Optimal. Leaf size=230 \[ \frac{5 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}-\frac{5 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}+\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{9/4}}-\frac{5 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{9/4}}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}+\frac{5 \sqrt{x}}{2 c^2} \]
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Rubi [A] time = 0.181984, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1584, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{5 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}-\frac{5 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}+\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{9/4}}-\frac{5 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{9/4}}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}+\frac{5 \sqrt{x}}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 288
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{15/2}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^{7/2}}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}+\frac{5 \int \frac{x^{3/2}}{b+c x^2} \, dx}{4 c}\\ &=\frac{5 \sqrt{x}}{2 c^2}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}-\frac{(5 b) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{4 c^2}\\ &=\frac{5 \sqrt{x}}{2 c^2}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 c^2}\\ &=\frac{5 \sqrt{x}}{2 c^2}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}-\frac{\left (5 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^2}-\frac{\left (5 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^2}\\ &=\frac{5 \sqrt{x}}{2 c^2}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}-\frac{\left (5 \sqrt{b}\right ) \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^{5/2}}-\frac{\left (5 \sqrt{b}\right ) \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^{5/2}}+\frac{\left (5 \sqrt [4]{b}\right ) \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} c^{9/4}}+\frac{\left (5 \sqrt [4]{b}\right ) \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} c^{9/4}}\\ &=\frac{5 \sqrt{x}}{2 c^2}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}+\frac{5 \sqrt [4]{b} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}-\frac{5 \sqrt [4]{b} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}-\frac{\left (5 \sqrt [4]{b}\right ) \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} c^{9/4}}+\frac{\left (5 \sqrt [4]{b}\right ) \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} c^{9/4}}\\ &=\frac{5 \sqrt{x}}{2 c^2}-\frac{x^{5/2}}{2 c \left (b+c x^2\right )}+\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{9/4}}-\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{9/4}}+\frac{5 \sqrt [4]{b} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}-\frac{5 \sqrt [4]{b} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.109156, size = 221, normalized size = 0.96 \[ \frac{\frac{32 c^{5/4} x^{5/2}}{b+c x^2}+\frac{40 b \sqrt [4]{c} \sqrt{x}}{b+c x^2}+5 \sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-5 \sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+10 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-10 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{16 c^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 158, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{x}}{{c}^{2}}}+{\frac{b}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}}{16\,{c}^{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{5\,\sqrt{2}}{8\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}}{8\,{c}^{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 = 1.36376, size = 443, normalized size = 1.93 \begin{align*} -\frac{20 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b}{c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{c^{4} \sqrt{-\frac{b}{c^{9}}} + x} c^{7} \left (-\frac{b}{c^{9}}\right )^{\frac{3}{4}} - c^{7} \sqrt{x} \left (-\frac{b}{c^{9}}\right )^{\frac{3}{4}}}{b}\right ) + 5 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b}{c^{9}}\right )^{\frac{1}{4}} \log \left (5 \, c^{2} \left (-\frac{b}{c^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) - 5 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b}{c^{9}}\right )^{\frac{1}{4}} \log \left (-5 \, c^{2} \left (-\frac{b}{c^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) - 4 \,{\left (4 \, c x^{2} + 5 \, b\right )} \sqrt{x}}{8 \,{\left (c^{3} x^{2} + b c^{2}\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 = 1.23465, size = 265, normalized size = 1.15 \begin{align*} -\frac{5 \, \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 )}{8 \, c^{3}} - \frac{5 \, \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 )}{8 \, c^{3}} - \frac{5 \, \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 )}{16 \, c^{3}} + \frac{5 \, \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 )}{16 \, c^{3}} + \frac{b \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} c^{2}} + \frac{2 \, \sqrt{x}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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