Optimal. Leaf size=146 \[ -\frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{21 c^{5/4} \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4} \]
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Rubi [A] time = 0.184986, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2021, 2024, 2032, 329, 220} \[ -\frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4} \]
Antiderivative was successfully verified.
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Rule 2021
Rule 2024
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \sqrt{x} \sqrt{b x^2+c x^4} \, dx &=\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4}+\frac{1}{7} (2 b) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4}-\frac{\left (2 b^2\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{21 c}\\ &=\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4}-\frac{\left (2 b^2 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{21 c \sqrt{b x^2+c x^4}}\\ &=\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4}-\frac{\left (4 b^2 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{21 c \sqrt{b x^2+c x^4}}\\ &=\frac{4 b \sqrt{b x^2+c x^4}}{21 c \sqrt{x}}+\frac{2}{7} x^{3/2} \sqrt{b x^2+c x^4}-\frac{2 b^{7/4} 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 )}{21 c^{5/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0339688, size = 86, normalized size = 0.59 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (\left (b+c x^2\right ) \sqrt{\frac{c x^2}{b}+1}-b \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )\right )}{7 c \sqrt{x} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.182, size = 145, normalized size = 1. \begin{align*} -{\frac{2}{ \left ( 21\,c{x}^{2}+21\,b \right ){c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ({b}^{2}\sqrt{-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 ) -3\,{c}^{3}{x}^{5}-5\,b{c}^{2}{x}^{3}-2\,{b}^{2}cx \right ){x}^{-{\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 \sqrt{c x^{4} + b x^{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 (\sqrt{c x^{4} + b x^{2}} \sqrt{x}, 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 \sqrt{x} \sqrt{x^{2} \left (b + c x^{2}\right )}\, 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 \sqrt{c x^{4} + b x^{2}} \sqrt{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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