Optimal. Leaf size=243 \[ -\frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{9 b \sqrt{x}}{2 c^3}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{9 x^{5/2}}{10 c^2} \]
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Rubi [A] time = 0.210042, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ -\frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{9 b \sqrt{x}}{2 c^3}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{9 x^{5/2}}{10 c^2} \]
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^{19/2}}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^{11/2}}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{9 \int \frac{x^{7/2}}{b+c x^2} \, dx}{4 c}\\ &=\frac{9 x^{5/2}}{10 c^2}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}-\frac{(9 b) \int \frac{x^{3/2}}{b+c x^2} \, dx}{4 c^2}\\ &=-\frac{9 b \sqrt{x}}{2 c^3}+\frac{9 x^{5/2}}{10 c^2}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{\left (9 b^2\right ) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{4 c^3}\\ &=-\frac{9 b \sqrt{x}}{2 c^3}+\frac{9 x^{5/2}}{10 c^2}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{\left (9 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 c^3}\\ &=-\frac{9 b \sqrt{x}}{2 c^3}+\frac{9 x^{5/2}}{10 c^2}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{\left (9 b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^3}+\frac{\left (9 b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 c^3}\\ &=-\frac{9 b \sqrt{x}}{2 c^3}+\frac{9 x^{5/2}}{10 c^2}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{\left (9 b^{3/2}\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^{7/2}}+\frac{\left (9 b^{3/2}\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^{7/2}}-\frac{\left (9 b^{5/4}\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^{13/4}}-\frac{\left (9 b^{5/4}\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^{13/4}}\\ &=-\frac{9 b \sqrt{x}}{2 c^3}+\frac{9 x^{5/2}}{10 c^2}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}-\frac{9 b^{5/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{\left (9 b^{5/4}\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^{13/4}}-\frac{\left (9 b^{5/4}\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^{13/4}}\\ &=-\frac{9 b \sqrt{x}}{2 c^3}+\frac{9 x^{5/2}}{10 c^2}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}\\ \end{align*}
Mathematica [A] time = 0.210854, size = 220, normalized size = 0.91 \[ \frac{\frac{8 \sqrt [4]{c} \sqrt{x} \left (-45 b^2-36 b c x^2+4 c^2 x^4\right )}{b+c x^2}-45 \sqrt{2} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-90 \sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{80 c^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 172, normalized size = 0.7 \begin{align*}{\frac{2}{5\,{c}^{2}}{x}^{{\frac{5}{2}}}}-4\,{\frac{b\sqrt{x}}{{c}^{3}}}-{\frac{{b}^{2}}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{9\,b\sqrt{2}}{16\,{c}^{3}}\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{9\,b\sqrt{2}}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{9\,b\sqrt{2}}{8\,{c}^{3}}\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 = 1.39966, size = 524, normalized size = 2.16 \begin{align*} \frac{180 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b c^{10} \sqrt{x} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{3}{4}} - \sqrt{c^{6} \sqrt{-\frac{b^{5}}{c^{13}}} + b^{2} x} c^{10} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{3}{4}}}{b^{5}}\right ) + 45 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \log \left (9 \, c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} + 9 \, b \sqrt{x}\right ) - 45 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \log \left (-9 \, c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} + 9 \, b \sqrt{x}\right ) + 4 \,{\left (4 \, c^{2} x^{4} - 36 \, b c x^{2} - 45 \, b^{2}\right )} \sqrt{x}}{40 \,{\left (c^{4} x^{2} + b c^{3}\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 = 1.17726, size = 292, normalized size = 1.2 \begin{align*} \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \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^{4}} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \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^{4}} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{b^{2} \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} c^{3}} + \frac{2 \,{\left (c^{8} x^{\frac{5}{2}} - 10 \, b c^{7} \sqrt{x}\right )}}{5 \, c^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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