Optimal. Leaf size=215 \[ \frac{c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}+\frac{c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 c}{b^2 \sqrt{x}}-\frac{2}{5 b x^{5/2}} \]
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Rubi [A] time = 0.187666, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {1584, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}+\frac{c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 c}{b^2 \sqrt{x}}-\frac{2}{5 b x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \left (b x^2+c x^4\right )} \, dx &=\int \frac{1}{x^{7/2} \left (b+c x^2\right )} \, dx\\ &=-\frac{2}{5 b x^{5/2}}-\frac{c \int \frac{1}{x^{3/2} \left (b+c x^2\right )} \, dx}{b}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 c}{b^2 \sqrt{x}}+\frac{c^2 \int \frac{\sqrt{x}}{b+c x^2} \, dx}{b^2}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 c}{b^2 \sqrt{x}}+\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 c}{b^2 \sqrt{x}}-\frac{c^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^2}+\frac{c^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 c}{b^2 \sqrt{x}}+\frac{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 )}{2 b^2}+\frac{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 )}{2 b^2}+\frac{c^{5/4} \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 )}{2 \sqrt{2} b^{9/4}}+\frac{c^{5/4} \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 )}{2 \sqrt{2} b^{9/4}}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 c}{b^2 \sqrt{x}}+\frac{c^{5/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}+\frac{c^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}-\frac{c^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 c}{b^2 \sqrt{x}}-\frac{c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}+\frac{c^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}+\frac{c^{5/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.0069805, size = 29, normalized size = 0.13 \[ -\frac{2 \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\frac{c x^2}{b}\right )}{5 b x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 152, normalized size = 0.7 \begin{align*} -{\frac{2}{5\,b}{x}^{-{\frac{5}{2}}}}+2\,{\frac{c}{{b}^{2}\sqrt{x}}}+{\frac{c\sqrt{2}}{4\,{b}^{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{c\sqrt{2}}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{c\sqrt{2}}{2\,{b}^{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 [A] time = 1.36692, size = 433, normalized size = 2.01 \begin{align*} -\frac{20 \, b^{2} x^{3} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{2} c^{4} \sqrt{x} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}} - \sqrt{-b^{5} c^{5} \sqrt{-\frac{c^{5}}{b^{9}}} + c^{8} x} b^{2} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}}}{c^{5}}\right ) - 5 \, b^{2} x^{3} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (b^{7} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{3}{4}} + c^{4} \sqrt{x}\right ) + 5 \, b^{2} x^{3} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-b^{7} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{3}{4}} + c^{4} \sqrt{x}\right ) - 4 \,{\left (5 \, c x^{2} - b\right )} \sqrt{x}}{10 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 67.5526, size = 196, normalized size = 0.91 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{9}{2}}} & \text{for}\: b = 0 \wedge c = 0 \\- \frac{2}{9 c x^{\frac{9}{2}}} & \text{for}\: b = 0 \\- \frac{2}{5 b x^{\frac{5}{2}}} & \text{for}\: c = 0 \\- \frac{2}{5 b x^{\frac{5}{2}}} + \frac{2 c}{b^{2} \sqrt{x}} - \frac{\left (-1\right )^{\frac{3}{4}} c^{6} \left (\frac{1}{c}\right )^{\frac{19}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{9}{4}}} + \frac{\left (-1\right )^{\frac{3}{4}} c^{6} \left (\frac{1}{c}\right )^{\frac{19}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{9}{4}}} + \frac{\left (-1\right )^{\frac{3}{4}} c^{6} \left (\frac{1}{c}\right )^{\frac{19}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{9}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13163, size = 270, normalized size = 1.26 \begin{align*} \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \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 )}{2 \, b^{3} c} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \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 )}{2 \, b^{3} c} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3} c} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3} c} + \frac{2 \,{\left (5 \, c x^{2} - b\right )}}{5 \, b^{2} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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