Optimal. Leaf size=86 \[ -\frac {1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {\sqrt [4]{a x^4+b} \left (b-4 a x^4\right )}{5 b x^5} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1489, 271, 264, 331, 298, 203, 206} \begin {gather*} -\frac {4 a \sqrt [4]{a x^4+b}}{5 b x}-\frac {1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {\sqrt [4]{a x^4+b}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 264
Rule 271
Rule 298
Rule 331
Rule 1489
Rubi steps
\begin {align*} \int \frac {-b+a x^8}{x^6 \left (b+a x^4\right )^{3/4}} \, dx &=\int \left (-\frac {b}{x^6 \left (b+a x^4\right )^{3/4}}+\frac {a x^2}{\left (b+a x^4\right )^{3/4}}\right ) \, dx\\ &=a \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx-b \int \frac {1}{x^6 \left (b+a x^4\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{b+a x^4}}{5 x^5}+\frac {1}{5} (4 a) \int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx+a \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\sqrt [4]{b+a x^4}}{5 x^5}-\frac {4 a \sqrt [4]{b+a x^4}}{5 b x}+\frac {1}{2} \sqrt {a} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \sqrt {a} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\sqrt [4]{b+a x^4}}{5 x^5}-\frac {4 a \sqrt [4]{b+a x^4}}{5 b x}-\frac {1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 84, normalized size = 0.98 \begin {gather*} \frac {1}{10} \left (-5 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+5 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {2 \sqrt [4]{a x^4+b} \left (b-4 a x^4\right )}{b x^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 86, normalized size = 1.00 \begin {gather*} \frac {\left (b-4 a x^4\right ) \sqrt [4]{b+a x^4}}{5 b x^5}-\frac {1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \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^{4} + b\right )}^{\frac {3}{4}} 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 \[\int \frac {a \,x^{8}-b}{x^{6} \left (a \,x^{4}+b \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 107, normalized size = 1.24 \begin {gather*} \frac {1}{4} \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {3}{4}}}\right )} - \frac {\frac {5 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} a}{x} - \frac {{\left (a x^{4} + b\right )}^{\frac {5}{4}}}{x^{5}}}{5 \, b} \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 {b-a\,x^8}{x^6\,{\left (a\,x^4+b\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.32, size = 105, normalized size = 1.22 \begin {gather*} - \frac {a^{\frac {5}{4}} \sqrt [4]{1 + \frac {b}{a x^{4}}} \Gamma \left (- \frac {5}{4}\right )}{4 b \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{a} \sqrt [4]{1 + \frac {b}{a x^{4}}} \Gamma \left (- \frac {5}{4}\right )}{16 x^{4} \Gamma \left (\frac {3}{4}\right )} + \frac {a x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 b^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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