Optimal. Leaf size=293 \[ \frac{2 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}} \]
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Rubi [A] time = 0.299394, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2020, 2025, 2032, 329, 305, 220, 1196} \[ \frac{2 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 c^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\sqrt{b x^2+c x^4}}{x^{9/2}} \, dx &=-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}}+\frac{1}{5} (2 c) \int \frac{1}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}+\frac{\left (2 c^2\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{5 b}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}+\frac{\left (2 c^2 x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{5 b \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}+\frac{\left (4 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 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}+\frac{\left (4 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 \sqrt{b} \sqrt{b x^2+c x^4}}-\frac{\left (4 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 \sqrt{b} \sqrt{b x^2+c x^4}}\\ &=\frac{4 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4}}{5 x^{7/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{5 b x^{3/2}}-\frac{4 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^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 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 )}{5 b^{3/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.013435, size = 57, normalized size = 0.19 \[ -\frac{2 \sqrt{x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{b}\right )}{5 x^{7/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.192, size = 224, normalized size = 0.8 \begin{align*}{\frac{2}{ \left ( 5\,c{x}^{2}+5\,b \right ) b}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 2\,\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-\sqrt{{ \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{ \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}bc-2\,{c}^{2}{x}^{4}-3\,bc{x}^{2}-{b}^{2} \right ){x}^{-{\frac{7}{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{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{9}{2}}}\,{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}}}{x^{\frac{9}{2}}}, 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{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{\frac{9}{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{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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