Optimal. Leaf size=76 \[ \frac {\tanh ^{-1}\left (\frac {x \left (a x^2-b\right )^{3/4}}{\sqrt {2} \left (b-a x^2\right )}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {x \left (a x^2-b\right )^{3/4}}{\sqrt {2} \left (b-a x^2\right )}\right )}{\sqrt {2}} \]
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Rubi [C] time = 26.56, antiderivative size = 2421, normalized size of antiderivative = 31.86, number of steps used = 24, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {1692, 234, 220, 401, 108, 409, 1217, 1707}
result too large to display
Warning: Unable to verify antiderivative.
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Rule 108
Rule 220
Rule 234
Rule 401
Rule 409
Rule 1217
Rule 1692
Rule 1707
Rubi steps
\begin {align*} \int \frac {x^2 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )} \, dx &=\int \left (\frac {a}{\left (-b+a x^2\right )^{3/4}}-\frac {2 \left (2 a b-\left (2 a^2-b\right ) x^2\right )}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )}\right ) \, dx\\ &=-\left (2 \int \frac {2 a b-\left (2 a^2-b\right ) x^2}{\left (-b+a x^2\right )^{3/4} \left (4 b-4 a x^2+x^4\right )} \, dx\right )+a \int \frac {1}{\left (-b+a x^2\right )^{3/4}} \, dx\\ &=-\left (2 \int \left (\frac {-2 a^2-2 a \sqrt {a^2-b}+b}{\left (-4 a-4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}}+\frac {-2 a^2+2 a \sqrt {a^2-b}+b}{\left (-4 a+4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}}\right ) \, dx\right )+\frac {\left (2 \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}\\ &=\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} x}+\left (2 \left (2 a^2-2 a \sqrt {a^2-b}-b\right )\right ) \int \frac {1}{\left (-4 a+4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx+\left (2 \left (2 a^2+2 a \sqrt {a^2-b}-b\right )\right ) \int \frac {1}{\left (-4 a-4 \sqrt {a^2-b}+2 x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx\\ &=\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} x}+\frac {\left (\left (2 a^2-2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {a x}{b}} \left (-4 a+4 \sqrt {a^2-b}+2 x\right ) (-b+a x)^{3/4}} \, dx,x,x^2\right )}{x}+\frac {\left (\left (2 a^2+2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {a x}{b}} \left (-4 a-4 \sqrt {a^2-b}+2 x\right ) (-b+a x)^{3/4}} \, dx,x,x^2\right )}{x}\\ &=\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} x}-\frac {\left (4 \left (2 a^2-2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a \left (-4 a+4 \sqrt {a^2-b}\right )-2 b-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\left (4 \left (2 a^2+2 a \sqrt {a^2-b}-b\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a \left (-4 a-4 \sqrt {a^2-b}\right )-2 b-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}\\ &=\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} x}-\frac {\sqrt {\frac {a x^2}{b}} \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\sqrt {\frac {a x^2}{b}} \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\sqrt {\frac {a x^2}{b}} \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}-\frac {\sqrt {\frac {a x^2}{b}} \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}\\ &=\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} x}-\frac {\left (\left (\frac {1}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}-\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\left (\frac {1}{2 a^2+2 a \sqrt {a^2-b}-b}-\frac {1}{b}\right ) \sqrt {b} x}+\frac {\left (\left (\frac {1}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}+\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\left (\frac {1}{2 a^2+2 a \sqrt {a^2-b}-b}-\frac {1}{b}\right ) \sqrt {b} x}-\frac {\left (\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1-\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2-a \sqrt {a^2-b}-b\right ) x}+\frac {\left (\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1+\frac {x^2}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2-a \sqrt {a^2-b}-b\right ) x}-\frac {\left (\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1-\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2+a \sqrt {a^2-b}-b\right ) x}+\frac {\left (\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt {b} \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (1+\frac {x^2}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \left (a^2+a \sqrt {a^2-b}-b\right ) x}+\frac {\left (\left (\frac {1}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}-\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {b} \left (\frac {1}{b}+\frac {1}{-2 a^2+2 a \sqrt {a^2-b}+b}\right ) x}-\frac {\left (\left (\frac {1}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}+\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {b} \left (\frac {1}{b}+\frac {1}{-2 a^2+2 a \sqrt {a^2-b}+b}\right ) x}\\ &=-\frac {\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt [4]{2 a^2-2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {a-\sqrt {a^2-b}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2 a^2-2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt {a-\sqrt {a^2-b}} \left (a^2-a \sqrt {a^2-b}-b\right ) x}-\frac {\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt [4]{2 a^2-2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a+\sqrt {a^2-b}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2 a^2-2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt {-a+\sqrt {a^2-b}} \left (a^2-a \sqrt {a^2-b}-b\right ) x}-\frac {\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt [4]{2 a^2+2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a-\sqrt {a^2-b}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2 a^2+2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt {-a-\sqrt {a^2-b}} \left (a^2+a \sqrt {a^2-b}-b\right ) x}-\frac {\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}-\sqrt {b}\right ) \left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}+\sqrt {b}\right ) \sqrt [4]{2 a^2+2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {a+\sqrt {a^2-b}} \sqrt [4]{-b+a x^2}}{\sqrt [4]{2 a^2+2 a \sqrt {a^2-b}-b} \sqrt {b} \sqrt {\frac {a x^2}{b}}}\right )}{4 \sqrt {2} \sqrt {a} \sqrt {a+\sqrt {a^2-b}} \left (a^2+a \sqrt {a^2-b}-b\right ) x}-\frac {\left (\frac {1}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}-\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \left (\frac {1}{2 a^2+2 a \sqrt {a^2-b}-b}-\frac {1}{b}\right ) b^{3/4} x}+\frac {\left (\frac {1}{\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}}+\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \left (\frac {1}{2 a^2+2 a \sqrt {a^2-b}-b}-\frac {1}{b}\right ) b^{3/4} x}+\frac {\sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} x}+\frac {\left (\frac {1}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}-\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{3/4} \left (\frac {1}{b}+\frac {1}{-2 a^2+2 a \sqrt {a^2-b}+b}\right ) x}-\frac {\left (\frac {1}{\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}}+\frac {1}{\sqrt {b}}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{3/4} \left (\frac {1}{b}+\frac {1}{-2 a^2+2 a \sqrt {a^2-b}+b}\right ) x}+\frac {\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}+\sqrt {b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (-\frac {\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}-\sqrt {b}\right )^2}{4 \sqrt {2 a^2-2 a \sqrt {a^2-b}-b} \sqrt {b}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{8 \left (a^2-a \sqrt {a^2-b}-b\right ) \sqrt [4]{b} x}+\frac {\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}-\sqrt {b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (\frac {\left (\sqrt {2 a^2-2 a \sqrt {a^2-b}-b}+\sqrt {b}\right )^2}{4 \sqrt {2 a^2-2 a \sqrt {a^2-b}-b} \sqrt {b}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{8 \left (a^2-a \sqrt {a^2-b}-b\right ) \sqrt [4]{b} x}+\frac {\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}+\sqrt {b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (-\frac {\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}-\sqrt {b}\right )^2}{4 \sqrt {2 a^2+2 a \sqrt {a^2-b}-b} \sqrt {b}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{8 \left (a^2+a \sqrt {a^2-b}-b\right ) \sqrt [4]{b} x}+\frac {\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}-\sqrt {b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {-b+a x^2}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^2}\right ) \Pi \left (\frac {\left (\sqrt {2 a^2+2 a \sqrt {a^2-b}-b}+\sqrt {b}\right )^2}{4 \sqrt {2 a^2+2 a \sqrt {a^2-b}-b} \sqrt {b}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{8 \left (a^2+a \sqrt {a^2-b}-b\right ) \sqrt [4]{b} x}\\ \end {align*}
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Mathematica [C] time = 0.60, size = 360, normalized size = 4.74 \begin {gather*} -\frac {\sqrt [4]{-b} \sqrt {\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-b} a+b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right )\right |-1\right )+\sqrt [4]{-b} \sqrt {\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-b} a+b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right )\right |-1\right )+\sqrt [4]{-b} \sqrt {\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-b} a+b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right )\right |-1\right )+\sqrt [4]{-b} \sqrt {\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-b} a+b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right )\right |-1\right )-\frac {a x^2 \left (1-\frac {a x^2}{b}\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {a x^2}{b}\right )}{\left (a x^2-b\right )^{3/4}}}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.13, size = 76, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {x \left (-b+a x^2\right )^{3/4}}{\sqrt {2} \left (b-a x^2\right )}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x \left (-b+a x^2\right )^{3/4}}{\sqrt {2} \left (b-a x^2\right )}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 113, normalized size = 1.49 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {x^{4} - 2 \, \sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} x^{3} + 4 \, a x^{2} + 4 \, \sqrt {a x^{2} - b} x^{2} - 4 \, \sqrt {2} {\left (a x^{2} - b\right )}^{\frac {3}{4}} x - 4 \, b}{x^{4} - 4 \, a x^{2} + 4 \, b}\right ) \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^{2} - 2 \, b\right )} x^{2}}{{\left (x^{4} - 4 \, a x^{2} + 4 \, b\right )} {\left (a x^{2} - b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a \,x^{2}-2 b \right )}{\left (a \,x^{2}-b \right )^{\frac {3}{4}} \left (x^{4}-4 a \,x^{2}+4 b \right )}\, dx\]
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^{2} - 2 \, b\right )} x^{2}}{{\left (x^{4} - 4 \, a x^{2} + 4 \, b\right )} {\left (a x^{2} - b\right )}^{\frac {3}{4}}}\,{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 {x^2\,\left (2\,b-a\,x^2\right )}{{\left (a\,x^2-b\right )}^{3/4}\,\left (x^4-4\,a\,x^2+4\,b\right )} \,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 {x^{2} \left (a x^{2} - 2 b\right )}{\left (a x^{2} - b\right )^{\frac {3}{4}} \left (- 4 a x^{2} + 4 b + x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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