Optimal. Leaf size=122 \[ \frac {\left (32 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{32 a^{3/4}}+\frac {\left (-32 a b-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{32 a^{3/4}}+\frac {\sqrt [4]{a x^4+b x^2} \left (4 a x^4+b x^2+32 b\right )}{16 x} \]
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Rubi [B] time = 0.49, antiderivative size = 321, normalized size of antiderivative = 2.63, number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2052, 2020, 2032, 329, 331, 298, 203, 206, 2021, 2024} \begin {gather*} \frac {3 b^2 x^{3/2} \left (a x^2+b\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{32 a^{3/4} \left (a x^4+b x^2\right )^{3/4}}-\frac {3 b^2 x^{3/2} \left (a x^2+b\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{32 a^{3/4} \left (a x^4+b x^2\right )^{3/4}}+\frac {1}{16} b x \sqrt [4]{a x^4+b x^2}+\frac {2 b \sqrt [4]{a x^4+b x^2}}{x}+\frac {\sqrt [4]{a} b x^{3/2} \left (a x^2+b\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\left (a x^4+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} b x^{3/2} \left (a x^2+b\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\left (a x^4+b x^2\right )^{3/4}}+\frac {1}{4} a x^3 \sqrt [4]{a x^4+b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 331
Rule 2020
Rule 2021
Rule 2024
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx &=\int \left (-\frac {b \sqrt [4]{b x^2+a x^4}}{x^2}+a x^2 \sqrt [4]{b x^2+a x^4}\right ) \, dx\\ &=a \int x^2 \sqrt [4]{b x^2+a x^4} \, dx-b \int \frac {\sqrt [4]{b x^2+a x^4}}{x^2} \, dx\\ &=\frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}+\frac {1}{8} (a b) \int \frac {x^4}{\left (b x^2+a x^4\right )^{3/4}} \, dx-(a b) \int \frac {x^2}{\left (b x^2+a x^4\right )^{3/4}} \, dx\\ &=\frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {1}{32} \left (3 b^2\right ) \int \frac {x^2}{\left (b x^2+a x^4\right )^{3/4}} \, dx-\frac {\left (a b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{\left (b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {\left (2 a b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{32 \left (b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {\left (2 a b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{16 \left (b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {\left (\sqrt {a} b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}+\frac {\left (\sqrt {a} b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{16 \left (b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}+\frac {\sqrt [4]{a} b x^{3/2} \left (b+a x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\sqrt [4]{a} b x^{3/2} \left (b+a x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{32 \sqrt {a} \left (b x^2+a x^4\right )^{3/4}}+\frac {\left (3 b^2 x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{32 \sqrt {a} \left (b x^2+a x^4\right )^{3/4}}\\ &=\frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}+\frac {\sqrt [4]{a} b x^{3/2} \left (b+a x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}+\frac {3 b^2 x^{3/2} \left (b+a x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{32 a^{3/4} \left (b x^2+a x^4\right )^{3/4}}-\frac {\sqrt [4]{a} b x^{3/2} \left (b+a x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {3 b^2 x^{3/2} \left (b+a x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{32 a^{3/4} \left (b x^2+a x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 108, normalized size = 0.89 \begin {gather*} \frac {\sqrt [4]{x^2 \left (a x^2+b\right )} \left (x^2 \left (\left (a x^2+b\right ) \sqrt [4]{\frac {a x^2}{b}+1}-b \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {a x^2}{b}\right )\right )+8 b \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\frac {a x^2}{b}\right )\right )}{4 x \sqrt [4]{\frac {a x^2}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 122, normalized size = 1.00 \begin {gather*} \frac {\left (32 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{32 a^{3/4}}+\frac {\left (-32 a b-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{32 a^{3/4}}+\frac {\sqrt [4]{a x^4+b x^2} \left (4 a x^4+b x^2+32 b\right )}{16 x} \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 [B] time = 0.38, size = 277, normalized size = 2.27 \begin {gather*} \frac {\frac {8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{4}} b^{3} + 3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} a b^{3}\right )} x^{4}}{b^{2}} + 256 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} b^{2} + \frac {2 \, \sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{\left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{\left (-a\right )^{\frac {3}{4}}}}{128 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{4}-b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{x^{2}}\, 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 {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{x^{2}}\,{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 {\left (b-a\,x^4\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}}{x^2} \,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 {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - b\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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