Optimal. Leaf size=725 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{11/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{2 \sqrt {2} b^{11/8}}-\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{11/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}+\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{2 \sqrt {2} b^{11/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{11/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{11/8}}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}{b \left (-a x \sqrt {a^2 x^2-b}-a^2 x^2+b\right )} \]
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Rubi [A] time = 0.91, antiderivative size = 746, normalized size of antiderivative = 1.03, number of steps used = 31, number of rules used = 14, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2120, 466, 471, 584, 301, 211, 1165, 628, 1162, 617, 204, 212, 206, 203} \begin {gather*} -\frac {\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 )}{4 \sqrt {2} b^{11/8}}+\frac {\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 )}{4 \sqrt {2} b^{11/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} b^{11/8}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}+1\right )}{2 \sqrt {2} b^{11/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}+\frac {2 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}{b \left (b-\left (\sqrt {a^2 x^2-b}+a x\right )^2\right )}-\frac {\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} (-b)^{11/8}}+\frac {\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} (-b)^{11/8}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}+\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{(-b)^{11/8}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 301
Rule 466
Rule 471
Rule 584
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2120
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x \left (-b+a^2 x^2\right )^{3/2}} \, dx &=8 \operatorname {Subst}\left (\int \frac {x^{9/4}}{\left (-b+x^2\right )^2 \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=32 \operatorname {Subst}\left (\int \frac {x^{12}}{\left (-b+x^8\right )^2 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {x^4 \left (5 b-3 x^8\right )}{\left (-b+x^8\right ) \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {x^4}{-b+x^8}-\frac {4 x^4}{b+x^8}\right ) \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {x^4}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}-\frac {8 \operatorname {Subst}\left (\int \frac {x^4}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{5/4}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{5/4}}+\frac {2 \operatorname {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 )}{(-b)^{5/4}}+\frac {2 \operatorname {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 )}{(-b)^{5/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 b^{5/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{2 b^{5/4}}+\frac {\operatorname {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 )}{2 b^{5/4}}+\frac {\operatorname {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 )}{2 b^{5/4}}\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\operatorname {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)^{11/8}}-\frac {\operatorname {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)^{11/8}}+\frac {\operatorname {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)^{5/4}}+\frac {\operatorname {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)^{5/4}}-\frac {\operatorname {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 )}{4 \sqrt {2} b^{11/8}}-\frac {\operatorname {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 )}{4 \sqrt {2} b^{11/8}}+\frac {\operatorname {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 )}{4 b^{5/4}}+\frac {\operatorname {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 )}{4 b^{5/4}}\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\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} (-b)^{11/8}}+\frac {\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} (-b)^{11/8}}-\frac {\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 )}{4 \sqrt {2} b^{11/8}}+\frac {\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 )}{4 \sqrt {2} b^{11/8}}+\frac {\sqrt {2} \operatorname {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 )}{(-b)^{11/8}}-\frac {\sqrt {2} \operatorname {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 )}{(-b)^{11/8}}+\frac {\operatorname {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 )}{2 \sqrt {2} b^{11/8}}-\frac {\operatorname {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 )}{2 \sqrt {2} b^{11/8}}\\ &=\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}+\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} b^{11/8}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 \sqrt {2} b^{11/8}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{11/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\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} (-b)^{11/8}}+\frac {\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} (-b)^{11/8}}-\frac {\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 )}{4 \sqrt {2} b^{11/8}}+\frac {\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 )}{4 \sqrt {2} b^{11/8}}\\ \end {align*}
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Mathematica [B] time = 3.95, size = 2000, normalized size = 2.76
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 3.23, size = 698, normalized size = 0.96 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{2 \sqrt {2} b^{11/8}}+\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{11/8}}-\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{11/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}\right )}{2 b^{11/8}}+\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{2 \sqrt {2} b^{11/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}+\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{b}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{11/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{b}}+\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{b}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{11/8}}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}{b \left (-a x \sqrt {a^2 x^2-b}-a^2 x^2+b\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 50, normalized size = 0.07 \begin {gather*} -\frac {\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{a^{2} b x^{2} - b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x \left (a^{2} x^{2}-b \right )^{\frac {3}{2}}}\, dx \end {gather*}
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 {{\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{{\left (a^{2} x^{2} - b\right )}^{\frac {3}{2}} x}\,{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 = -0.00 \begin {gather*} \int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}{x\,{\left (a^2\,x^2-b\right )}^{3/2}} \,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 {\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}}{x \left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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