Optimal. Leaf size=82 \[ \frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{3/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{3/4}}+\frac {\left (x^4+4\right ) \sqrt [4]{a x^4+b}}{4 x} \]
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Rubi [A] time = 0.06, antiderivative size = 92, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1487, 451, 331, 298, 203, 206} \begin {gather*} \frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{3/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{3/4}}+\frac {\sqrt [4]{a x^4+b}}{x}+\frac {1}{4} x^3 \sqrt [4]{a x^4+b} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 331
Rule 451
Rule 1487
Rubi steps
\begin {align*} \int \frac {-b+a x^8}{x^2 \left (b+a x^4\right )^{3/4}} \, dx &=\frac {1}{4} x^3 \sqrt [4]{b+a x^4}+\frac {\int \frac {-4 a b-3 a b x^4}{x^2 \left (b+a x^4\right )^{3/4}} \, dx}{4 a}\\ &=\frac {\sqrt [4]{b+a x^4}}{x}+\frac {1}{4} x^3 \sqrt [4]{b+a x^4}-\frac {1}{4} (3 b) \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{b+a x^4}}{x}+\frac {1}{4} x^3 \sqrt [4]{b+a x^4}-\frac {1}{4} (3 b) \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}}{x}+\frac {1}{4} x^3 \sqrt [4]{b+a x^4}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt {a}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt {a}}\\ &=\frac {\sqrt [4]{b+a x^4}}{x}+\frac {1}{4} x^3 \sqrt [4]{b+a x^4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 82, normalized size = 1.00 \begin {gather*} \frac {2 a^{3/4} \left (x^4+4\right ) \sqrt [4]{a x^4+b}+3 b x \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-3 b x \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{3/4} x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 82, normalized size = 1.00 \begin {gather*} \frac {\left (4+x^4\right ) \sqrt [4]{b+a x^4}}{4 x}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}} \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^{2}}\,{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 \[\int \frac {a \,x^{8}-b}{x^{2} \left (a \,x^{4}+b \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 127, normalized size = 1.55 \begin {gather*} -\frac {1}{16} \, a {\left (\frac {3 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {b \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 )}}{a} + \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} b}{{\left (a^{2} - \frac {{\left (a x^{4} + b\right )} a}{x^{4}}\right )} x}\right )} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{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 {b-a\,x^8}{x^2\,{\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 = 2.37, size = 70, normalized size = 0.85 \begin {gather*} - \frac {\sqrt [4]{a} \sqrt [4]{1 + \frac {b}{a x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} + \frac {a x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 b^{\frac {3}{4}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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