Optimal. Leaf size=289 \[ \frac{x^{3/2} (7 b B-3 A c)}{6 b c^2}-\frac{(7 b B-3 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{11/4}}-\frac{(7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} \sqrt [4]{b} c^{11/4}}-\frac{x^{7/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
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Rubi [A] time = 0.229241, 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, 297, 1162, 617, 204, 1165, 628} \[ \frac{x^{3/2} (7 b B-3 A c)}{6 b c^2}-\frac{(7 b B-3 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{11/4}}-\frac{(7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} \sqrt [4]{b} c^{11/4}}-\frac{x^{7/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 321
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{13/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^{5/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{(b B-A c) x^{7/2}}{2 b c \left (b+c x^2\right )}+\frac{\left (\frac{7 b B}{2}-\frac{3 A c}{2}\right ) \int \frac{x^{5/2}}{b+c x^2} \, dx}{2 b c}\\ &=\frac{(7 b B-3 A c) x^{3/2}}{6 b c^2}-\frac{(b B-A c) x^{7/2}}{2 b c \left (b+c x^2\right )}-\frac{(7 b B-3 A c) \int \frac{\sqrt{x}}{b+c x^2} \, dx}{4 c^2}\\ &=\frac{(7 b B-3 A c) x^{3/2}}{6 b c^2}-\frac{(b B-A c) x^{7/2}}{2 b c \left (b+c x^2\right )}-\frac{(7 b B-3 A c) \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 c^2}\\ &=\frac{(7 b B-3 A c) x^{3/2}}{6 b c^2}-\frac{(b B-A c) x^{7/2}}{2 b c \left (b+c x^2\right )}+\frac{(7 b B-3 A c) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^{5/2}}-\frac{(7 b B-3 A c) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^{5/2}}\\ &=\frac{(7 b B-3 A c) x^{3/2}}{6 b c^2}-\frac{(b B-A c) x^{7/2}}{2 b c \left (b+c x^2\right )}-\frac{(7 b B-3 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 c^3}-\frac{(7 b B-3 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 c^3}-\frac{(7 b B-3 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} \sqrt [4]{b} c^{11/4}}-\frac{(7 b B-3 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} \sqrt [4]{b} c^{11/4}}\\ &=\frac{(7 b B-3 A c) x^{3/2}}{6 b c^2}-\frac{(b B-A c) x^{7/2}}{2 b c \left (b+c x^2\right )}-\frac{(7 b B-3 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}-\frac{(7 b B-3 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} \sqrt [4]{b} c^{11/4}}+\frac{(7 b B-3 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} \sqrt [4]{b} c^{11/4}}\\ &=\frac{(7 b B-3 A c) x^{3/2}}{6 b c^2}-\frac{(b B-A c) x^{7/2}}{2 b c \left (b+c x^2\right )}+\frac{(7 b B-3 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{11/4}}-\frac{(7 b B-3 A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{11/4}}-\frac{(7 b B-3 A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}+\frac{(7 b B-3 A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.197619, size = 136, normalized size = 0.47 \[ \frac{2 x^{3/2} (b B-A c) \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right )}{3 b c^2}+\frac{(3 A c-6 b B) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b}}\right )+(6 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b}}\right )+2 \sqrt [4]{-b} B c^{3/4} x^{3/2}}{3 \sqrt [4]{-b} c^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 317, normalized size = 1.1 \begin{align*}{\frac{2\,B}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{A}{2\,c \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{Bb}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{7\,\sqrt{2}Bb}{16\,{c}^{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{7\,\sqrt{2}Bb}{8\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{7\,\sqrt{2}Bb}{8\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}A}{16\,{c}^{2}}\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{3\,\sqrt{2}A}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}A}{8\,{c}^{2}}\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 = 2.66426, size = 2141, normalized size = 7.41 \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 = 1.349, size = 382, normalized size = 1.32 \begin{align*} \frac{2 \, B x^{\frac{3}{2}}}{3 \, c^{2}} + \frac{B b x^{\frac{3}{2}} - A c x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} c^{2}} - \frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \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 c^{5}} - \frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \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 c^{5}} + \frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \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 c^{5}} - \frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \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 c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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