Optimal. Leaf size=199 \[ \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) x \sqrt [4]{2 a b+c} \sqrt [4]{a^2 x^8+b^2+c x^4}}{x^2 \sqrt {2 a b+c}-i \sqrt {a^2 x^8+b^2+c x^4}}\right )}{\sqrt [4]{2 a b+c}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a^2 x^8+b^2+c x^4}}{\sqrt [4]{2 a b+c}}+\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \sqrt [4]{2 a b+c}}{x \sqrt [4]{a^2 x^8+b^2+c x^4}}\right )}{\sqrt [4]{2 a b+c}} \]
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Rubi [F] time = 0.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {b+a x^4}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {b+a x^4}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx &=\int \left (\frac {1}{\sqrt [4]{b^2+c x^4+a^2 x^8}}+\frac {2 b}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}}\right ) \, dx\\ &=(2 b) \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx+\int \frac {1}{\sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx\\ &=(2 b) \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx+\frac {\left (\sqrt [4]{1+\frac {2 a^2 x^4}{c-\sqrt {-4 a^2 b^2+c^2}}} \sqrt [4]{1+\frac {2 a^2 x^4}{c+\sqrt {-4 a^2 b^2+c^2}}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {2 a^2 x^4}{c-\sqrt {-4 a^2 b^2+c^2}}} \sqrt [4]{1+\frac {2 a^2 x^4}{c+\sqrt {-4 a^2 b^2+c^2}}}} \, dx}{\sqrt [4]{b^2+c x^4+a^2 x^8}}\\ &=\frac {x \sqrt [4]{1+\frac {2 a^2 x^4}{c-\sqrt {-4 a^2 b^2+c^2}}} \sqrt [4]{1+\frac {2 a^2 x^4}{c+\sqrt {-4 a^2 b^2+c^2}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {2 a^2 x^4}{c-\sqrt {-4 a^2 b^2+c^2}},-\frac {2 a^2 x^4}{c+\sqrt {-4 a^2 b^2+c^2}}\right )}{\sqrt [4]{b^2+c x^4+a^2 x^8}}+(2 b) \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+a x^4}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.22, size = 207, normalized size = 1.04 \begin {gather*} -\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tan ^{-1}\left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \sqrt [4]{2 a b+c}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a^2 x^8+b^2+c x^4}}{\sqrt [4]{2 a b+c}}}{x \sqrt [4]{a^2 x^8+b^2+c x^4}}\right )}{\sqrt [4]{2 a b+c}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a^2 x^8+b^2+c x^4}}{\sqrt [4]{2 a b+c}}+\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \sqrt [4]{2 a b+c}}{x \sqrt [4]{a^2 x^8+b^2+c x^4}}\right )}{\sqrt [4]{2 a b+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^{4} + b}{{\left (a^{2} x^{8} + c x^{4} + b^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \,x^{4}+b}{\left (a \,x^{4}-b \right ) \left (a^{2} x^{8}+c \,x^{4}+b^{2}\right )^{\frac {1}{4}}}\, dx \end {gather*}
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^{4} + b}{{\left (a^{2} x^{8} + c x^{4} + b^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}\,{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 = -0.01 \begin {gather*} \int -\frac {a\,x^4+b}{\left (b-a\,x^4\right )\,{\left (a^2\,x^8+b^2+c\,x^4\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{\left (a x^{4} - b\right ) \sqrt [4]{a^{2} x^{8} + b^{2} + c x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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