Optimal. Leaf size=211 \[ -a^{5/4} \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+a^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )-\frac {1}{2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {a b \log (x)-a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^2 \log (x) \text {$\#$1}^4+b \log (x) \text {$\#$1}^4+a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-2 \text {$\#$1}^7}\& \right ] \]
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Rubi [A]
time = 0.41, antiderivative size = 200, normalized size of antiderivative = 0.95, number of steps
used = 8, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {2081, 1283,
1542, 525, 524} \begin {gather*} \frac {4 b x \sqrt [4]{a x^4-b x^2} F_1\left (\frac {3}{4};1,-\frac {5}{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 {4 b x \sqrt [4]{a x^4-b x^2} F_1\left (\frac {3}{4};1,-\frac {5}{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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 1283
Rule 1542
Rule 2081
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^2\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} \left (-b+a x^2\right )^{5/4}}{b-a x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-b+a x^4\right )^{5/4}}{b-a x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {\left (2 \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {2 x^2 \left (-b+a x^4\right )^{5/4}}{\sqrt {a^2-4 b} \left (a+\sqrt {a^2-4 b}-2 x^4\right )}-\frac {2 x^2 \left (-b+a x^4\right )^{5/4}}{\sqrt {a^2-4 b} \left (-a+\sqrt {a^2-4 b}+2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=-\frac {\left (4 \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-b+a x^4\right )^{5/4}}{a+\sqrt {a^2-4 b}-2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (4 \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-b+a x^4\right )^{5/4}}{-a+\sqrt {a^2-4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {\left (4 b \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-\frac {a x^4}{b}\right )^{5/4}}{a+\sqrt {a^2-4 b}-2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {\left (4 b \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-\frac {a x^4}{b}\right )^{5/4}}{-a+\sqrt {a^2-4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}\\ &=\frac {4 b x \sqrt [4]{-b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};\frac {2 x^2}{a-\sqrt {a^2-4 b}},\frac {a x^2}{b}\right )}{3 \left (a^2-a \sqrt {a^2-4 b}-4 b\right ) \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {4 b x \sqrt [4]{-b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {5}{4};\frac {7}{4};\frac {2 x^2}{a+\sqrt {a^2-4 b}},\frac {a x^2}{b}\right )}{3 \left (a^2+a \sqrt {a^2-4 b}-4 b\right ) \sqrt [4]{1-\frac {a x^2}{b}}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 249, normalized size = 1.18 \begin {gather*} \frac {\sqrt [4]{-b x^2+a x^4} \left (4 a^{5/4} \left (-\text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )\right )+\text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a b \log (x)+2 a b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-2 a^2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{4 \sqrt {x} \sqrt [4]{-b+a x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}{x^{4}-a \,x^{2}+b}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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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} - 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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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 6.42, size = 184, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \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 ) + \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \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 ) + \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \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 ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \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 ) \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^2\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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