Optimal. Leaf size=279 \[ -\frac{221 c^2}{16 b^5 \sqrt{x}}-\frac{221 c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{21/4}}+\frac{221 c}{80 b^4 x^{5/2}}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}-\frac{221}{144 b^3 x^{9/2}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.267987, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1584, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{221 c^2}{16 b^5 \sqrt{x}}-\frac{221 c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{21/4}}+\frac{221 c}{80 b^4 x^{5/2}}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}-\frac{221}{144 b^3 x^{9/2}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 290
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{1}{x^{11/2} \left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17 \int \frac{1}{x^{11/2} \left (b+c x^2\right )^2} \, dx}{8 b}\\ &=\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}+\frac{221 \int \frac{1}{x^{11/2} \left (b+c x^2\right )} \, dx}{32 b^2}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}-\frac{(221 c) \int \frac{1}{x^{7/2} \left (b+c x^2\right )} \, dx}{32 b^3}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{221 c}{80 b^4 x^{5/2}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}+\frac{\left (221 c^2\right ) \int \frac{1}{x^{3/2} \left (b+c x^2\right )} \, dx}{32 b^4}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}-\frac{\left (221 c^3\right ) \int \frac{\sqrt{x}}{b+c x^2} \, dx}{32 b^5}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}-\frac{\left (221 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{16 b^5}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}+\frac{\left (221 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^5}-\frac{\left (221 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^5}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}-\frac{\left (221 c^2\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^5}-\frac{\left (221 c^2\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^5}-\frac{\left (221 c^{9/4}\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^{21/4}}-\frac{\left (221 c^{9/4}\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^{21/4}}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}-\frac{221 c^{9/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}-\frac{\left (221 c^{9/4}\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^{21/4}}+\frac{\left (221 c^{9/4}\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^{21/4}}\\ &=-\frac{221}{144 b^3 x^{9/2}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}+\frac{221 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^{9/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^{9/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}\\ \end{align*}
Mathematica [C] time = 0.0080898, size = 29, normalized size = 0.1 \[ -\frac{2 \, _2F_1\left (-\frac{9}{4},3;-\frac{5}{4};-\frac{c x^2}{b}\right )}{9 b^3 x^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 209, normalized size = 0.8 \begin{align*} -{\frac{2}{9\,{b}^{3}}{x}^{-{\frac{9}{2}}}}-12\,{\frac{{c}^{2}}{{b}^{5}\sqrt{x}}}+{\frac{6\,c}{5\,{b}^{4}}{x}^{-{\frac{5}{2}}}}-{\frac{29\,{c}^{4}}{16\,{b}^{5} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{33\,{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{221\,{c}^{2}\sqrt{2}}{128\,{b}^{5}}\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{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{221\,{c}^{2}\sqrt{2}}{64\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{221\,{c}^{2}\sqrt{2}}{64\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \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 [A] time = 1.42449, size = 828, normalized size = 2.97 \begin{align*} \frac{39780 \,{\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}} \arctan \left (-\frac{10793861 \, b^{5} c^{7} \sqrt{x} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}} - \sqrt{-116507435287321 \, b^{11} c^{9} \sqrt{-\frac{c^{9}}{b^{21}}} + 116507435287321 \, c^{14} x} b^{5} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}}}{10793861 \, c^{9}}\right ) - 9945 \,{\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}} \log \left (10793861 \, b^{16} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{3}{4}} + 10793861 \, c^{7} \sqrt{x}\right ) + 9945 \,{\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}} \log \left (-10793861 \, b^{16} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{3}{4}} + 10793861 \, c^{7} \sqrt{x}\right ) - 4 \,{\left (9945 \, c^{4} x^{8} + 17901 \, b c^{3} x^{6} + 7072 \, b^{2} c^{2} x^{4} - 544 \, b^{3} c x^{2} + 160 \, b^{4}\right )} \sqrt{x}}{2880 \,{\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )}} \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 = 1.16819, size = 312, normalized size = 1.12 \begin{align*} -\frac{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \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{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \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{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6}} - \frac{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6}} - \frac{29 \, c^{4} x^{\frac{7}{2}} + 33 \, b c^{3} x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{5}} - \frac{2 \,{\left (270 \, c^{2} x^{4} - 27 \, b c x^{2} + 5 \, b^{2}\right )}}{45 \, b^{5} x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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