Optimal. Leaf size=230 \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4}}-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )} \]
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Rubi [A] time = 0.184947, 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, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4}}-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )} \]
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 )^2} \, dx &=\int \frac{1}{x^{5/2} \left (b+c x^2\right )^2} \, dx\\ &=\frac{1}{2 b x^{3/2} \left (b+c x^2\right )}+\frac{7 \int \frac{1}{x^{5/2} \left (b+c x^2\right )} \, dx}{4 b}\\ &=-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )}-\frac{(7 c) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{4 b^2}\\ &=-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )}-\frac{(7 c) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{2 b^2}\\ &=-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )}-\frac{(7 c) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 b^{5/2}}-\frac{(7 c) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{4 b^{5/2}}\\ &=-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )}-\frac{\left (7 \sqrt{c}\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 b^{5/2}}-\frac{\left (7 \sqrt{c}\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 b^{5/2}}+\frac{\left (7 c^{3/4}\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} b^{11/4}}+\frac{\left (7 c^{3/4}\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} b^{11/4}}\\ &=-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )}+\frac{7 c^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}-\frac{\left (7 c^{3/4}\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} b^{11/4}}+\frac{\left (7 c^{3/4}\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} b^{11/4}}\\ &=-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4}}+\frac{7 c^{3/4} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.0069592, size = 29, normalized size = 0.13 \[ -\frac{2 \, _2F_1\left (-\frac{3}{4},2;\frac{1}{4};-\frac{c x^2}{b}\right )}{3 b^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 161, normalized size = 0.7 \begin{align*} -{\frac{2}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{c}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{7\,c\sqrt{2}}{16\,{b}^{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{7\,c\sqrt{2}}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{7\,c\sqrt{2}}{8\,{b}^{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 = 1.49951, size = 518, normalized size = 2.25 \begin{align*} -\frac{84 \,{\left (b^{2} c x^{4} + b^{3} x^{2}\right )} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{8} c \sqrt{x} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{3}{4}} - \sqrt{b^{6} \sqrt{-\frac{c^{3}}{b^{11}}} + c^{2} x} b^{8} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{3}{4}}}{c^{3}}\right ) + 21 \,{\left (b^{2} c x^{4} + b^{3} x^{2}\right )} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} \log \left (7 \, b^{3} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} + 7 \, c \sqrt{x}\right ) - 21 \,{\left (b^{2} c x^{4} + b^{3} x^{2}\right )} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-7 \, b^{3} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} + 7 \, c \sqrt{x}\right ) + 4 \,{\left (7 \, c x^{2} + 4 \, b\right )} \sqrt{x}}{24 \,{\left (b^{2} c x^{4} + b^{3} x^{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 = 1.18647, size = 265, normalized size = 1.15 \begin{align*} -\frac{7 \, \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 \, b^{3}} - \frac{7 \, \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 \, b^{3}} - \frac{7 \, \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 \, b^{3}} + \frac{7 \, \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 \, b^{3}} - \frac{c \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} b^{2}} - \frac{2}{3 \, b^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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