Optimal. Leaf size=192 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} \sqrt [4]{b} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} \sqrt [4]{b} c^{3/4}} \]
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Rubi [A] time = 0.142039, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {1584, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} \sqrt [4]{b} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} \sqrt [4]{b} c^{3/4}} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{b x^2+c x^4} \, dx &=\int \frac{\sqrt{x}}{b+c x^2} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{c}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{c}}\\ &=\frac{\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 c}+\frac{\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 c}+\frac{\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} \sqrt [4]{b} c^{3/4}}+\frac{\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} \sqrt [4]{b} c^{3/4}}\\ &=\frac{\log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}-\frac{\log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}+\frac{\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} \sqrt [4]{b} c^{3/4}}-\frac{\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} \sqrt [4]{b} c^{3/4}}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} \sqrt [4]{b} c^{3/4}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} \sqrt [4]{b} c^{3/4}}+\frac{\log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}-\frac{\log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{b} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0250279, size = 54, normalized size = 0.28 \[ \frac{b \left (\tan ^{-1}\left (\frac{b \sqrt [4]{c} \sqrt{x}}{(-b)^{5/4}}\right )+\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b}}\right )\right )}{(-b)^{5/4} c^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 132, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}}{4\,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{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}}{2\,c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}}{2\,c}\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 = 1.33029, size = 332, normalized size = 1.73 \begin{align*} -2 \, \left (-\frac{1}{b c^{3}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{-b c \sqrt{-\frac{1}{b c^{3}}} + x} c \left (-\frac{1}{b c^{3}}\right )^{\frac{1}{4}} - c \sqrt{x} \left (-\frac{1}{b c^{3}}\right )^{\frac{1}{4}}\right ) + \frac{1}{2} \, \left (-\frac{1}{b c^{3}}\right )^{\frac{1}{4}} \log \left (b c^{2} \left (-\frac{1}{b c^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) - \frac{1}{2} \, \left (-\frac{1}{b c^{3}}\right )^{\frac{1}{4}} \log \left (-b c^{2} \left (-\frac{1}{b c^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 72.7057, size = 170, normalized size = 0.89 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: b = 0 \wedge c = 0 \\\frac{2 x^{\frac{3}{2}}}{3 b} & \text{for}\: c = 0 \\- \frac{2}{c \sqrt{x}} & \text{for}\: b = 0 \\- \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 \sqrt [4]{b} c^{2} \left (\frac{1}{c}\right )^{\frac{5}{4}}} + \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 \sqrt [4]{b} c^{2} \left (\frac{1}{c}\right )^{\frac{5}{4}}} + \frac{\left (-1\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{\sqrt [4]{b} c^{2} \left (\frac{1}{c}\right )^{\frac{5}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13058, size = 246, normalized size = 1.28 \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 c^{3}} + \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 c^{3}} - \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 c^{3}} + \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 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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