Optimal. Leaf size=270 \[ \frac {\sqrt [4]{a x^4+2 b} \left (-2 a x^4-4 b-5 c x^4\right )}{20 b x^5}-\frac {c \text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4 a-\text {$\#$1}^4 c+2 a^2-2 a b+a c\& ,\frac {-\text {$\#$1}^4 c \log \left (\sqrt [4]{a x^4+2 b}-\text {$\#$1} x\right )-2 \text {$\#$1}^4 a \log \left (\sqrt [4]{a x^4+2 b}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 a \log (x)+\text {$\#$1}^4 c \log (x)+2 a^2 \log \left (\sqrt [4]{a x^4+2 b}-\text {$\#$1} x\right )+a c \log \left (\sqrt [4]{a x^4+2 b}-\text {$\#$1} x\right )-2 a b \log \left (\sqrt [4]{a x^4+2 b}-\text {$\#$1} x\right )-2 a^2 \log (x)+2 a b \log (x)-a c \log (x)}{-4 \text {$\#$1}^7+4 \text {$\#$1}^3 a+\text {$\#$1}^3 c}\& \right ]}{16 b} \]
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Rubi [B] time = 8.24, antiderivative size = 1308, normalized size of antiderivative = 4.84, number of steps used = 43, number of rules used = 14, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6728, 264, 277, 331, 298, 203, 206, 1528, 511, 510, 1518, 494, 205, 208} \begin {gather*} \frac {a c^2 \sqrt [4]{a x^4+2 b} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c-\sqrt {c^2+16 a b}},-\frac {a x^4}{2 b}\right ) x^3}{3\ 2^{3/4} b \left (16 a b+c \left (c-\sqrt {c^2+16 a b}\right )\right ) \sqrt [4]{\frac {a x^4}{b}+2}}+\frac {a c^2 \sqrt [4]{a x^4+2 b} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c+\sqrt {c^2+16 a b}},-\frac {a x^4}{2 b}\right ) x^3}{3\ 2^{3/4} b \left (16 a b+c \left (c+\sqrt {c^2+16 a b}\right )\right ) \sqrt [4]{\frac {a x^4}{b}+2}}-\frac {\sqrt [4]{a} (2 b-c) c \left (\sqrt {c^2+16 a b}-c\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{\sqrt {c^2+16 a b}-c} \sqrt [4]{a x^4+2 b}}\right )}{8 b \sqrt {c^2+16 a b} \left (4 b-c+\sqrt {c^2+16 a b}\right )^{3/4}}-\frac {a^{5/4} c \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{\sqrt {c^2+16 a b}-c} \sqrt [4]{a x^4+2 b}}\right )}{\sqrt {c^2+16 a b} \sqrt [4]{\sqrt {c^2+16 a b}-c} \left (4 b-c+\sqrt {c^2+16 a b}\right )^{3/4}}+\frac {\sqrt [4]{a} (2 b-c) c \left (c+\sqrt {c^2+16 a b}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{c+\sqrt {c^2+16 a b}} \sqrt [4]{a x^4+2 b}}\right )}{8 b \sqrt {c^2+16 a b} \left (-4 b+c+\sqrt {c^2+16 a b}\right )^{3/4}}-\frac {a^{5/4} c \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{c+\sqrt {c^2+16 a b}} \sqrt [4]{a x^4+2 b}}\right )}{\sqrt {c^2+16 a b} \sqrt [4]{c+\sqrt {c^2+16 a b}} \left (-4 b+c+\sqrt {c^2+16 a b}\right )^{3/4}}+\frac {\sqrt [4]{a} (2 b-c) c \left (\sqrt {c^2+16 a b}-c\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{\sqrt {c^2+16 a b}-c} \sqrt [4]{a x^4+2 b}}\right )}{8 b \sqrt {c^2+16 a b} \left (4 b-c+\sqrt {c^2+16 a b}\right )^{3/4}}+\frac {a^{5/4} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{\sqrt {c^2+16 a b}-c} \sqrt [4]{a x^4+2 b}}\right )}{\sqrt {c^2+16 a b} \sqrt [4]{\sqrt {c^2+16 a b}-c} \left (4 b-c+\sqrt {c^2+16 a b}\right )^{3/4}}-\frac {\sqrt [4]{a} (2 b-c) c \left (c+\sqrt {c^2+16 a b}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{c+\sqrt {c^2+16 a b}} \sqrt [4]{a x^4+2 b}}\right )}{8 b \sqrt {c^2+16 a b} \left (-4 b+c+\sqrt {c^2+16 a b}\right )^{3/4}}+\frac {a^{5/4} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {c^2+16 a b}} x}{\sqrt [4]{c+\sqrt {c^2+16 a b}} \sqrt [4]{a x^4+2 b}}\right )}{\sqrt {c^2+16 a b} \sqrt [4]{c+\sqrt {c^2+16 a b}} \left (-4 b+c+\sqrt {c^2+16 a b}\right )^{3/4}}-\frac {c \sqrt [4]{a x^4+2 b}}{4 b x}-\frac {\left (a x^4+2 b\right )^{5/4}}{10 b x^5} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 203
Rule 205
Rule 206
Rule 208
Rule 264
Rule 277
Rule 298
Rule 331
Rule 494
Rule 510
Rule 511
Rule 1518
Rule 1528
Rule 6728
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{2 b+a x^4} \left (-4 b+a x^8\right )}{x^6 \left (-4 b+c x^4+a x^8\right )} \, dx &=\int \left (\frac {\sqrt [4]{2 b+a x^4}}{x^6}+\frac {c \sqrt [4]{2 b+a x^4}}{4 b x^2}+\frac {c x^2 \sqrt [4]{2 b+a x^4} \left (c+a x^4\right )}{4 b \left (4 b-c x^4-a x^8\right )}\right ) \, dx\\ &=\frac {c \int \frac {\sqrt [4]{2 b+a x^4}}{x^2} \, dx}{4 b}+\frac {c \int \frac {x^2 \sqrt [4]{2 b+a x^4} \left (c+a x^4\right )}{4 b-c x^4-a x^8} \, dx}{4 b}+\int \frac {\sqrt [4]{2 b+a x^4}}{x^6} \, dx\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}+\frac {c \int \left (-\frac {c x^2 \sqrt [4]{2 b+a x^4}}{-4 b+c x^4+a x^8}-\frac {a x^6 \sqrt [4]{2 b+a x^4}}{-4 b+c x^4+a x^8}\right ) \, dx}{4 b}+\frac {(a c) \int \frac {x^2}{\left (2 b+a x^4\right )^{3/4}} \, dx}{4 b}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}-\frac {(a c) \int \frac {x^6 \sqrt [4]{2 b+a x^4}}{-4 b+c x^4+a x^8} \, dx}{4 b}+\frac {(a c) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{4 b}-\frac {c^2 \int \frac {x^2 \sqrt [4]{2 b+a x^4}}{-4 b+c x^4+a x^8} \, dx}{4 b}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}+\frac {c \int \frac {x^2 \left (-4 a b-a (2 b-c) x^4\right )}{\left (2 b+a x^4\right )^{3/4} \left (-4 b+c x^4+a x^8\right )} \, dx}{4 b}+\frac {\left (\sqrt {a} c\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b}-\frac {\left (\sqrt {a} c\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b}-\frac {(a c) \int \frac {x^2}{\left (2 b+a x^4\right )^{3/4}} \, dx}{4 b}-\frac {c^2 \int \left (\frac {2 a x^2 \sqrt [4]{2 b+a x^4}}{\sqrt {16 a b+c^2} \left (c-\sqrt {16 a b+c^2}+2 a x^4\right )}-\frac {2 a x^2 \sqrt [4]{2 b+a x^4}}{\sqrt {16 a b+c^2} \left (c+\sqrt {16 a b+c^2}+2 a x^4\right )}\right ) \, dx}{4 b}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}-\frac {\sqrt [4]{a} c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b}+\frac {\sqrt [4]{a} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b}+\frac {c \int \left (-\frac {4 a b x^2}{\left (2 b+a x^4\right )^{3/4} \left (-4 b+c x^4+a x^8\right )}-\frac {a (2 b-c) x^6}{\left (2 b+a x^4\right )^{3/4} \left (-4 b+c x^4+a x^8\right )}\right ) \, dx}{4 b}-\frac {(a c) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{4 b}-\frac {\left (a c^2\right ) \int \frac {x^2 \sqrt [4]{2 b+a x^4}}{c-\sqrt {16 a b+c^2}+2 a x^4} \, dx}{2 b \sqrt {16 a b+c^2}}+\frac {\left (a c^2\right ) \int \frac {x^2 \sqrt [4]{2 b+a x^4}}{c+\sqrt {16 a b+c^2}+2 a x^4} \, dx}{2 b \sqrt {16 a b+c^2}}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}-\frac {\sqrt [4]{a} c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b}+\frac {\sqrt [4]{a} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b}-(a c) \int \frac {x^2}{\left (2 b+a x^4\right )^{3/4} \left (-4 b+c x^4+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]{2 b+a x^4}}\right )}{8 b}+\frac {\left (\sqrt {a} c\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b}-\frac {(a (2 b-c) c) \int \frac {x^6}{\left (2 b+a x^4\right )^{3/4} \left (-4 b+c x^4+a x^8\right )} \, dx}{4 b}-\frac {\left (a c^2 \sqrt [4]{2 b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{2 b}}}{c-\sqrt {16 a b+c^2}+2 a x^4} \, dx}{2 b \sqrt {16 a b+c^2} \sqrt [4]{1+\frac {a x^4}{2 b}}}+\frac {\left (a c^2 \sqrt [4]{2 b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{2 b}}}{c+\sqrt {16 a b+c^2}+2 a x^4} \, dx}{2 b \sqrt {16 a b+c^2} \sqrt [4]{1+\frac {a x^4}{2 b}}}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c-\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c-\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c+\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c+\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}-(a c) \int \left (\frac {2 a x^2}{\sqrt {16 a b+c^2} \left (2 b+a x^4\right )^{3/4} \left (c-\sqrt {16 a b+c^2}+2 a x^4\right )}-\frac {2 a x^2}{\sqrt {16 a b+c^2} \left (2 b+a x^4\right )^{3/4} \left (c+\sqrt {16 a b+c^2}+2 a x^4\right )}\right ) \, dx-\frac {(a (2 b-c) c) \int \left (\frac {\left (-c+\sqrt {16 a b+c^2}\right ) x^2}{\sqrt {16 a b+c^2} \left (2 b+a x^4\right )^{3/4} \left (c-\sqrt {16 a b+c^2}+2 a x^4\right )}+\frac {\left (c+\sqrt {16 a b+c^2}\right ) x^2}{\sqrt {16 a b+c^2} \left (2 b+a x^4\right )^{3/4} \left (c+\sqrt {16 a b+c^2}+2 a x^4\right )}\right ) \, dx}{4 b}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c-\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c-\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c+\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c+\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}-\frac {\left (2 a^2 c\right ) \int \frac {x^2}{\left (2 b+a x^4\right )^{3/4} \left (c-\sqrt {16 a b+c^2}+2 a x^4\right )} \, dx}{\sqrt {16 a b+c^2}}+\frac {\left (2 a^2 c\right ) \int \frac {x^2}{\left (2 b+a x^4\right )^{3/4} \left (c+\sqrt {16 a b+c^2}+2 a x^4\right )} \, dx}{\sqrt {16 a b+c^2}}-\frac {\left (a (2 b-c) c \left (1-\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \int \frac {x^2}{\left (2 b+a x^4\right )^{3/4} \left (c-\sqrt {16 a b+c^2}+2 a x^4\right )} \, dx}{4 b}-\frac {\left (a (2 b-c) c \left (1+\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \int \frac {x^2}{\left (2 b+a x^4\right )^{3/4} \left (c+\sqrt {16 a b+c^2}+2 a x^4\right )} \, dx}{4 b}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c-\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c-\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c+\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c+\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}-\frac {\left (2 a^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{c-\sqrt {16 a b+c^2}-\left (-4 a b+a \left (c-\sqrt {16 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2}}+\frac {\left (2 a^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+\sqrt {16 a b+c^2}-\left (-4 a b+a \left (c+\sqrt {16 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2}}-\frac {\left (a (2 b-c) c \left (1-\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{c-\sqrt {16 a b+c^2}-\left (-4 a b+a \left (c-\sqrt {16 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{4 b}-\frac {\left (a (2 b-c) c \left (1+\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+\sqrt {16 a b+c^2}-\left (-4 a b+a \left (c+\sqrt {16 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{4 b}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c-\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c-\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c+\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c+\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}+\frac {\left (a^{3/2} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {16 a b+c^2}}-\sqrt {a} \sqrt {4 b-c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt {4 b-c+\sqrt {16 a b+c^2}}}-\frac {\left (a^{3/2} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {16 a b+c^2}}+\sqrt {a} \sqrt {4 b-c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt {4 b-c+\sqrt {16 a b+c^2}}}+\frac {\left (\sqrt {a} (2 b-c) c \left (1-\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {16 a b+c^2}}-\sqrt {a} \sqrt {4 b-c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {4 b-c+\sqrt {16 a b+c^2}}}-\frac {\left (\sqrt {a} (2 b-c) c \left (1-\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {16 a b+c^2}}+\sqrt {a} \sqrt {4 b-c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {4 b-c+\sqrt {16 a b+c^2}}}+\frac {\left (a^{3/2} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {16 a b+c^2}}-\sqrt {a} \sqrt {-4 b+c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt {-4 b+c+\sqrt {16 a b+c^2}}}-\frac {\left (a^{3/2} c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {16 a b+c^2}}+\sqrt {a} \sqrt {-4 b+c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt {-4 b+c+\sqrt {16 a b+c^2}}}-\frac {\left (\sqrt {a} (2 b-c) c \left (1+\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {16 a b+c^2}}-\sqrt {a} \sqrt {-4 b+c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {-4 b+c+\sqrt {16 a b+c^2}}}+\frac {\left (\sqrt {a} (2 b-c) c \left (1+\frac {c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {16 a b+c^2}}+\sqrt {a} \sqrt {-4 b+c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {-4 b+c+\sqrt {16 a b+c^2}}}\\ &=-\frac {c \sqrt [4]{2 b+a x^4}}{4 b x}-\frac {\left (2 b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c-\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c-\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}+\frac {a c^2 x^3 \sqrt [4]{2 b+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 a x^4}{c+\sqrt {16 a b+c^2}},-\frac {a x^4}{2 b}\right )}{3\ 2^{3/4} b \left (16 a b+c \left (c+\sqrt {16 a b+c^2}\right )\right ) \sqrt [4]{2+\frac {a x^4}{b}}}-\frac {a^{5/4} c \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt [4]{-c+\sqrt {16 a b+c^2}} \left (4 b-c+\sqrt {16 a b+c^2}\right )^{3/4}}-\frac {\sqrt [4]{a} (2 b-c) c \left (-c+\sqrt {16 a b+c^2}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {16 a b+c^2} \left (4 b-c+\sqrt {16 a b+c^2}\right )^{3/4}}-\frac {a^{5/4} c \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt [4]{c+\sqrt {16 a b+c^2}} \left (-4 b+c+\sqrt {16 a b+c^2}\right )^{3/4}}+\frac {\sqrt [4]{a} (2 b-c) c \left (c+\sqrt {16 a b+c^2}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {16 a b+c^2} \left (-4 b+c+\sqrt {16 a b+c^2}\right )^{3/4}}+\frac {a^{5/4} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt [4]{-c+\sqrt {16 a b+c^2}} \left (4 b-c+\sqrt {16 a b+c^2}\right )^{3/4}}+\frac {\sqrt [4]{a} (2 b-c) c \left (-c+\sqrt {16 a b+c^2}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {16 a b+c^2} \left (4 b-c+\sqrt {16 a b+c^2}\right )^{3/4}}+\frac {a^{5/4} c \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{\sqrt {16 a b+c^2} \sqrt [4]{c+\sqrt {16 a b+c^2}} \left (-4 b+c+\sqrt {16 a b+c^2}\right )^{3/4}}-\frac {\sqrt [4]{a} (2 b-c) c \left (c+\sqrt {16 a b+c^2}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{c+\sqrt {16 a b+c^2}} \sqrt [4]{2 b+a x^4}}\right )}{8 b \sqrt {16 a b+c^2} \left (-4 b+c+\sqrt {16 a b+c^2}\right )^{3/4}}\\ \end {align*}
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Mathematica [F] time = 9.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{2 b+a x^4} \left (-4 b+a x^8\right )}{x^6 \left (-4 b+c x^4+a x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 271, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{2 b+a x^4} \left (-4 b-2 a x^4-5 c x^4\right )}{20 b x^5}-\frac {c \text {RootSum}\left [2 a^2-2 a b+a c-4 a \text {$\#$1}^4-c \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {2 a^2 \log (x)-2 a b \log (x)+a c \log (x)-2 a^2 \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right )+2 a b \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right )-a c \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right )-2 a \log (x) \text {$\#$1}^4-c \log (x) \text {$\#$1}^4+2 a \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+c \log \left (\sqrt [4]{2 b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-4 a \text {$\#$1}^3-c \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ]}{16 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} - 4 \, b\right )} {\left (a x^{4} + 2 \, b\right )}^{\frac {1}{4}}}{{\left (a x^{8} + c x^{4} - 4 \, 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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+2 b \right )^{\frac {1}{4}} \left (a \,x^{8}-4 b \right )}{x^{6} \left (a \,x^{8}+c \,x^{4}-4 b \right )}\, dx\]
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 {{\left (a x^{8} - 4 \, b\right )} {\left (a x^{4} + 2 \, b\right )}^{\frac {1}{4}}}{{\left (a x^{8} + c x^{4} - 4 \, 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+2\,b\right )}^{1/4}\,\left (4\,b-a\,x^8\right )}{x^6\,\left (a\,x^8+c\,x^4-4\,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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