Integrand size = 42, antiderivative size = 526 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {4 \sqrt {-b+a^2 x^2} \left (418 a b^2 x-561 a^3 b x^3+132 a^5 x^5\right )+4 \left (-152 b^3+682 a^2 b^2 x^2-627 a^4 b x^4+132 a^6 x^6\right )}{429 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\sqrt {2+\sqrt {2}} b^{13/8} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2-\sqrt {2}} b^{13/8} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2+\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )-\sqrt {2-\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \]
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Time = 0.40 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.89, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2145, 477, 472, 307, 217, 1179, 642, 1176, 631, 210, 218, 212, 209} \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=-2 (-b)^{13/8} \arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )-\sqrt {2} (-b)^{13/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )+\sqrt {2} (-b)^{13/8} \arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )-2 (-b)^{13/8} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )-\frac {b^3}{22 \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}}+\frac {5 b^2}{6 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}+\frac {1}{26} \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4}-\frac {1}{2} b \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}-\frac {(-b)^{13/8} \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2}}+\frac {(-b)^{13/8} \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2}} \]
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Rule 209
Rule 210
Rule 212
Rule 217
Rule 218
Rule 307
Rule 472
Rule 477
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2145
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \text {Subst}\left (\int \frac {\left (-b+x^2\right )^4}{x^{15/4} \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {\left (-b+x^8\right )^4}{x^{12} \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {b^3}{x^{12}}-\frac {5 b^2}{x^4}-5 b x^4+x^{12}+\frac {16 b^2 x^4}{b+x^8}\right ) \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^3}{22 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\frac {5 b^2}{6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}-\frac {1}{2} b \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+\frac {1}{26} \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}+\left (8 b^2\right ) \text {Subst}\left (\int \frac {x^4}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^3}{22 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\frac {5 b^2}{6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}-\frac {1}{2} b \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+\frac {1}{26} \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}-\left (4 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )+\left (4 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^3}{22 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\frac {5 b^2}{6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}-\frac {1}{2} b \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+\frac {1}{26} \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}-\left (2 (-b)^{7/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\left (2 (-b)^{7/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )+\left (2 (-b)^{7/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-b}-x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )+\left (2 (-b)^{7/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{-b}+x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^3}{22 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\frac {5 b^2}{6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}-\frac {1}{2} b \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+\frac {1}{26} \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}-2 (-b)^{13/8} \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-2 (-b)^{13/8} \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {(-b)^{13/8} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}+2 x}{-\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {(-b)^{13/8} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}-2 x}{-\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+(-b)^{7/4} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )+(-b)^{7/4} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^3}{22 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\frac {5 b^2}{6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}-\frac {1}{2} b \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+\frac {1}{26} \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}-2 (-b)^{13/8} \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-2 (-b)^{13/8} \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {(-b)^{13/8} \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+\frac {(-b)^{13/8} \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+\left (\sqrt {2} (-b)^{13/8}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\left (\sqrt {2} (-b)^{13/8}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right ) \\ & = -\frac {b^3}{22 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\frac {5 b^2}{6 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}-\frac {1}{2} b \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}+\frac {1}{26} \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}-2 (-b)^{13/8} \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\sqrt {2} (-b)^{13/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\sqrt {2} (-b)^{13/8} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-2 (-b)^{13/8} \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {(-b)^{13/8} \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+\frac {(-b)^{13/8} \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 1.43 (sec) , antiderivative size = 469, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {4 \left (-152 b^3+682 a^2 b^2 x^2-627 a^4 b x^4+132 a^6 x^6+11 \sqrt {-b+a^2 x^2} \left (38 a b^2 x-51 a^3 b x^3+12 a^5 x^5\right )\right )}{429 \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}+\sqrt {2+\sqrt {2}} b^{13/8} \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2-\sqrt {2}} b^{13/8} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )-\sqrt {2-\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+\sqrt {2+\sqrt {2}} b^{13/8} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \]
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\[\int \frac {\left (a^{2} x^{2}-b \right )^{\frac {3}{2}} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x}d x\]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + \left (i + 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \left (i - 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + \left (i - 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \left (i + 1\right ) \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}}\right ) - \frac {4}{429} \, {\left (3 \, a^{3} x^{3} - 38 \, a b x - 4 \, {\left (9 \, a^{2} x^{2} - 38 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} - \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + \left (-b^{13}\right )^{\frac {5}{8}}\right ) - i \, \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + i \, \left (-b^{13}\right )^{\frac {5}{8}}\right ) + i \, \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - i \, \left (-b^{13}\right )^{\frac {5}{8}}\right ) + \left (-b^{13}\right )^{\frac {1}{8}} \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \left (-b^{13}\right )^{\frac {5}{8}}\right ) \]
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\[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\int \frac {\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}{x}\, dx \]
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\[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,{\left (a^2\,x^2-b\right )}^{3/2}}{x} \,d x \]
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