Optimal. Leaf size=218 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{x^{3/2}}{2 c \left (b+c x^2\right )} \]
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Rubi [A] time = 0.165619, antiderivative size = 218, 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, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{x^{3/2}}{2 c \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 288
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{13/2}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^{5/2}}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{x^{3/2}}{2 c \left (b+c x^2\right )}+\frac{3 \int \frac{\sqrt{x}}{b+c x^2} \, dx}{4 c}\\ &=-\frac{x^{3/2}}{2 c \left (b+c x^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 c}\\ &=-\frac{x^{3/2}}{2 c \left (b+c x^2\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^{3/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^{3/2}}\\ &=-\frac{x^{3/2}}{2 c \left (b+c x^2\right )}+\frac{3 \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 c^2}+\frac{3 \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 c^2}+\frac{3 \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} \sqrt [4]{b} c^{7/4}}+\frac{3 \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} \sqrt [4]{b} c^{7/4}}\\ &=-\frac{x^{3/2}}{2 c \left (b+c x^2\right )}+\frac{3 \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{3 \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}+\frac{3 \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} \sqrt [4]{b} c^{7/4}}-\frac{3 \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} \sqrt [4]{b} c^{7/4}}\\ &=-\frac{x^{3/2}}{2 c \left (b+c x^2\right )}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{7/4}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} \sqrt [4]{b} c^{7/4}}+\frac{3 \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}-\frac{3 \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{b} c^{7/4}}\\ \end{align*}
Mathematica [C] time = 0.016464, size = 43, normalized size = 0.2 \[ \frac{2 x^{3/2} \left (\frac{\, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{c x^2}{b}\right )}{b}-\frac{1}{b+c x^2}\right )}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 149, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,c \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{3\,\sqrt{2}}{16\,{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{3\,\sqrt{2}}{8\,{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{3\,\sqrt{2}}{8\,{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 = 1.41057, size = 447, normalized size = 2.05 \begin{align*} -\frac{12 \,{\left (c^{2} x^{2} + b c\right )} \left (-\frac{1}{b c^{7}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{-b c^{3} \sqrt{-\frac{1}{b c^{7}}} + x} c^{2} \left (-\frac{1}{b c^{7}}\right )^{\frac{1}{4}} - c^{2} \sqrt{x} \left (-\frac{1}{b c^{7}}\right )^{\frac{1}{4}}\right ) - 3 \,{\left (c^{2} x^{2} + b c\right )} \left (-\frac{1}{b c^{7}}\right )^{\frac{1}{4}} \log \left (b c^{5} \left (-\frac{1}{b c^{7}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) + 3 \,{\left (c^{2} x^{2} + b c\right )} \left (-\frac{1}{b c^{7}}\right )^{\frac{1}{4}} \log \left (-b c^{5} \left (-\frac{1}{b c^{7}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) + 4 \, x^{\frac{3}{2}}}{8 \,{\left (c^{2} x^{2} + b c\right )}} \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 = 1.18814, size = 269, normalized size = 1.23 \begin{align*} -\frac{x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} c} + \frac{3 \, \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 )}{8 \, b c^{4}} + \frac{3 \, \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 )}{8 \, b c^{4}} - \frac{3 \, \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 )}{16 \, b c^{4}} + \frac{3 \, \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 )}{16 \, b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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