Optimal. Leaf size=163 \[ \frac {2 c x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};-\frac {2 c x^4}{b-\sqrt {b^2-4 a c}}\right )}{(m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}}\right )}{(m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )} \]
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Rubi [A] time = 0.14, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1375, 364} \[ \frac {2 c x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};-\frac {2 c x^4}{b-\sqrt {b^2-4 a c}}\right )}{(m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}}\right )}{(m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1375
Rubi steps
\begin {align*} \int \frac {x^m}{a+b x^4+c x^8} \, dx &=\frac {c \int \frac {x^m}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {x^m}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {2 c x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};-\frac {2 c x^4}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) (1+m)}-\frac {2 c x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m)}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 82, normalized size = 0.50 \[ \frac {x^m \text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\& ,\frac {\left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right )}{2 \text {$\#$1}^7 c+\text {$\#$1}^3 b}\& \right ]}{4 m} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{c x^{8} + b x^{4} + a}, x\right ) \]
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 \[ \int \frac {x^{m}}{c x^{8} + b x^{4} + a}\,{d x} \]
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 \[ \int \frac {x^{m}}{c \,x^{8}+b \,x^{4}+a}\, dx \]
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 \[ \int \frac {x^{m}}{c x^{8} + b x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m}{c\,x^8+b\,x^4+a} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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