Optimal. Leaf size=346 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4 a+2 a^2-b\& ,\frac {-\text {$\#$1}^4 a^3 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a^3 \log (x)+\text {$\#$1}^4 a b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^4 a b \log (x)+\text {$\#$1}^4 b \log (x)+2 a^4 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-a^2 b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-2 a^4 \log (x)+a^2 b \log (x)}{3 \text {$\#$1}^3 a-2 \text {$\#$1}^7}\& \right ]+\frac {1}{4} \left (\sqrt [4]{a} b-4 a^{9/4}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )+\frac {1}{4} \left (4 a^{9/4}-\sqrt [4]{a} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )+\frac {1}{2} a x \sqrt [4]{a x^4-b x^2} \]
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Rubi [A] time = 1.83, antiderivative size = 405, normalized size of antiderivative = 1.17, number of steps used = 17, number of rules used = 11, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.268, Rules used = {2056, 6728, 279, 329, 331, 298, 203, 206, 466, 511, 510} \begin {gather*} -\frac {2 x \left (a^3-a^2 \sqrt {a^2+4 b}+2 a b-2 b\right ) \sqrt [4]{a x^4-b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {2 x^2}{a-\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (-a \sqrt {a^2+4 b}+a^2+4 b\right ) \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {2 x \left (a^2+\frac {a^3+2 a b-2 b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{a x^4-b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {2 x^2}{a+\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (\sqrt {a^2+4 b}+a\right ) \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {1}{2} a x \sqrt [4]{a x^4-b x^2}+\frac {\sqrt [4]{a} b \sqrt [4]{a x^4-b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt {x} \sqrt [4]{a x^2-b}}-\frac {\sqrt [4]{a} b \sqrt [4]{a x^4-b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt {x} \sqrt [4]{a x^2-b}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 329
Rule 331
Rule 466
Rule 510
Rule 511
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx &=\frac {\sqrt [4]{-b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2} \left (-b+a x^4\right )}{-b-a x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {\sqrt [4]{-b x^2+a x^4} \int \left (a \sqrt {x} \sqrt [4]{-b+a x^2}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \left (b-a b-a^2 x^2\right )}{-b-a x^2+x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=-\frac {\sqrt [4]{-b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2} \left (b-a b-a^2 x^2\right )}{-b-a x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (a \sqrt [4]{-b x^2+a x^4}\right ) \int \sqrt {x} \sqrt [4]{-b+a x^2} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\sqrt [4]{-b x^2+a x^4} \int \left (\frac {\left (-a^2+\frac {-a^3+2 b-2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}}{-a-\sqrt {a^2+4 b}+2 x^2}+\frac {\left (-a^2-\frac {-a^3+2 b-2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}}{-a+\sqrt {a^2+4 b}+2 x^2}\right ) \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (a b \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (-b+a x^2\right )^{3/4}} \, dx}{4 \sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (a b \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (\left (-a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2}}{-a-\sqrt {a^2+4 b}+2 x^2} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (\left (-a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2}}{-a+\sqrt {a^2+4 b}+2 x^2} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (a b \sqrt [4]{-b x^2+a x^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 \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (2 \left (-a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+a x^4}}{-a-\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (2 \left (-a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+a x^4}}{-a+\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (\sqrt {a} b \sqrt [4]{-b x^2+a x^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 \sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (\sqrt {a} b \sqrt [4]{-b x^2+a x^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 \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (2 \left (-a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{-a-\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {\left (2 \left (-a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{-a+\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}\\ &=\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {2 \left (a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) x \sqrt [4]{-b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {2 x^2}{a-\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (a-\sqrt {a^2+4 b}\right ) \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {2 \left (a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) x \sqrt [4]{-b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {2 x^2}{a+\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (a+\sqrt {a^2+4 b}\right ) \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {\sqrt [4]{a} b \sqrt [4]{-b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\sqrt [4]{a} b \sqrt [4]{-b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-b+a x^2}}\\ \end {align*}
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Mathematica [A] time = 2.75, size = 447, normalized size = 1.29 \begin {gather*} \frac {\sqrt [4]{a x^4-b x^2} \left (4 a^3 x^5+(4-3 a) b^2 x+a b x^3 \left (a-x^2-4\right )\right ) \left (\frac {3 \sqrt [4]{a} \left (4 a^2-b\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )\right )}{x^{3/2} \sqrt [4]{a x^2-b}}+\frac {8 \left (a^4+\frac {-a^5-2 a^3 b+a^2 b+2 a b^2-2 b^2}{\sqrt {a^2+4 b}}-a b\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {b \left (\frac {a}{b}+\frac {2}{\sqrt {a^2+4 b}-a}\right ) x^2}{b-a x^2}\right )}{\left (a-\sqrt {a^2+4 b}\right ) \left (a x^2-b\right )}+\frac {8 \left (a^4+\frac {a^5+2 a^3 b-a^2 b-2 a b^2+2 b^2}{\sqrt {a^2+4 b}}-a b\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {b \left (\frac {a}{b}-\frac {2}{a+\sqrt {a^2+4 b}}\right ) x^2}{b-a x^2}\right )}{\left (\sqrt {a^2+4 b}+a\right ) \left (a x^2-b\right )}\right )}{12 \left (-4 a^3 x^4+(3 a-4) b^2+a b x^2 \left (-a+x^2+4\right )\right )}+\frac {1}{2} a x \sqrt [4]{a x^4-b x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.16, size = 346, normalized size = 1.00 \begin {gather*} \frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}+\frac {1}{4} \left (-4 a^{9/4}+\sqrt [4]{a} b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{4} \left (4 a^{9/4}-\sqrt [4]{a} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a^4 \log (x)-a^2 b \log (x)-2 a^4 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+a^2 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^3 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4+a b \log (x) \text {$\#$1}^4+a^3 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}^3+2 \text {$\#$1}^7}\&\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(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \[\int \frac {\left (a \,x^{4}-b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}{x^{4}-a \,x^{2}-b}\, dx\]
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^{4} - b\right )}}{x^{4} - a x^{2} - b}\,{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 = -0.00 \begin {gather*} \int \frac {\left (b-a\,x^4\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}}{-x^4+a\,x^2+b} \,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 )}{- a x^{2} - b + x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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