Optimal. Leaf size=332 \[ \frac{c^{5/4} (9 b B-13 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}+\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{17/4}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )} \]
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Rubi [A] time = 0.28566, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1584, 457, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{c^{5/4} (9 b B-13 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}+\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{17/4}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )^2} \, dx &=\int \frac{A+B x^2}{x^{11/2} \left (b+c x^2\right )^2} \, dx\\ &=-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac{\left (-\frac{9 b B}{2}+\frac{13 A c}{2}\right ) \int \frac{1}{x^{11/2} \left (b+c x^2\right )} \, dx}{2 b c}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac{(9 b B-13 A c) \int \frac{1}{x^{7/2} \left (b+c x^2\right )} \, dx}{4 b^2}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}-\frac{(c (9 b B-13 A c)) \int \frac{1}{x^{3/2} \left (b+c x^2\right )} \, dx}{4 b^3}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac{\left (c^2 (9 b B-13 A c)\right ) \int \frac{\sqrt{x}}{b+c x^2} \, dx}{4 b^4}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac{\left (c^2 (9 b B-13 A c)\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 b^4}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}-\frac{\left (c^{3/2} (9 b B-13 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 b^4}+\frac{\left (c^{3/2} (9 b B-13 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 b^4}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac{(c (9 b B-13 A c)) \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 )}{8 b^4}+\frac{(c (9 b B-13 A c)) \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 )}{8 b^4}+\frac{\left (c^{5/4} (9 b B-13 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 )}{8 \sqrt{2} b^{17/4}}+\frac{\left (c^{5/4} (9 b B-13 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 )}{8 \sqrt{2} b^{17/4}}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}+\frac{\left (c^{5/4} (9 b B-13 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 )}{4 \sqrt{2} b^{17/4}}-\frac{\left (c^{5/4} (9 b B-13 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 )}{4 \sqrt{2} b^{17/4}}\\ &=\frac{9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac{9 b B-13 A c}{10 b^3 x^{5/2}}+\frac{c (9 b B-13 A c)}{2 b^4 \sqrt{x}}-\frac{b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}-\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}+\frac{c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{17/4}}+\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}-\frac{c^{5/4} (9 b B-13 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{17/4}}\\ \end{align*}
Mathematica [C] time = 0.4141, size = 176, normalized size = 0.53 \[ \frac{2 c^2 x^{3/2} (b B-A c) \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right )}{3 b^5}-\frac{2 (b B-2 A c)}{5 b^3 x^{5/2}}+\frac{2 c (2 b B-3 A c)}{b^4 \sqrt{x}}-\frac{2 A}{9 b^2 x^{9/2}}+\frac{c^{5/4} (2 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b}}\right )}{(-b)^{17/4}}+\frac{c^{5/4} (3 A c-2 b B) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b}}\right )}{(-b)^{17/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 372, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{9\,{b}^{2}}{x}^{-{\frac{9}{2}}}}+{\frac{4\,Ac}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}-6\,{\frac{A{c}^{2}}{{b}^{4}\sqrt{x}}}+4\,{\frac{Bc}{{b}^{3}\sqrt{x}}}-{\frac{A{c}^{3}}{2\,{b}^{4} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{{c}^{2}B}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{13\,{c}^{2}\sqrt{2}A}{16\,{b}^{4}}\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{13\,{c}^{2}\sqrt{2}A}{8\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{13\,{c}^{2}\sqrt{2}A}{8\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}B}{16\,{b}^{3}}\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{9\,c\sqrt{2}B}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}B}{8\,{b}^{3}}\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 [B] time = 2.63345, size = 2475, normalized size = 7.45 \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 = 1.37606, size = 443, normalized size = 1.33 \begin{align*} \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\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 )}{8 \, b^{5} c} + \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\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 )}{8 \, b^{5} c} - \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{5} c} + \frac{\sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{5} c} + \frac{B b c^{2} x^{\frac{3}{2}} - A c^{3} x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b^{4}} + \frac{2 \,{\left (90 \, B b c x^{4} - 135 \, A c^{2} x^{4} - 9 \, B b^{2} x^{2} + 18 \, A b c x^{2} - 5 \, A b^{2}\right )}}{45 \, b^{4} x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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