Optimal. Leaf size=204 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{7/4}}-\frac{2}{3 b x^{3/2}} \]
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Rubi [A] time = 0.167133, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {1584, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{7/4}}-\frac{2}{3 b x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (b x^2+c x^4\right )} \, dx &=\int \frac{1}{x^{5/2} \left (b+c x^2\right )} \, dx\\ &=-\frac{2}{3 b x^{3/2}}-\frac{c \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{b}\\ &=-\frac{2}{3 b x^{3/2}}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b}\\ &=-\frac{2}{3 b x^{3/2}}-\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^{3/2}}-\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^{3/2}}\\ &=-\frac{2}{3 b x^{3/2}}-\frac{\sqrt{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^{3/2}}-\frac{\sqrt{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^{3/2}}+\frac{c^{3/4} \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^{7/4}}+\frac{c^{3/4} \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^{7/4}}\\ &=-\frac{2}{3 b x^{3/2}}+\frac{c^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \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^{7/4}}+\frac{c^{3/4} \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^{7/4}}\\ &=-\frac{2}{3 b x^{3/2}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4}}+\frac{c^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}\\ \end{align*}
Mathematica [C] time = 0.0065507, size = 29, normalized size = 0.14 \[ -\frac{2 \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{c x^2}{b}\right )}{3 b x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 143, normalized size = 0.7 \begin{align*} -{\frac{2}{3\,b}{x}^{-{\frac{3}{2}}}}-{\frac{c\sqrt{2}}{4\,{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{c\sqrt{2}}{2\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{c\sqrt{2}}{2\,{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 [A] time = 1.39365, size = 387, normalized size = 1.9 \begin{align*} -\frac{12 \, b x^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{5} c \sqrt{x} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{3}{4}} - \sqrt{b^{4} \sqrt{-\frac{c^{3}}{b^{7}}} + c^{2} x} b^{5} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{3}{4}}}{c^{3}}\right ) + 3 \, b x^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (b^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} + c \sqrt{x}\right ) - 3 \, b x^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (-b^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} + c \sqrt{x}\right ) + 4 \, \sqrt{x}}{6 \, b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 41.386, size = 184, normalized size = 0.9 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: b = 0 \wedge c = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} & \text{for}\: c = 0 \\- \frac{2}{7 c x^{\frac{7}{2}}} & \text{for}\: b = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} + \frac{\sqrt [4]{-1} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{7}{4}} c^{2} \left (\frac{1}{c}\right )^{\frac{11}{4}}} - \frac{\sqrt [4]{-1} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{7}{4}} c^{2} \left (\frac{1}{c}\right )^{\frac{11}{4}}} + \frac{\sqrt [4]{-1} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{7}{4}} c^{2} \left (\frac{1}{c}\right )^{\frac{11}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13961, size = 240, normalized size = 1.18 \begin{align*} -\frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{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^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{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^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2}} - \frac{2}{3 \, b x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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