Optimal. Leaf size=343 \[ -\frac{x^{9/2} (13 b B-5 A c)}{16 b c^2 \left (b+c x^2\right )}+\frac{9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac{9 \sqrt{x} (13 b B-5 A c)}{16 c^4}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{17/4}}-\frac{x^{13/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.278437, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {1584, 457, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{x^{9/2} (13 b B-5 A c)}{16 b c^2 \left (b+c x^2\right )}+\frac{9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac{9 \sqrt{x} (13 b B-5 A c)}{16 c^4}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{17/4}}-\frac{x^{13/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 288
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{23/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x^{11/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}+\frac{\left (\frac{13 b B}{2}-\frac{5 A c}{2}\right ) \int \frac{x^{11/2}}{\left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac{(9 (13 b B-5 A c)) \int \frac{x^{7/2}}{b+c x^2} \, dx}{32 b c^2}\\ &=\frac{9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac{(9 (13 b B-5 A c)) \int \frac{x^{3/2}}{b+c x^2} \, dx}{32 c^3}\\ &=-\frac{9 (13 b B-5 A c) \sqrt{x}}{16 c^4}+\frac{9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac{(9 b (13 b B-5 A c)) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{32 c^4}\\ &=-\frac{9 (13 b B-5 A c) \sqrt{x}}{16 c^4}+\frac{9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac{(9 b (13 b B-5 A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{16 c^4}\\ &=-\frac{9 (13 b B-5 A c) \sqrt{x}}{16 c^4}+\frac{9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac{\left (9 \sqrt{b} (13 b B-5 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 c^4}+\frac{\left (9 \sqrt{b} (13 b B-5 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 c^4}\\ &=-\frac{9 (13 b B-5 A c) \sqrt{x}}{16 c^4}+\frac{9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac{\left (9 \sqrt{b} (13 b B-5 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 c^{9/2}}+\frac{\left (9 \sqrt{b} (13 b B-5 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 c^{9/2}}-\frac{\left (9 \sqrt [4]{b} (13 b B-5 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} c^{17/4}}-\frac{\left (9 \sqrt [4]{b} (13 b B-5 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} c^{17/4}}\\ &=-\frac{9 (13 b B-5 A c) \sqrt{x}}{16 c^4}+\frac{9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}+\frac{\left (9 \sqrt [4]{b} (13 b B-5 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} c^{17/4}}-\frac{\left (9 \sqrt [4]{b} (13 b B-5 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} c^{17/4}}\\ &=-\frac{9 (13 b B-5 A c) \sqrt{x}}{16 c^4}+\frac{9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac{(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac{(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{17/4}}-\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}+\frac{9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{17/4}}\\ \end{align*}
Mathematica [A] time = 0.47518, size = 435, normalized size = 1.27 \[ \frac{-\frac{160 A b^2 c^{5/4} \sqrt{x}}{\left (b+c x^2\right )^2}-90 \sqrt{2} \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{680 A b c^{5/4} \sqrt{x}}{b+c x^2}+225 \sqrt{2} A \sqrt [4]{b} c \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-225 \sqrt{2} A \sqrt [4]{b} c \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+1280 A c^{5/4} \sqrt{x}+\frac{160 b^3 B \sqrt [4]{c} \sqrt{x}}{\left (b+c x^2\right )^2}-\frac{1000 b^2 B \sqrt [4]{c} \sqrt{x}}{b+c x^2}-585 \sqrt{2} b^{5/4} B \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+585 \sqrt{2} b^{5/4} B \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-3840 b B \sqrt [4]{c} \sqrt{x}+256 B c^{5/4} x^{5/2}}{640 c^{17/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 381, normalized size = 1.1 \begin{align*}{\frac{2\,B}{5\,{c}^{3}}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{{c}^{3}}}-6\,{\frac{Bb\sqrt{x}}{{c}^{4}}}+{\frac{17\,Ab}{16\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{25\,B{b}^{2}}{16\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{13\,A{b}^{2}}{16\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{21\,B{b}^{3}}{16\,{c}^{4} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{45\,\sqrt{2}A}{64\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{45\,\sqrt{2}A}{64\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{45\,\sqrt{2}A}{128\,{c}^{3}}\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{117\,b\sqrt{2}B}{64\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{117\,b\sqrt{2}B}{64\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{117\,b\sqrt{2}B}{128\,{c}^{4}}\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 = 2.57405, size = 1955, normalized size = 5.7 \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 = 1.31172, size = 433, normalized size = 1.26 \begin{align*} \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{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 )}{64 \, c^{5}} + \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{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 )}{64 \, c^{5}} + \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, c^{5}} - \frac{9 \, \sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, c^{5}} - \frac{25 \, B b^{2} c x^{\frac{5}{2}} - 17 \, A b c^{2} x^{\frac{5}{2}} + 21 \, B b^{3} \sqrt{x} - 13 \, A b^{2} c \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} c^{4}} + \frac{2 \,{\left (B c^{12} x^{\frac{5}{2}} - 15 \, B b c^{11} \sqrt{x} + 5 \, A c^{12} \sqrt{x}\right )}}{5 \, c^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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