Optimal. Leaf size=264 \[ -\frac{165 c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}-\frac{165 c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{19/4}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac{165}{112 b^3 x^{7/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.231428, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1584, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{165 c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}-\frac{165 c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{19/4}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac{165}{112 b^3 x^{7/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 290
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{1}{x^{9/2} \left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15 \int \frac{1}{x^{9/2} \left (b+c x^2\right )^2} \, dx}{8 b}\\ &=\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac{165 \int \frac{1}{x^{9/2} \left (b+c x^2\right )} \, dx}{32 b^2}\\ &=-\frac{165}{112 b^3 x^{7/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac{(165 c) \int \frac{1}{x^{5/2} \left (b+c x^2\right )} \, dx}{32 b^3}\\ &=-\frac{165}{112 b^3 x^{7/2}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac{\left (165 c^2\right ) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{32 b^4}\\ &=-\frac{165}{112 b^3 x^{7/2}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac{\left (165 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{16 b^4}\\ &=-\frac{165}{112 b^3 x^{7/2}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac{\left (165 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{9/2}}+\frac{\left (165 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{32 b^{9/2}}\\ &=-\frac{165}{112 b^3 x^{7/2}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac{\left (165 c^{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 )}{64 b^{9/2}}+\frac{\left (165 c^{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 )}{64 b^{9/2}}-\frac{\left (165 c^{7/4}\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 )}{64 \sqrt{2} b^{19/4}}-\frac{\left (165 c^{7/4}\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 )}{64 \sqrt{2} b^{19/4}}\\ &=-\frac{165}{112 b^3 x^{7/2}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac{165 c^{7/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}+\frac{\left (165 c^{7/4}\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 )}{32 \sqrt{2} b^{19/4}}-\frac{\left (165 c^{7/4}\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 )}{32 \sqrt{2} b^{19/4}}\\ &=-\frac{165}{112 b^3 x^{7/2}}+\frac{55 c}{16 b^4 x^{3/2}}+\frac{1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac{15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac{165 c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{19/4}}-\frac{165 c^{7/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}+\frac{165 c^{7/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} b^{19/4}}\\ \end{align*}
Mathematica [C] time = 0.0063249, size = 29, normalized size = 0.11 \[ -\frac{2 \, _2F_1\left (-\frac{7}{4},3;-\frac{3}{4};-\frac{c x^2}{b}\right )}{7 b^3 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 198, normalized size = 0.8 \begin{align*} -{\frac{2}{7\,{b}^{3}}{x}^{-{\frac{7}{2}}}}+2\,{\frac{c}{{b}^{4}{x}^{3/2}}}+{\frac{23\,{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{27\,{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}+{\frac{165\,{c}^{2}\sqrt{2}}{128\,{b}^{5}}\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{165\,{c}^{2}\sqrt{2}}{64\,{b}^{5}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{165\,{c}^{2}\sqrt{2}}{64\,{b}^{5}}\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 = 1.40631, size = 689, normalized size = 2.61 \begin{align*} \frac{4620 \,{\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac{c^{7}}{b^{19}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{14} c^{2} \sqrt{x} \left (-\frac{c^{7}}{b^{19}}\right )^{\frac{3}{4}} - \sqrt{b^{10} \sqrt{-\frac{c^{7}}{b^{19}}} + c^{4} x} b^{14} \left (-\frac{c^{7}}{b^{19}}\right )^{\frac{3}{4}}}{c^{7}}\right ) + 1155 \,{\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac{c^{7}}{b^{19}}\right )^{\frac{1}{4}} \log \left (165 \, b^{5} \left (-\frac{c^{7}}{b^{19}}\right )^{\frac{1}{4}} + 165 \, c^{2} \sqrt{x}\right ) - 1155 \,{\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac{c^{7}}{b^{19}}\right )^{\frac{1}{4}} \log \left (-165 \, b^{5} \left (-\frac{c^{7}}{b^{19}}\right )^{\frac{1}{4}} + 165 \, c^{2} \sqrt{x}\right ) + 4 \,{\left (385 \, c^{3} x^{6} + 605 \, b c^{2} x^{4} + 160 \, b^{2} c x^{2} - 32 \, b^{3}\right )} \sqrt{x}}{448 \,{\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\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 = 1.19944, size = 302, normalized size = 1.14 \begin{align*} \frac{165 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \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 \, b^{5}} + \frac{165 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \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 \, b^{5}} + \frac{165 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5}} - \frac{165 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5}} + \frac{23 \, c^{3} x^{\frac{5}{2}} + 27 \, b c^{2} \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{4}} + \frac{2 \,{\left (7 \, c x^{2} - b\right )}}{7 \, b^{4} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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