Integrand size = 26, antiderivative size = 106 \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}} \]
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Time = 0.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.60, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2081, 108, 107, 504, 1225, 226, 1713, 209, 212} \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\frac {\sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a x-b}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a x-b}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{a x^3-b x^2}} \]
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Rule 107
Rule 108
Rule 209
Rule 212
Rule 226
Rule 504
Rule 1225
Rule 1713
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {x} (-2 b+a x) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}} \\ & = \frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {\frac {a x}{b}} (-2 b+a x) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}} \\ & = -\frac {\left (4 \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {x^2}{\left (a b-a 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 (2 \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}-x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt [4]{-b x^2+a x^3}}+\frac {\left (2 \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}+x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt [4]{-b x^2+a x^3}} \\ & = \frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\left (\sqrt {b}+x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt {b} \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\left (\sqrt {b}-x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt {b} \sqrt [4]{-b x^2+a x^3}} \\ & = -\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-2 x^2} \, dx,x,\frac {\sqrt [4]{-b+a x}}{\sqrt {\frac {a x}{b}}}\right )}{a \sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+2 x^2} \, dx,x,\frac {\sqrt [4]{-b+a x}}{\sqrt {\frac {a x}{b}}}\right )}{a \sqrt [4]{-b x^2+a x^3}} \\ & = \frac {\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-b+a x}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-b+a x}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\frac {\sqrt {x} \sqrt [4]{-b+a x} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x}}{\sqrt {a} \sqrt {x}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x}}{\sqrt {a} \sqrt {x}}\right )\right )}{\sqrt {2} \sqrt {a} b^{3/4} \sqrt [4]{x^2 (-b+a x)}} \]
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\[\int \frac {1}{\left (a x -2 b \right ) \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}}}d x\]
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Timed out. \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x - b\right )} \left (a x - 2 b\right )}\, dx \]
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\[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x - 2 \, b\right )}} \,d x } \]
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\[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x - 2 \, b\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=-\int \frac {1}{\left (2\,b-a\,x\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/4}} \,d x \]
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