Optimal. Leaf size=320 \[ -\frac{21 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{10 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{21 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.364085, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2023, 2025, 2032, 329, 305, 220, 1196} \[ -\frac{21 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{21 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{b x^2+c x^4}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2023
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}+\frac{7 \int \frac{1}{x^{5/2} \sqrt{b x^2+c x^4}} \, dx}{2 b}\\ &=\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}-\frac{(21 c) \int \frac{1}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx}{10 b^2}\\ &=\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac{\left (21 c^2\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{10 b^3}\\ &=\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac{\left (21 c^2 x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{10 b^3 \sqrt{b x^2+c x^4}}\\ &=\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac{\left (21 c^2 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{5 b^3 \sqrt{b x^2+c x^4}}\\ &=\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac{\left (21 c^{3/2} x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{5 b^{5/2} \sqrt{b x^2+c x^4}}+\frac{\left (21 c^{3/2} x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{b}}}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{5 b^{5/2} \sqrt{b x^2+c x^4}}\\ &=\frac{1}{b x^{3/2} \sqrt{b x^2+c x^4}}-\frac{21 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b^3 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{7 \sqrt{b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac{21 c \sqrt{b x^2+c x^4}}{5 b^3 x^{3/2}}+\frac{21 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{b x^2+c x^4}}-\frac{21 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0170526, size = 60, normalized size = 0.19 \[ -\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left (-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-\frac{c x^2}{b}\right )}{5 b x^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.197, size = 222, normalized size = 0.7 \begin{align*} -{\frac{c{x}^{2}+b}{10\,{b}^{3}}\sqrt{x} \left ( 42\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}bc-21\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}bc-42\,{c}^{2}{x}^{4}-28\,bc{x}^{2}+4\,{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{x}}{c^{2} x^{9} + 2 \, b c x^{7} + b^{2} x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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