Optimal. Leaf size=98 \[ \frac {\sqrt [4]{a x^4+b x^2} \left (a x^2+8 b\right )}{2 x}+\frac {7}{4} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )-\frac {7}{4} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 170, normalized size of antiderivative = 1.73, number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2038, 2004, 2032, 329, 331, 298, 203, 206} \begin {gather*} -\frac {7}{2} a x \sqrt [4]{a x^4+b x^2}+\frac {7 \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 )}{4 \left (a x^4+b x^2\right )^{3/4}}-\frac {7 \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 )}{4 \left (a x^4+b x^2\right )^{3/4}}+\frac {4 \left (a x^4+b x^2\right )^{5/4}}{x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 331
Rule 2004
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {\left (-2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx &=\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}-(7 a) \int \sqrt [4]{b x^2+a x^4} \, dx\\ &=-\frac {7}{2} a x \sqrt [4]{b x^2+a x^4}+\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}-\frac {1}{4} (7 a b) \int \frac {x^2}{\left (b x^2+a x^4\right )^{3/4}} \, dx\\ &=-\frac {7}{2} a x \sqrt [4]{b x^2+a x^4}+\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}-\frac {\left (7 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}{4 \left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {7}{2} a x \sqrt [4]{b x^2+a x^4}+\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}-\frac {\left (7 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 )}{2 \left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {7}{2} a x \sqrt [4]{b x^2+a x^4}+\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}-\frac {\left (7 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 )}{2 \left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {7}{2} a x \sqrt [4]{b x^2+a x^4}+\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}-\frac {\left (7 \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 )}{4 \left (b x^2+a x^4\right )^{3/4}}+\frac {\left (7 \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 )}{4 \left (b x^2+a x^4\right )^{3/4}}\\ &=-\frac {7}{2} a x \sqrt [4]{b x^2+a x^4}+\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}+\frac {7 \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 )}{4 \left (b x^2+a x^4\right )^{3/4}}-\frac {7 \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 )}{4 \left (b x^2+a x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 71, normalized size = 0.72 \begin {gather*} \frac {2 \sqrt [4]{x^2 \left (a x^2+b\right )} \left (6 \left (a x^2+b\right )-\frac {7 a x^2 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {a x^2}{b}\right )}{\sqrt [4]{\frac {a x^2}{b}+1}}\right )}{3 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 98, normalized size = 1.00 \begin {gather*} \frac {\left (8 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{2 x}+\frac {7}{4} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )-\frac {7}{4} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+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 [B] time = 0.20, size = 221, normalized size = 2.26 \begin {gather*} \frac {8 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} a b x^{2} - 14 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \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 ) - 14 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \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 ) - 7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \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 ) + 7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \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 ) + 64 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} b^{2}}{16 \, b} \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 \[\int \frac {\left (a \,x^{2}-2 b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{x^{2}}\, dx\]
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^{2} - 2 \, b\right )}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 89, normalized size = 0.91 \begin {gather*} \frac {2\,a\,x\,{\left (a\,x^4+b\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {a\,x^2}{b}\right )}{3\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/4}}+\frac {4\,b\,{\left (a\,x^4+b\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {a\,x^2}{b}\right )}{x\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/4}} \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^{2} - 2 b\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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