt [4]{-b+a x^4}}{x^2} \, dx}{b}+\int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a \int \left (\frac {b x^2 \sqrt [4]{-b+a x^4}}{b+2 a x^8}+\frac {2 c x^6 \sqrt [4]{-b+a x^4}}{b+2 a x^8}\right ) \, dx}{b}+\frac {(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{b}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-a \int \frac {x^2 \sqrt [4]{-b+a x^4}}{b+2 a x^8} \, dx+\frac {(a c) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{b}-\frac {(2 a c) \int \frac {x^6 \sqrt [4]{-b+a x^4}}{b+2 a x^8} \, dx}{b}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-a \int \left (-\frac {a x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4\right )}-\frac {a x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}\right ) \, dx+\frac {c \int \frac {x^2 \left (a b+2 a b x^4\right )}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )} \, dx}{b}+\frac {\left (\sqrt {a} c\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}-\frac {\left (\sqrt {a} c\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}-\frac {(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{b}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {\sqrt [4]{a} c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\sqrt [4]{a} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {(-a)^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4} \, dx}{\sqrt {2} \sqrt {b}}+\frac {(-a)^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4} \, dx}{\sqrt {2} \sqrt {b}}+\frac {c \int \left (\frac {a b x^2}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )}+\frac {2 a b x^6}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )}\right ) \, dx}{b}-\frac {(a c) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{b}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {\sqrt [4]{a} c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\sqrt [4]{a} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )} \, dx+(2 a c) \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+2 a x^8\right )} \, dx-\frac {\left (\sqrt {a} c\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\left (\sqrt {a} c\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b}+\frac {\left ((-a)^{3/2} \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4} \, dx}{\sqrt {2} \sqrt {b} \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {\left ((-a)^{3/2} \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4} \, dx}{\sqrt {2} \sqrt {b} \sqrt [4]{1-\frac {a x^4}{b}}}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+(a c) \int \left (-\frac {a x^2}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4\right ) \left (-b+a x^4\right )^{3/4}}-\frac {a x^2}{\sqrt {2} \sqrt {-a} \sqrt {b} \left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}\right ) \, dx+(2 a c) \int \left (\frac {x^2}{2 \left (-b+a x^4\right )^{3/4} \left (-\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}+\frac {x^2}{2 \left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )}\right ) \, dx\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (-\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )} \, dx+(a c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )} \, dx-\frac {\left ((-a)^{3/2} c\right ) \int \frac {x^2}{\left (\sqrt {2} \sqrt {-a} \sqrt {b}-2 a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx}{\sqrt {2} \sqrt {b}}-\frac {\left ((-a)^{3/2} c\right ) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (\sqrt {2} \sqrt {-a} \sqrt {b}+2 a x^4\right )} \, dx}{\sqrt {2} \sqrt {b}}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+(a c) \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt {2} \sqrt {-a} \sqrt {b}-\left (-\sqrt {2} \sqrt {-a} a \sqrt {b}+2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+(a c) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}+2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {\left ((-a)^{3/2} c\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}-2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {2} \sqrt {b}}-\frac {\left ((-a)^{3/2} c\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}+2 a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {2} \sqrt {b}}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} b}-\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} b}+\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} b}-\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} b}-\frac {\left (\sqrt {-a} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} \sqrt {b}}+\frac {\left (\sqrt {-a} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} \sqrt {b}}+\frac {\left (\sqrt {-a} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} \sqrt {b}}-\frac {\left (\sqrt {-a} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} \sqrt {b}}\\ &=-\frac {c \sqrt [4]{-b+a x^4}}{b x}+\frac {\left (-b+a x^4\right )^{5/4}}{5 b x^5}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a x^3 \sqrt [4]{-b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {2} \sqrt {-a} x^4}{\sqrt {b}},\frac {a x^4}{b}\right )}{6 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {\sqrt {-a} c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}-\frac {\sqrt {-a} c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a-2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2\ 2^{5/8} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} b}+\frac {\sqrt {-a} c \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+2 \sqrt {-a} \sqrt {b}} x}{\sqrt [8]{2} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [8]{2} \left (\sqrt {2} a+2 \sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {b}}\\ \end {align*}
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Mathematica [F] time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )}{x^6 \left (b+2 a x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.53, size = 214, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-b+a x^4} \left (-b+a x^4-5 c x^4\right )}{5 b x^5}-\frac {a \text {RootSum}\left [a^2+2 a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a c \log (x)+2 b c \log (x)-a c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 b c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-b \log (x) \text {$\#$1}^4-c \log (x) \text {$\#$1}^4+b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 b} \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 {{\left (a x^{8} + c x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (2 \, a x^{8} + b\right )} x^{6}}\,{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 \[\int \frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}+c \,x^{4}+b \right )}{x^{6} \left (2 a \,x^{8}+b \right )}\, dx\]
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 {{\left (a x^{8} + c x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (2 \, a x^{8} + b\right )} x^{6}}\,{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 {{\left (a\,x^4-b\right )}^{1/4}\,\left (a\,x^8+c\,x^4+b\right )}{x^6\,\left (2\,a\,x^8+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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