Integrand size = 40, antiderivative size = 62 \[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {2 b}{5 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{3 a^2} \]
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Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2145, 14} \[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {2 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{3 a^2}-\frac {2 b}{5 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}} \]
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Rule 14
Rule 2145
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b+x^2}{x^{9/4}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^2} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b}{x^{9/4}}+\frac {1}{\sqrt [4]{x}}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^2} \\ & = -\frac {2 b}{5 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{3 a^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {4 \left (-4 b+5 a x \left (a x+\sqrt {-b+a^2 x^2}\right )\right )}{15 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}} \]
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\[\int \frac {x}{\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}d x\]
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none
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {4 \, {\left (3 \, a^{2} x^{2} - 3 \, \sqrt {a^{2} x^{2} - b} a x - 4 \, b\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{15 \, a^{2} b} \]
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\[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {x}{\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \]
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\[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int { \frac {x}{\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {x}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,\sqrt {a^2\,x^2-b}} \,d x \]
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