Optimal. Leaf size=251 \[ -\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac{45 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{13/4}}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}+\frac{45 \sqrt{x}}{16 c^3} \]
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Rubi [A] time = 0.210625, antiderivative size = 251, 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} \[ -\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac{45 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{13/4}}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}+\frac{45 \sqrt{x}}{16 c^3} \]
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^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x^{11/2}}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}+\frac{9 \int \frac{x^{7/2}}{\left (b+c x^2\right )^2} \, dx}{8 c}\\ &=-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac{45 \int \frac{x^{3/2}}{b+c x^2} \, dx}{32 c^2}\\ &=\frac{45 \sqrt{x}}{16 c^3}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac{(45 b) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{32 c^3}\\ &=\frac{45 \sqrt{x}}{16 c^3}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac{(45 b) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{16 c^3}\\ &=\frac{45 \sqrt{x}}{16 c^3}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac{\left (45 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 c^3}-\frac{\left (45 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 c^3}\\ &=\frac{45 \sqrt{x}}{16 c^3}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac{\left (45 \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 )}{64 c^{7/2}}-\frac{\left (45 \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 )}{64 c^{7/2}}+\frac{\left (45 \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 )}{64 \sqrt{2} c^{13/4}}+\frac{\left (45 \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 )}{64 \sqrt{2} c^{13/4}}\\ &=\frac{45 \sqrt{x}}{16 c^3}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac{45 \sqrt [4]{b} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}-\frac{\left (45 \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 )}{32 \sqrt{2} c^{13/4}}+\frac{\left (45 \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 )}{32 \sqrt{2} c^{13/4}}\\ &=\frac{45 \sqrt{x}}{16 c^3}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{13/4}}+\frac{45 \sqrt [4]{b} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}\\ \end{align*}
Mathematica [A] time = 0.261666, size = 220, normalized size = 0.88 \[ \frac{\frac{8 \sqrt [4]{c} \sqrt{x} \left (45 b^2+81 b c x^2+32 c^2 x^4\right )}{\left (b+c x^2\right )^2}+45 \sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-45 \sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+90 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-90 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{128 c^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 178, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{x}}{{c}^{3}}}+{\frac{17\,b}{16\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{13\,{b}^{2}}{16\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{45\,\sqrt{2}}{128\,{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{45\,\sqrt{2}}{64\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{45\,\sqrt{2}}{64\,{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 = 1.38763, size = 579, normalized size = 2.31 \begin{align*} -\frac{180 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{c^{6} \sqrt{-\frac{b}{c^{13}}} + x} c^{10} \left (-\frac{b}{c^{13}}\right )^{\frac{3}{4}} - c^{10} \sqrt{x} \left (-\frac{b}{c^{13}}\right )^{\frac{3}{4}}}{b}\right ) + 45 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} \log \left (45 \, c^{3} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} + 45 \, \sqrt{x}\right ) - 45 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} \log \left (-45 \, c^{3} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} + 45 \, \sqrt{x}\right ) - 4 \,{\left (32 \, c^{2} x^{4} + 81 \, b c x^{2} + 45 \, b^{2}\right )} \sqrt{x}}{64 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} 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 = 1.22288, size = 281, normalized size = 1.12 \begin{align*} -\frac{45 \, \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 )}{64 \, c^{4}} - \frac{45 \, \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 )}{64 \, c^{4}} - \frac{45 \, \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 )}{128 \, c^{4}} + \frac{45 \, \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 )}{128 \, c^{4}} + \frac{2 \, \sqrt{x}}{c^{3}} + \frac{17 \, b c x^{\frac{5}{2}} + 13 \, b^{2} \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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