Optimal. Leaf size=540 \[ \frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )+\frac {112 a^6 x^6-1652 a^4 b x^4+1034 a^2 b^2 x^2+\sqrt {a^2 x^2-b} \left (112 a^5 x^5-1596 a^3 b x^3+250 a b^2 x\right )+63 b^3}{63 x \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}} \]
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Rubi [A] time = 0.70, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2120, 466, 468, 570, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {11 a b^2}{28 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/4}}+\frac {a \left (b-\left (\sqrt {a^2 x^2-b}+a x\right )^2\right )^3}{4 \left (\sqrt {a^2 x^2-b}+a x\right )^{7/4} \left (\left (\sqrt {a^2 x^2-b}+a x\right )^2+b\right )}+\frac {13}{36} a \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}-7 a b \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\frac {a (-b)^{9/8} \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 )}{4 \sqrt {2}}-\frac {a (-b)^{9/8} \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 )}{4 \sqrt {2}}-\frac {1}{2} a (-b)^{9/8} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}}-\frac {a (-b)^{9/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{2 \sqrt {2}}-\frac {1}{2} a (-b)^{9/8} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 214
Rule 466
Rule 468
Rule 570
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2120
Rubi steps
\begin {align*} \int \frac {\left (-b+a^2 x^2\right )^{3/2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{x^2} \, dx &=\frac {1}{4} a \operatorname {Subst}\left (\int \frac {\left (-b+x^2\right )^4}{x^{11/4} \left (b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=a \operatorname {Subst}\left (\int \frac {\left (-b+x^8\right )^4}{x^8 \left (b+x^8\right )^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )\\ &=\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {a \operatorname {Subst}\left (\int \frac {\left (-b+x^8\right )^2 \left (-22 b^2-26 b x^8\right )}{x^8 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{8 b}\\ &=\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {a \operatorname {Subst}\left (\int \left (56 b^2-\frac {22 b^3}{x^8}-26 b x^8-\frac {16 b^3}{b+x^8}\right ) \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{8 b}\\ &=-\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\left (a (-b)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\left (a (-b)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )\\ &=-\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} \left (a (-b)^{5/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{2} \left (a (-b)^{5/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )-\frac {1}{2} \left (a (-b)^{5/4}\right ) \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 )-\frac {1}{2} \left (a (-b)^{5/4}\right ) \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 )\\ &=-\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} a (-b)^{9/8} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {1}{2} a (-b)^{9/8} \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {\left (a (-b)^{9/8}\right ) \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}}+\frac {\left (a (-b)^{9/8}\right ) \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}}-\frac {1}{4} \left (a (-b)^{5/4}\right ) \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 )-\frac {1}{4} \left (a (-b)^{5/4}\right ) \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 )\\ &=-\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} a (-b)^{9/8} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {1}{2} a (-b)^{9/8} \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \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 )}{4 \sqrt {2}}-\frac {a (-b)^{9/8} \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 )}{4 \sqrt {2}}-\frac {\left (a (-b)^{9/8}\right ) \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}}+\frac {\left (a (-b)^{9/8}\right ) \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}}\\ &=-\frac {11 a b^2}{28 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}-7 a b \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\frac {13}{36} a \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}+\frac {a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}{4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4} \left (b+\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {1}{2} a (-b)^{9/8} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}}-\frac {a (-b)^{9/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{2 \sqrt {2}}-\frac {1}{2} a (-b)^{9/8} \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\frac {a (-b)^{9/8} \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 )}{4 \sqrt {2}}-\frac {a (-b)^{9/8} \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 )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 3.90, size = 226, normalized size = 0.42 \begin {gather*} \frac {\left (a^2 x^2-b\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (56 a^5 x^5-812 a^3 b x^3-63 b^2 \sqrt {a^2 x^2-b}+126 a b x \left (2 a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right ) \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^2}{b}\right )-784 a^2 b x^2 \sqrt {a^2 x^2-b}+56 a^4 x^4 \sqrt {a^2 x^2-b}+313 a b^2 x\right )}{63 x \left (b-a x \left (\sqrt {a^2 x^2-b}+a x\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.03, size = 514, normalized size = 0.95 \begin {gather*} \frac {63 b^3+1034 a^2 b^2 x^2-1652 a^4 b x^4+112 a^6 x^6+\sqrt {-b+a^2 x^2} \left (250 a b^2 x-1596 a^3 b x^3+112 a^5 x^5\right )}{63 x \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}-\frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \tan ^{-1}\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 )-\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \tan ^{-1}\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 )+\frac {1}{4} \sqrt {2+\sqrt {2}} a b^{9/8} \tanh ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a b^{9/8} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 770, normalized size = 1.43 \begin {gather*} \frac {252 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \arctan \left (-\frac {a^{8} b^{9} + \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} - \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (-a^{8} b^{9}\right )^{\frac {1}{4}}}}{a^{8} b^{9}}\right ) + 252 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \arctan \left (\frac {a^{8} b^{9} - \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {7}{8}} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} + \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (-a^{8} b^{9}\right )^{\frac {1}{4}}}}{a^{8} b^{9}}\right ) + 63 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} + 4 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + 4 \, \left (-a^{8} b^{9}\right )^{\frac {1}{4}}\right ) - 63 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} - 4 \, \sqrt {2} \left (-a^{8} b^{9}\right )^{\frac {1}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + 4 \, \left (-a^{8} b^{9}\right )^{\frac {1}{4}}\right ) + 504 \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \arctan \left (-\frac {\left (-a^{8} b^{9}\right )^{\frac {7}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \left (-a^{8} b^{9}\right )^{\frac {7}{8}} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} a^{2} b^{2} + \left (-a^{8} b^{9}\right )^{\frac {1}{4}}}}{a^{8} b^{9}}\right ) + 126 \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b + \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) - 126 \, \left (-a^{8} b^{9}\right )^{\frac {1}{8}} x \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} a b - \left (-a^{8} b^{9}\right )^{\frac {1}{8}}\right ) - 8 \, {\left (4 \, a^{3} x^{3} + 439 \, a b x - {\left (32 \, a^{2} x^{2} + 63 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{504 \, x} \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] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} x^{2}-b \right )^{\frac {3}{2}} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x^{2}}\, dx\]
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*} \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^{2}}\,{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}\,{\left (a^2\,x^2-b\right )}^{3/2}}{x^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}} \left (a^{2} x^{2} - b\right )^{\frac {3}{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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