Optimal. Leaf size=289 \[ \frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{(3 b B-7 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\frac{(3 b B-7 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\frac{b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )} \]
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Rubi [A] time = 0.225164, 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, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{(3 b B-7 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\frac{(3 b B-7 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\frac{b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{A+B x^2}{x^{5/2} \left (b+c x^2\right )^2} \, dx\\ &=-\frac{b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}+\frac{\left (-\frac{3 b B}{2}+\frac{7 A c}{2}\right ) \int \frac{1}{x^{5/2} \left (b+c x^2\right )} \, dx}{2 b c}\\ &=\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \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}{4 b^2}\\ &=\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \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 )}{2 b^2}\\ &=\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \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 )}{4 b^{5/2}}+\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 )}{4 b^{5/2}}\\ &=\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \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 )}{8 b^{5/2} \sqrt{c}}+\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 )}{8 b^{5/2} \sqrt{c}}-\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 )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\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 )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}\\ &=\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \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 )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\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 )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\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 )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\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 )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}\\ &=\frac{3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac{b B-A c}{2 b c x^{3/2} \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 )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\frac{(3 b B-7 A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4} \sqrt [4]{c}}-\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 )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}+\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 )}{8 \sqrt{2} b^{11/4} \sqrt [4]{c}}\\ \end{align*}
Mathematica [A] time = 0.423149, size = 355, normalized size = 1.23 \[ \frac{-\frac{24 A b^{3/4} c \sqrt{x}}{b+c x^2}-\frac{32 A b^{3/4}}{x^{3/2}}+\frac{6 \sqrt{2} (7 A c-3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} (3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt [4]{c}}+21 \sqrt{2} A c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-21 \sqrt{2} A c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+\frac{24 b^{7/4} B \sqrt{x}}{b+c x^2}-\frac{9 \sqrt{2} b B \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{9 \sqrt{2} b B \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}}{48 b^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 317, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{Ac}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{B}{2\,b \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{7\,\sqrt{2}Ac}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{7\,\sqrt{2}Ac}{16\,{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{7\,\sqrt{2}Ac}{8\,{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}{8\,{b}^{2}}\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}{16\,{b}^{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 ) }+{\frac{3\,\sqrt{2}B}{8\,{b}^{2}}\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 = 2.4927, size = 1700, normalized size = 5.88 \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 = 1.26993, size = 382, normalized size = 1.32 \begin{align*} \frac{\sqrt{2}{\left (3 \, \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 )}{8 \, b^{3} c} + \frac{\sqrt{2}{\left (3 \, \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 )}{8 \, b^{3} c} + \frac{\sqrt{2}{\left (3 \, \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 )}{16 \, b^{3} c} - \frac{\sqrt{2}{\left (3 \, \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 )}{16 \, b^{3} c} + \frac{B b \sqrt{x} - A c \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} b^{2}} - \frac{2 \, A}{3 \, b^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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