Optimal. Leaf size=289 \[ \frac{(5 b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}+\frac{(5 b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{3/4} c^{9/4}}+\frac{\sqrt{x} (5 b B-A c)}{2 b c^2}-\frac{x^{5/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
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Rubi [A] time = 0.233747, antiderivative size = 289, 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, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{(5 b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}+\frac{(5 b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{3/4} c^{9/4}}+\frac{\sqrt{x} (5 b B-A c)}{2 b c^2}-\frac{x^{5/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 321
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 )^2} \, dx &=\int \frac{x^{3/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}+\frac{\left (\frac{5 b B}{2}-\frac{A c}{2}\right ) \int \frac{x^{3/2}}{b+c x^2} \, dx}{2 b c}\\ &=\frac{(5 b B-A c) \sqrt{x}}{2 b c^2}-\frac{(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac{(5 b B-A c) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{4 c^2}\\ &=\frac{(5 b B-A c) \sqrt{x}}{2 b c^2}-\frac{(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac{(5 b B-A c) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 c^2}\\ &=\frac{(5 b B-A c) \sqrt{x}}{2 b c^2}-\frac{(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac{(5 b B-A c) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 \sqrt{b} c^2}-\frac{(5 b B-A c) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 \sqrt{b} c^2}\\ &=\frac{(5 b B-A c) \sqrt{x}}{2 b c^2}-\frac{(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac{(5 b B-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 \sqrt{b} c^{5/2}}-\frac{(5 b B-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 \sqrt{b} c^{5/2}}+\frac{(5 b B-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 )}{8 \sqrt{2} b^{3/4} c^{9/4}}+\frac{(5 b B-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 )}{8 \sqrt{2} b^{3/4} c^{9/4}}\\ &=\frac{(5 b B-A c) \sqrt{x}}{2 b c^2}-\frac{(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}+\frac{(5 b B-A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-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 )}{4 \sqrt{2} b^{3/4} c^{9/4}}+\frac{(5 b B-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 )}{4 \sqrt{2} b^{3/4} c^{9/4}}\\ &=\frac{(5 b B-A c) \sqrt{x}}{2 b c^2}-\frac{(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}+\frac{(5 b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{3/4} c^{9/4}}+\frac{(5 b B-A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}-\frac{(5 b B-A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{3/4} c^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.390944, size = 354, normalized size = 1.22 \[ \frac{\frac{2 \sqrt{2} (5 b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{b^{3/4}}-\frac{2 \sqrt{2} (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{b^{3/4}}-\frac{\sqrt{2} A c \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{3/4}}+\frac{\sqrt{2} A c \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{3/4}}-\frac{8 A c^{5/4} \sqrt{x}}{b+c x^2}+\frac{8 b B \sqrt [4]{c} \sqrt{x}}{b+c x^2}+5 \sqrt{2} \sqrt [4]{b} B \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-5 \sqrt{2} \sqrt [4]{b} B \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+32 B \sqrt [4]{c} \sqrt{x}}{16 c^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 323, normalized size = 1.1 \begin{align*} 2\,{\frac{B\sqrt{x}}{{c}^{2}}}-{\frac{A}{2\,c \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{Bb}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{\sqrt{2}A}{8\,bc}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{\sqrt{2}A}{8\,bc}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{\sqrt{2}A}{16\,bc}\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{5\,\sqrt{2}B}{8\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}B}{8\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}B}{16\,{c}^{2}}\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 = 2.52275, size = 1594, normalized size = 5.52 \begin{align*} \frac{4 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{b^{2} c^{4} \sqrt{-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}} +{\left (25 \, B^{2} b^{2} - 10 \, A B b c + A^{2} c^{2}\right )} x} b^{2} c^{7} \left (-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac{3}{4}} +{\left (5 \, B b^{3} c^{7} - A b^{2} c^{8}\right )} \sqrt{x} \left (-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac{3}{4}}}{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}\right ) +{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (b c^{2} \left (-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac{1}{4}} -{\left (5 \, B b - A c\right )} \sqrt{x}\right ) -{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (-b c^{2} \left (-\frac{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac{1}{4}} -{\left (5 \, B b - A c\right )} \sqrt{x}\right ) + 4 \,{\left (4 \, B c x^{2} + 5 \, B b - A c\right )} \sqrt{x}}{8 \,{\left (c^{3} x^{2} + b c^{2}\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 = 1.22107, size = 382, normalized size = 1.32 \begin{align*} \frac{2 \, B \sqrt{x}}{c^{2}} - \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - \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 )}{8 \, b c^{3}} - \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - \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 )}{8 \, b c^{3}} - \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - \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 )}{16 \, b c^{3}} + \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - \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 )}{16 \, b c^{3}} + \frac{B b \sqrt{x} - A c \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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