Optimal. Leaf size=204 \[ -\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{7/4}}+\frac{2 x^{3/2}}{3 c} \]
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Rubi [A] time = 0.185877, 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, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{7/4}}+\frac{2 x^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 321
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{9/2}}{b x^2+c x^4} \, dx &=\int \frac{x^{5/2}}{b+c x^2} \, dx\\ &=\frac{2 x^{3/2}}{3 c}-\frac{b \int \frac{\sqrt{x}}{b+c x^2} \, dx}{c}\\ &=\frac{2 x^{3/2}}{3 c}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{c}\\ &=\frac{2 x^{3/2}}{3 c}+\frac{b \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{c^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{c^{3/2}}\\ &=\frac{2 x^{3/2}}{3 c}-\frac{b \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^2}-\frac{b \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^2}-\frac{b^{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} c^{7/4}}-\frac{b^{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} c^{7/4}}\\ &=\frac{2 x^{3/2}}{3 c}-\frac{b^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}-\frac{b^{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} c^{7/4}}+\frac{b^{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} c^{7/4}}\\ &=\frac{2 x^{3/2}}{3 c}+\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{7/4}}-\frac{b^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{b^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}\\ \end{align*}
Mathematica [A] time = 0.0208927, size = 78, normalized size = 0.38 \[ \frac{(-b)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b}}\right )}{c^{7/4}}-\frac{(-b)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-b}}\right )}{c^{7/4}}+\frac{2 x^{3/2}}{3 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 143, normalized size = 0.7 \begin{align*}{\frac{2}{3\,c}{x}^{{\frac{3}{2}}}}-{\frac{b\sqrt{2}}{4\,{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{b\sqrt{2}}{2\,{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{b\sqrt{2}}{2\,{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 [A] time = 1.28749, size = 377, normalized size = 1.85 \begin{align*} \frac{12 \, c \left (-\frac{b^{3}}{c^{7}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{2} c^{2} \sqrt{x} \left (-\frac{b^{3}}{c^{7}}\right )^{\frac{1}{4}} - \sqrt{-b^{3} c^{3} \sqrt{-\frac{b^{3}}{c^{7}}} + b^{4} x} c^{2} \left (-\frac{b^{3}}{c^{7}}\right )^{\frac{1}{4}}}{b^{3}}\right ) - 3 \, c \left (-\frac{b^{3}}{c^{7}}\right )^{\frac{1}{4}} \log \left (c^{5} \left (-\frac{b^{3}}{c^{7}}\right )^{\frac{3}{4}} + b^{2} \sqrt{x}\right ) + 3 \, c \left (-\frac{b^{3}}{c^{7}}\right )^{\frac{1}{4}} \log \left (-c^{5} \left (-\frac{b^{3}}{c^{7}}\right )^{\frac{3}{4}} + b^{2} \sqrt{x}\right ) + 4 \, x^{\frac{3}{2}}}{6 \, c} \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 = 1.14407, size = 240, normalized size = 1.18 \begin{align*} \frac{2 \, x^{\frac{3}{2}}}{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 \, c^{4}} - \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 \, c^{4}} + \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 \, c^{4}} - \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 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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