Optimal. Leaf size=105 \[ \frac {\left (-8 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{5/4}}+\frac {\left (-8 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{5/4}}+\frac {\left (8 a+3 x^4\right ) \left (a x^4+b\right )^{3/4}}{12 a x^3} \]
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Rubi [A] time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1486, 451, 240, 212, 206, 203} \begin {gather*} -\frac {\left (8 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{5/4}}-\frac {\left (8 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{5/4}}+\frac {x \left (a x^4+b\right )^{3/4}}{4 a}+\frac {2 \left (a x^4+b\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 451
Rule 1486
Rubi steps
\begin {align*} \int \frac {-2 b-2 a x^4+x^8}{x^4 \sqrt [4]{b+a x^4}} \, dx &=\frac {x \left (b+a x^4\right )^{3/4}}{4 a}+\frac {\int \frac {-8 a b-\left (8 a^2+b\right ) x^4}{x^4 \sqrt [4]{b+a x^4}} \, dx}{4 a}\\ &=\frac {2 \left (b+a x^4\right )^{3/4}}{3 x^3}+\frac {x \left (b+a x^4\right )^{3/4}}{4 a}+\frac {\left (-8 a^2-b\right ) \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx}{4 a}\\ &=\frac {2 \left (b+a x^4\right )^{3/4}}{3 x^3}+\frac {x \left (b+a x^4\right )^{3/4}}{4 a}+\frac {\left (-8 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 a}\\ &=\frac {2 \left (b+a x^4\right )^{3/4}}{3 x^3}+\frac {x \left (b+a x^4\right )^{3/4}}{4 a}+\frac {\left (-8 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 a}+\frac {\left (-8 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 a}\\ &=\frac {2 \left (b+a x^4\right )^{3/4}}{3 x^3}+\frac {x \left (b+a x^4\right )^{3/4}}{4 a}-\frac {\left (8 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{5/4}}-\frac {\left (8 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{5/4}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 102, normalized size = 0.97 \begin {gather*} \frac {-3 x^3 \left (8 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-3 x^3 \left (8 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+2 \sqrt [4]{a} \left (8 a+3 x^4\right ) \left (a x^4+b\right )^{3/4}}{24 a^{5/4} x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 105, normalized size = 1.00 \begin {gather*} \frac {\left (8 a+3 x^4\right ) \left (b+a x^4\right )^{3/4}}{12 a x^3}+\frac {\left (-8 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{5/4}}+\frac {\left (-8 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{5/4}} \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 {x^{8} - 2 \, a x^{4} - 2 \, b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} x^{4}}\,{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 \[\int \frac {x^{8}-2 a \,x^{4}-2 b}{x^{4} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 192, normalized size = 1.83 \begin {gather*} \frac {1}{2} \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{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 {1}{4}}}\right )} + \frac {b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{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 {1}{4}}}\right )}}{16 \, a} + \frac {2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{3 \, x^{3}} - \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} b}{4 \, {\left (a^{2} - \frac {{\left (a x^{4} + b\right )} a}{x^{4}}\right )} x^{3}} \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 {-x^8+2\,a\,x^4+2\,b}{x^4\,{\left (a\,x^4+b\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.95, size = 107, normalized size = 1.02 \begin {gather*} - \frac {a^{\frac {3}{4}} \left (1 + \frac {b}{a x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{2 \Gamma \left (\frac {1}{4}\right )} - \frac {a x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{2 \sqrt [4]{b} \Gamma \left (\frac {5}{4}\right )} + \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 \sqrt [4]{b} \Gamma \left (\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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