Optimal. Leaf size=342 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {-\text {$\#$1}^4 a^2 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a^2 \log (x)-\text {$\#$1}^4 a b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\text {$\#$1}^4 b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 a b \log (x)+\text {$\#$1}^4 b \log (x)+a^3 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+a^2 b \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-a^3 \log (x)-a^2 b \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{4 a}+\frac {\left (4 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{4 a^{7/4}}+\frac {\left (-4 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{4 a^{7/4}}+\frac {x \sqrt [4]{a x^4-b x^2}}{2 a} \]
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Rubi [A] time = 1.29, antiderivative size = 345, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 11, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {2056, 6725, 279, 329, 331, 298, 203, 206, 466, 511, 510} \begin {gather*} \frac {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 a^{7/4} \sqrt {x} \sqrt [4]{a x^2-b}}-\frac {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 a^{7/4} \sqrt {x} \sqrt [4]{a x^2-b}}-\frac {x \left (\frac {\sqrt {-a} a}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4-b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {-a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 a \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {x \left (\frac {(-a)^{3/2}}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4-b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {-a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 a \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {x \sqrt [4]{a x^4-b x^2}}{2 a} \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 6725
Rubi steps
\begin {align*} \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}}{b+a 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^2+x^4\right )}{b+a x^4} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {\sqrt [4]{-b x^2+a x^4} \int \left (\frac {\sqrt {x} \sqrt [4]{-b+a x^2}}{a}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \left ((1+a) b+a^2 x^2\right )}{a \left (b+a x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {\sqrt [4]{-b x^2+a x^4} \int \sqrt {x} \sqrt [4]{-b+a x^2} \, dx}{a \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 ((1+a) b+a^2 x^2\right )}{b+a x^4} \, dx}{a \sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}-\frac {\sqrt [4]{-b x^2+a x^4} \int \left (-\frac {\sqrt {-a} \left (a^2 \sqrt {b}+\sqrt {-a} (1+a) b\right ) \sqrt {x} \sqrt [4]{-b+a x^2}}{2 a \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^2\right )}+\frac {\sqrt {-a} \left (a^2 \sqrt {b}-\sqrt {-a} (1+a) b\right ) \sqrt {x} \sqrt [4]{-b+a x^2}}{2 a \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^2\right )}\right ) \, dx}{a \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (b \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (-b+a x^2\right )^{3/4}} \, dx}{4 a \sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-a} x^2} \, dx}{2 a \sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2}}{\sqrt {b}-\sqrt {-a} x^2} \, dx}{2 a \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (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 a \sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+a x^4}}{\sqrt {b}+\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+a x^4}}{\sqrt {b}-\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (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 a \sqrt {x} \sqrt [4]{-b+a x^2}}\\ &=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}-\frac {\left (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 a^{3/2} \sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (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 a^{3/2} \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {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}}}{\sqrt {b}+\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {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}}}{\sqrt {b}-\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}\\ &=\frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}-\frac {\left (1+\frac {1}{a}+\frac {\sqrt {-a}}{\sqrt {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 {\sqrt {-a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {\left (1+\frac {1}{a}-\frac {\sqrt {-a}}{\sqrt {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 {\sqrt {-a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {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 a^{7/4} \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {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 a^{7/4} \sqrt {x} \sqrt [4]{-b+a x^2}}\\ \end {align*}
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Mathematica [B] time = 2.49, size = 756, normalized size = 2.21 \begin {gather*} \frac {x \sqrt [4]{a x^4-b x^2} \left (8-\frac {\left (\frac {b}{x^2}-a\right )^{3/4} \left (\frac {8 \left (a^2+a b+b\right ) \left (\sqrt [4]{\sqrt {-a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )-\sqrt [4]{\sqrt {-a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )+\sqrt [4]{\sqrt {-a}-\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )-\sqrt [4]{\sqrt {-a}+\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )\right )}{(-a)^{3/8} \sqrt {b}}+\frac {\sqrt {2} \left (4 a^2+b\right ) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{\frac {b}{x^2}-a}+\sqrt {\frac {b}{x^2}-a}+\sqrt {a}\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{\frac {b}{x^2}-a}+\sqrt {\frac {b}{x^2}-a}+\sqrt {a}\right )\right )}{a^{3/4}}+\frac {2 \sqrt {2} \left (4 a^2+b\right ) \left (\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{a}}\right )-\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{a}}+1\right )\right )}{a^{3/4}}+\frac {8 (-a)^{9/8} (a+b) \left (\left (\sqrt {-a}+\sqrt {b}\right )^{3/4} \left (\tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}-\sqrt {b}}}\right )\right )-\left (\sqrt {-a}-\sqrt {b}\right )^{3/4} \left (\tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [8]{-a} \sqrt [4]{\sqrt {-a}+\sqrt {b}}}\right )\right )\right )}{\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right )^{3/4} \left (\sqrt {-a}+\sqrt {b}\right )^{3/4}}\right )}{a x^2-b}\right )}{16 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 341, normalized size = 1.00 \begin {gather*} \frac {x \sqrt [4]{-b x^2+a x^4}}{2 a}+\frac {\left (4 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\left (-4 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{7/4}}-\frac {\text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a^3 \log (x)+a^2 b \log (x)-a^3 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^2 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4+a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{4 a} \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 {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}}{a x^{4} + b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-a \,x^{2}-b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}{a \,x^{4}+b}\, dx\]
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 (x^{4} - a x^{2} - b\right )}}{a x^{4} + 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 (a\,x^4-b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )}{a\,x^4+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^{2} - b + x^{4}\right )}{a x^{4} + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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