Optimal. Leaf size=59 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}} \]
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Rubi [C] time = 1.50, antiderivative size = 460, normalized size of antiderivative = 7.80, number of steps used = 18, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6728, 246, 245, 1438, 430, 429, 511, 510} \begin {gather*} -\frac {x \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^8}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}-\frac {x \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^8}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}+\frac {a x^5 \left (c-\sqrt {4 a b+c^2}\right ) \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^8}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^8}{b}\right )}{5 \left (c \left (c-\sqrt {4 a b+c^2}\right )+2 a b\right ) \sqrt [4]{b-a x^8}}+\frac {a x^5 \left (\sqrt {4 a b+c^2}+c\right ) \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^8}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^8}{b}\right )}{5 \left (c \left (\sqrt {4 a b+c^2}+c\right )+2 a b\right ) \sqrt [4]{b-a x^8}}+\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}} \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 6728
Rubi steps
\begin {align*} \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{b-a x^8}}+\frac {2 b-c x^4}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{b-a x^8}} \, dx+\int \frac {2 b-c x^4}{\sqrt [4]{b-a x^8} \left (-b+c x^4+a x^8\right )} \, dx\\ &=\frac {\sqrt [4]{1-\frac {a x^8}{b}} \int \frac {1}{\sqrt [4]{1-\frac {a x^8}{b}}} \, dx}{\sqrt [4]{b-a x^8}}+\int \left (\frac {-c+\sqrt {4 a b+c^2}}{\left (c-\sqrt {4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b-a x^8}}+\frac {-c-\sqrt {4 a b+c^2}}{\left (c+\sqrt {4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b-a x^8}}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}+\left (-c-\sqrt {4 a b+c^2}\right ) \int \frac {1}{\left (c+\sqrt {4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b-a x^8}} \, dx+\left (-c+\sqrt {4 a b+c^2}\right ) \int \frac {1}{\left (c-\sqrt {4 a b+c^2}+2 a x^4\right ) \sqrt [4]{b-a x^8}} \, dx\\ &=\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}+\left (-c-\sqrt {4 a b+c^2}\right ) \int \left (\frac {c+\sqrt {4 a b+c^2}}{2 \sqrt [4]{b-a x^8} \left (2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^8\right )}+\frac {a x^4}{\sqrt [4]{b-a x^8} \left (-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^8\right )}\right ) \, dx+\left (-c+\sqrt {4 a b+c^2}\right ) \int \left (\frac {-c+\sqrt {4 a b+c^2}}{2 \sqrt [4]{b-a x^8} \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^8\right )}+\frac {a x^4}{\sqrt [4]{b-a x^8} \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^8\right )}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}-\left (a \left (c-\sqrt {4 a b+c^2}\right )\right ) \int \frac {x^4}{\sqrt [4]{b-a x^8} \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^8\right )} \, dx+\frac {1}{2} \left (c-\sqrt {4 a b+c^2}\right )^2 \int \frac {1}{\sqrt [4]{b-a x^8} \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^8\right )} \, dx-\left (a \left (c+\sqrt {4 a b+c^2}\right )\right ) \int \frac {x^4}{\sqrt [4]{b-a x^8} \left (-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^8\right )} \, dx-\frac {1}{2} \left (c+\sqrt {4 a b+c^2}\right )^2 \int \frac {1}{\sqrt [4]{b-a x^8} \left (2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^8\right )} \, dx\\ &=\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}-\frac {\left (a \left (c-\sqrt {4 a b+c^2}\right ) \sqrt [4]{1-\frac {a x^8}{b}}\right ) \int \frac {x^4}{\left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx}{\sqrt [4]{b-a x^8}}+\frac {\left (\left (c-\sqrt {4 a b+c^2}\right )^2 \sqrt [4]{1-\frac {a x^8}{b}}\right ) \int \frac {1}{\left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx}{2 \sqrt [4]{b-a x^8}}-\frac {\left (a \left (c+\sqrt {4 a b+c^2}\right ) \sqrt [4]{1-\frac {a x^8}{b}}\right ) \int \frac {x^4}{\left (-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx}{\sqrt [4]{b-a x^8}}-\frac {\left (\left (c+\sqrt {4 a b+c^2}\right )^2 \sqrt [4]{1-\frac {a x^8}{b}}\right ) \int \frac {1}{\left (2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx}{2 \sqrt [4]{b-a x^8}}\\ &=-\frac {x \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^8}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}-\frac {x \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^8}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}+\frac {a \left (c-\sqrt {4 a b+c^2}\right ) x^5 \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^8}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^8}{b}\right )}{5 \left (2 a b+c \left (c-\sqrt {4 a b+c^2}\right )\right ) \sqrt [4]{b-a x^8}}+\frac {a \left (c+\sqrt {4 a b+c^2}\right ) x^5 \sqrt [4]{1-\frac {a x^8}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^8}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^8}{b}\right )}{5 \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [4]{b-a x^8}}+\frac {x \sqrt [4]{1-\frac {a x^8}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^8}{b}\right )}{\sqrt [4]{b-a x^8}}\\ \end {align*}
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Mathematica [F] time = 1.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+a x^8}{\sqrt [4]{b-a x^8} \left (-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 = 12.31, size = 59, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{b-a x^8}}\right )}{2 \sqrt [4]{c}} \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^{8} + b}{{\left (a x^{8} + c x^{4} - b\right )} {\left (-a x^{8} + 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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{8}+b}{\left (-a \,x^{8}+b \right )^{\frac {1}{4}} \left (a \,x^{8}+c \,x^{4}-b \right )}\, dx\]
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 {a x^{8} + b}{{\left (a x^{8} + c x^{4} - b\right )} {\left (-a x^{8} + 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 = -0.02 \begin {gather*} \int \frac {a\,x^8+b}{{\left (b-a\,x^8\right )}^{1/4}\,\left (a\,x^8+c\,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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