Optimal. Leaf size=146 \[ -\frac{2 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}} \]
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Rubi [A] time = 0.182391, 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 = {2020, 2025, 2032, 329, 220} \[ -\frac{2 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{b x^2+c x^4}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2025
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\sqrt{b x^2+c x^4}}{x^{11/2}} \, dx &=-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}}+\frac{1}{7} (2 c) \int \frac{1}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{\left (2 c^2\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{21 b}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{\left (2 c^2 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{21 b \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{\left (4 c^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 b \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 x^{9/2}}-\frac{4 c \sqrt{b x^2+c x^4}}{21 b x^{5/2}}-\frac{2 c^{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 b^{5/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0154649, size = 57, normalized size = 0.39 \[ -\frac{2 \sqrt{x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{b}\right )}{7 x^{9/2} \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 142, normalized size = 1. \begin{align*} -{\frac{2}{ \left ( 21\,c{x}^{2}+21\,b \right ) b}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \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 ) \sqrt{-bc}{x}^{3}c+2\,{c}^{2}{x}^{4}+5\,bc{x}^{2}+3\,{b}^{2} \right ){x}^{-{\frac{9}{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{11}{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{11}{2}}}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{4} + b x^{2}}}{x^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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