Optimal. Leaf size=526 \[ \sqrt {2+\sqrt {2}} b^{13/8} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )+\sqrt {2-\sqrt {2}} b^{13/8} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )+\sqrt {2+\sqrt {2}} b^{13/8} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )-\sqrt {2-\sqrt {2}} b^{13/8} \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )+\frac {4 \left (132 a^6 x^6-627 a^4 b x^4+682 a^2 b^2 x^2-152 b^3\right )+4 \sqrt {a^2 x^2-b} \left (132 a^5 x^5-561 a^3 b x^3+418 a b^2 x\right )}{429 \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}} \]
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Rubi [A] time = 0.66, antiderivative size = 468, normalized size of antiderivative = 0.89, number of steps used = 17, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2120, 466, 461, 301, 211, 1165, 628, 1162, 617, 204, 212, 206, 203} \begin {gather*} -\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}}-2 (-b)^{13/8} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )-\sqrt {2} (-b)^{13/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )+\sqrt {2} (-b)^{13/8} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )-2 (-b)^{13/8} \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 301
Rule 461
Rule 466
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} \, dx &=\frac {1}{8} \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \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 )+\left (2 (-b)^{7/4}\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 {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} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-2 (-b)^{13/8} \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\frac {(-b)^{13/8} \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}}-\frac {(-b)^{13/8} \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)^{7/4} \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)^{7/4} \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 {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} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-2 (-b)^{13/8} \tanh ^{-1}\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 ) \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 )-\left (\sqrt {2} (-b)^{13/8}\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 )\\ &=-\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} \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-\sqrt {2} (-b)^{13/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )+\sqrt {2} (-b)^{13/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )-2 (-b)^{13/8} \tanh ^{-1}\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*}
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Mathematica [B] time = 23.49, size = 14841, normalized size = 28.21 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.78, size = 501, normalized size = 0.95 \begin {gather*} -\sqrt {2+\sqrt {2}} b^{13/8} \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 )+\sqrt {2-\sqrt {2}} b^{13/8} \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 )-\sqrt {2-\sqrt {2}} b^{13/8} \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 )+\sqrt {2+\sqrt {2}} b^{13/8} \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 )+\frac {4 \left (132 a^6 x^6-627 a^4 b x^4+682 a^2 b^2 x^2-152 b^3\right )+4 \sqrt {a^2 x^2-b} \left (132 a^5 x^5-561 a^3 b x^3+418 a b^2 x\right )}{429 \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 683, normalized size = 1.30 \begin {gather*} 2 \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \arctan \left (-\frac {b^{13} + \sqrt {2} \left (-b^{13}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \sqrt {2} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{16} - \left (-b^{13}\right )^{\frac {1}{4}} b^{13} - \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8}} \left (-b^{13}\right )^{\frac {3}{8}}}{b^{13}}\right ) + 2 \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \arctan \left (\frac {b^{13} - \sqrt {2} \left (-b^{13}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} + \sqrt {2} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{16} - \left (-b^{13}\right )^{\frac {1}{4}} b^{13} + \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8}} \left (-b^{13}\right )^{\frac {3}{8}}}{b^{13}}\right ) + \frac {1}{2} \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{16} - 4 \, \left (-b^{13}\right )^{\frac {1}{4}} b^{13} + 4 \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8}\right ) - \frac {1}{2} \, \sqrt {2} \left (-b^{13}\right )^{\frac {1}{8}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{16} - 4 \, \left (-b^{13}\right )^{\frac {1}{4}} b^{13} - 4 \, \sqrt {2} \left (-b^{13}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{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}} - 4 \, \left (-b^{13}\right )^{\frac {1}{8}} \arctan \left (-\frac {\left (-b^{13}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{8} - \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{16} - \left (-b^{13}\right )^{\frac {1}{4}} b^{13}} \left (-b^{13}\right )^{\frac {3}{8}}}{b^{13}}\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 ) + \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 ) \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^{2} x^{2}-b \right )^{\frac {3}{2}} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}{x}\, 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^{2} x^{2} - b\right )}^{\frac {3}{2}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}{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}\,{\left (a^2\,x^2-b\right )}^{3/2}}{x} \,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}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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