Optimal. Leaf size=365 \[ -\frac{15 c^{7/4} (11 b B-19 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}-\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{23/4}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}+\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.320727, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {1584, 457, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{15 c^{7/4} (11 b B-19 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}-\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{23/4}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}+\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 290
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{A+B x^2}{\sqrt{x} \left (b x^2+c x^4\right )^3} \, dx &=\int \frac{A+B x^2}{x^{13/2} \left (b+c x^2\right )^3} \, dx\\ &=-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}+\frac{\left (-\frac{11 b B}{2}+\frac{19 A c}{2}\right ) \int \frac{1}{x^{13/2} \left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}-\frac{(15 (11 b B-19 A c)) \int \frac{1}{x^{13/2} \left (b+c x^2\right )} \, dx}{32 b^2 c}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}+\frac{(15 (11 b B-19 A c)) \int \frac{1}{x^{9/2} \left (b+c x^2\right )} \, dx}{32 b^3}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}-\frac{(15 c (11 b B-19 A c)) \int \frac{1}{x^{5/2} \left (b+c x^2\right )} \, dx}{32 b^4}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}+\frac{\left (15 c^2 (11 b B-19 A c)\right ) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{32 b^5}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}+\frac{\left (15 c^2 (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{16 b^5}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}+\frac{\left (15 c^2 (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{11/2}}+\frac{\left (15 c^2 (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{11/2}}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}+\frac{\left (15 c^{3/2} (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 b^{11/2}}+\frac{\left (15 c^{3/2} (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 b^{11/2}}-\frac{\left (15 c^{7/4} (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} b^{23/4}}-\frac{\left (15 c^{7/4} (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} b^{23/4}}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}-\frac{15 c^{7/4} (11 b B-19 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}+\frac{\left (15 c^{7/4} (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{23/4}}-\frac{\left (15 c^{7/4} (11 b B-19 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{23/4}}\\ &=\frac{15 (11 b B-19 A c)}{176 b^3 c x^{11/2}}-\frac{15 (11 b B-19 A c)}{112 b^4 x^{7/2}}+\frac{5 c (11 b B-19 A c)}{16 b^5 x^{3/2}}-\frac{b B-A c}{4 b c x^{11/2} \left (b+c x^2\right )^2}-\frac{11 b B-19 A c}{16 b^2 c x^{11/2} \left (b+c x^2\right )}-\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{23/4}}-\frac{15 c^{7/4} (11 b B-19 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}+\frac{15 c^{7/4} (11 b B-19 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{23/4}}\\ \end{align*}
Mathematica [A] time = 0.581209, size = 467, normalized size = 1.28 \[ \frac{-\frac{19096 A b^{3/4} c^3 \sqrt{x}}{b+c x^2}-\frac{2464 A b^{7/4} c^3 \sqrt{x}}{\left (b+c x^2\right )^2}-\frac{39424 A b^{3/4} c^2}{x^{3/2}}+\frac{8448 A b^{7/4} c}{x^{7/2}}-\frac{1792 A b^{11/4}}{x^{11/2}}+2310 \sqrt{2} c^{7/4} (19 A c-11 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+2310 \sqrt{2} c^{7/4} (11 b B-19 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+21945 \sqrt{2} A c^{11/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-21945 \sqrt{2} A c^{11/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+\frac{14168 b^{7/4} B c^2 \sqrt{x}}{b+c x^2}+\frac{2464 b^{11/4} B c^2 \sqrt{x}}{\left (b+c x^2\right )^2}+\frac{19712 b^{7/4} B c}{x^{3/2}}-\frac{2816 b^{11/4} B}{x^{7/2}}-12705 \sqrt{2} b B c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+12705 \sqrt{2} b B c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{9856 b^{23/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 420, normalized size = 1.2 \begin{align*} -{\frac{2\,A}{11\,{b}^{3}}{x}^{-{\frac{11}{2}}}}+{\frac{6\,Ac}{7\,{b}^{4}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,{b}^{3}}{x}^{-{\frac{7}{2}}}}-4\,{\frac{A{c}^{2}}{{b}^{5}{x}^{3/2}}}+2\,{\frac{Bc}{{b}^{4}{x}^{3/2}}}-{\frac{31\,{c}^{4}A}{16\,{b}^{5} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{23\,{c}^{3}B}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{35\,A{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}+{\frac{27\,{c}^{2}B}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{285\,{c}^{3}\sqrt{2}A}{64\,{b}^{6}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{285\,{c}^{3}\sqrt{2}A}{64\,{b}^{6}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{285\,{c}^{3}\sqrt{2}A}{128\,{b}^{6}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{165\,{c}^{2}\sqrt{2}B}{64\,{b}^{5}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{165\,{c}^{2}\sqrt{2}B}{64\,{b}^{5}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{165\,{c}^{2}\sqrt{2}B}{128\,{b}^{5}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.5948, size = 2233, normalized size = 6.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39349, size = 474, normalized size = 1.3 \begin{align*} \frac{15 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{1}{4}} B b c - 19 \, \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{6}} + \frac{15 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{1}{4}} B b c - 19 \, \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{6}} + \frac{15 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{1}{4}} B b c - 19 \, \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6}} - \frac{15 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{1}{4}} B b c - 19 \, \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6}} + \frac{23 \, B b c^{3} x^{\frac{5}{2}} - 31 \, A c^{4} x^{\frac{5}{2}} + 27 \, B b^{2} c^{2} \sqrt{x} - 35 \, A b c^{3} \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{5}} + \frac{2 \,{\left (77 \, B b c x^{4} - 154 \, A c^{2} x^{4} - 11 \, B b^{2} x^{2} + 33 \, A b c x^{2} - 7 \, A b^{2}\right )}}{77 \, b^{5} x^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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