Optimal. Leaf size=114 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b x^2}}{\sqrt {a x^3-b x^2}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b x^2}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b x^2}}\right ) \]
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Rubi [C] time = 14.51, antiderivative size = 2624, normalized size of antiderivative = 23.02, number of steps used = 21, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2056, 6728, 107, 106, 490, 1217, 220, 1707}
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Warning: Unable to verify antiderivative.
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Rule 106
Rule 107
Rule 220
Rule 490
Rule 1217
Rule 1707
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {-2 b+a x}{\left (-b+a x+x^2\right ) \sqrt [4]{-b x^2+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {-2 b+a x}{\sqrt {x} \sqrt [4]{-b+a x} \left (-b+a x+x^2\right )} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x}\right ) \int \left (\frac {a-\sqrt {a^2+4 b}}{\sqrt {x} \left (a-\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}}+\frac {a+\sqrt {a^2+4 b}}{\sqrt {x} \left (a+\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}}\right ) \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a-\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a+\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {\frac {a x}{b}} \left (a-\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {\frac {a x}{b}} \left (a+\sqrt {a^2+4 b}+2 x\right ) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (4 \left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 b-a \left (a-\sqrt {a^2+4 b}\right )-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}-\frac {\left (4 \left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 b-a \left (a+\sqrt {a^2+4 b}\right )-2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \sqrt {b} \left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \sqrt {b} \left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {2} \sqrt {b} \left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {2} \sqrt {b} \left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}\\ &=\frac {2^{3/4} \sqrt {b} \left (a^3+4 a b+\left (a^2+2 b\right ) \sqrt {a^2+4 b}\right ) \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {b} \sqrt {-a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {-a+\sqrt {a^2+4 b}} \sqrt [4]{-b+a x}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{-b x^2+a x^3}}+\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2 \sqrt [4]{-b+a x} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {-b+a x}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{-b x^2+a x^3}}\\ \end {align*}
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Mathematica [C] time = 7.43, size = 611, normalized size = 5.36 \begin {gather*} \frac {i \sqrt {2} a \sqrt {\frac {a x}{a x-b}} (a x-b)^{3/4} \left (\left (-a^2+\sqrt {a^4+4 a^2 b}-4 b\right ) \sqrt {-\frac {a^2+\sqrt {a^4+4 a^2 b}+2 b}{b^2}} \Pi \left (-\frac {i \sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2-2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )+\left (a^2-\sqrt {a^4+4 a^2 b}+4 b\right ) \sqrt {-\frac {a^2+\sqrt {a^4+4 a^2 b}+2 b}{b^2}} \Pi \left (\frac {i \sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2-2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )+\sqrt {\frac {-a^2+\sqrt {a^4+4 a^2 b}-2 b}{b^2}} \left (a^2+\sqrt {a^4+4 a^2 b}+4 b\right ) \left (\Pi \left (-\frac {i \sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2+2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2+2 b+\sqrt {a^4+4 b a^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {i \sqrt {b}}}{\sqrt [4]{a x-b}}\right )\right |-1\right )\right )\right )}{\left (i \sqrt {b}\right )^{5/2} \sqrt {a^4+4 a^2 b} \sqrt {\frac {-a^2+\sqrt {a^4+4 a^2 b}-2 b}{b^2}} \sqrt {-\frac {a^2+\sqrt {a^4+4 a^2 b}+2 b}{b^2}} \sqrt [4]{x^2 (a x-b)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 114, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^3}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^3}}{x^2+\sqrt {-b x^2+a x^3}}\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {a x -2 b}{\left (a x +x^{2}-b \right ) \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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 {a x - 2 \, b}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} - b\right )}}\,{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 {2\,b-a\,x}{{\left (a\,x^3-b\,x^2\right )}^{1/4}\,\left (x^2+a\,x-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 {a x - 2 b}{\sqrt [4]{x^{2} \left (a x - b\right )} \left (a x - b + x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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