Optimal. Leaf size=293 \[ -\frac{3 (7 A c+b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (7 A c+b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}-\frac{3 (7 A c+b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (7 A c+b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{11/4} c^{5/4}}+\frac{\sqrt{x} (7 A c+b B)}{16 b^2 c \left (b+c x^2\right )}-\frac{\sqrt{x} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.230511, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1584, 457, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 (7 A c+b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (7 A c+b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}-\frac{3 (7 A c+b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (7 A c+b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{11/4} c^{5/4}}+\frac{\sqrt{x} (7 A c+b B)}{16 b^2 c \left (b+c x^2\right )}-\frac{\sqrt{x} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{11/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{A+B x^2}{\sqrt{x} \left (b+c x^2\right )^3} \, dx\\ &=-\frac{(b B-A c) \sqrt{x}}{4 b c \left (b+c x^2\right )^2}+\frac{\left (\frac{b B}{2}+\frac{7 A c}{2}\right ) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac{(b B-A c) \sqrt{x}}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+7 A c) \sqrt{x}}{16 b^2 c \left (b+c x^2\right )}+\frac{(3 (b B+7 A c)) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{32 b^2 c}\\ &=-\frac{(b B-A c) \sqrt{x}}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+7 A c) \sqrt{x}}{16 b^2 c \left (b+c x^2\right )}+\frac{(3 (b B+7 A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{16 b^2 c}\\ &=-\frac{(b B-A c) \sqrt{x}}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+7 A c) \sqrt{x}}{16 b^2 c \left (b+c x^2\right )}+\frac{(3 (b B+7 A c)) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{5/2} c}+\frac{(3 (b B+7 A c)) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{5/2} c}\\ &=-\frac{(b B-A c) \sqrt{x}}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+7 A c) \sqrt{x}}{16 b^2 c \left (b+c x^2\right )}+\frac{(3 (b B+7 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 )}{64 b^{5/2} c^{3/2}}+\frac{(3 (b B+7 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 )}{64 b^{5/2} c^{3/2}}-\frac{(3 (b B+7 A c)) \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^{11/4} c^{5/4}}-\frac{(3 (b B+7 A c)) \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^{11/4} c^{5/4}}\\ &=-\frac{(b B-A c) \sqrt{x}}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+7 A c) \sqrt{x}}{16 b^2 c \left (b+c x^2\right )}-\frac{3 (b B+7 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (b B+7 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}+\frac{(3 (b B+7 A c)) \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^{11/4} c^{5/4}}-\frac{(3 (b B+7 A c)) \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^{11/4} c^{5/4}}\\ &=-\frac{(b B-A c) \sqrt{x}}{4 b c \left (b+c x^2\right )^2}+\frac{(b B+7 A c) \sqrt{x}}{16 b^2 c \left (b+c x^2\right )}-\frac{3 (b B+7 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (b B+7 A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{11/4} c^{5/4}}-\frac{3 (b B+7 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}+\frac{3 (b B+7 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{11/4} c^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.297919, size = 230, normalized size = 0.78 \[ \frac{\frac{(7 A c+b B) \left (7 \left (b+c x^2\right ) \left (8 b^{3/4} \sqrt [4]{c} \sqrt{x}-3 \sqrt{2} \left (b+c x^2\right ) \left (\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )\right )\right )+32 b^{7/4} \sqrt [4]{c} \sqrt{x}\right )}{b^{11/4} \sqrt [4]{c}}-256 B \sqrt{x}}{896 c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 325, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 7\,Ac+Bb \right ){x}^{5/2}}{{b}^{2}}}+1/32\,{\frac{ \left ( 11\,Ac-3\,Bb \right ) \sqrt{x}}{bc}} \right ) }+{\frac{21\,\sqrt{2}A}{64\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}A}{128\,{b}^{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{21\,\sqrt{2}A}{64\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}B}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}B}{128\,{b}^{2}c}\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{3\,\sqrt{2}B}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+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 = 2.54239, size = 1782, normalized size = 6.08 \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 = 1.31727, size = 396, normalized size = 1.35 \begin{align*} \frac{3 \, \sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b + 7 \, \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 \, b^{3} c^{2}} + \frac{3 \, \sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b + 7 \, \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 \, b^{3} c^{2}} + \frac{3 \, \sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b + 7 \, \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 \, b^{3} c^{2}} - \frac{3 \, \sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b + 7 \, \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 \, b^{3} c^{2}} + \frac{B b c x^{\frac{5}{2}} + 7 \, A c^{2} x^{\frac{5}{2}} - 3 \, B b^{2} \sqrt{x} + 11 \, A b c \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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