Optimal. Leaf size=114 \[ -\sqrt {2} \text {ArcTan}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^3}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^3}}{x^2+\sqrt {-b x^2+a x^3}}\right ) \]
[Out]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 11.76, 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 = {2081,
6860, 108, 107, 504, 1231, 226, 1721} \begin {gather*} -\frac {\left (a+\sqrt {a^2+4 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}};2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {b} \sqrt {\sqrt {a^2+4 b}-a} \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\sqrt {a^2+4 b}-a} \sqrt [4]{a x-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}+\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}};2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) \sqrt [4]{a x^3-b x^2}}+\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \Pi \left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}};2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt [4]{a x-b} \sqrt {\frac {a x}{\left (\sqrt {b}+\sqrt {a x-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x-b}\right ) \Pi \left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}};2 \text {ArcTan}\left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^3-b x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 107
Rule 108
Rule 226
Rule 504
Rule 1231
Rule 1721
Rule 2081
Rule 6860
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.27, size = 116, normalized size = 1.02 \begin {gather*} \frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x} \left (-\text {ArcTan}\left (\frac {-x+\sqrt {-b+a x}}{\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x}}{x+\sqrt {-b+a x}}\right )\right )}{\sqrt [4]{x^2 (-b+a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, 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*} \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 {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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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