Optimal. Leaf size=129 \[ \frac{\left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\sqrt{x} \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 c^{3/2} \sqrt{a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.0933199, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1918, 1914, 1107, 621, 206} \[ \frac{\left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\sqrt{x} \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 c^{3/2} \sqrt{a x+b x^3+c x^5}} \]
Antiderivative was successfully verified.
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Rule 1918
Rule 1914
Rule 1107
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \sqrt{x} \sqrt{a x+b x^3+c x^5} \, dx &=\frac{\left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\left (b^2-4 a c\right ) \int \frac{x^{3/2}}{\sqrt{a x+b x^3+c x^5}} \, dx}{8 c}\\ &=\frac{\left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\left (\left (b^2-4 a c\right ) \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{x}{\sqrt{a+b x^2+c x^4}} \, dx}{8 c \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\left (\left (b^2-4 a c\right ) \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 c \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\left (\left (b^2-4 a c\right ) \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{8 c \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (b+2 c x^2\right ) \sqrt{a x+b x^3+c x^5}}{8 c \sqrt{x}}-\frac{\left (b^2-4 a c\right ) \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 c^{3/2} \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [A] time = 0.0775967, size = 126, normalized size = 0.98 \[ \frac{\sqrt{x \left (a+b x^2+c x^4\right )} \left (\frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{4 c}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{2 \sqrt{x} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 157, normalized size = 1.2 \begin{align*}{\frac{1}{16}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( 4\,{x}^{2}{c}^{3/2}\sqrt{c{x}^{4}+b{x}^{2}+a}+4\,\ln \left ( 1/2\,{\frac{2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b}{\sqrt{c}}} \right ) ac-\ln \left ({\frac{1}{2} \left ( 2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b \right ){\frac{1}{\sqrt{c}}}} \right ){b}^{2}+2\,b\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c} \right ){c}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \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^{5} + b x^{3} + a x} \sqrt{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39673, size = 547, normalized size = 4.24 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{5} + 8 \, b c x^{3} + 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c x^{2} + b\right )} \sqrt{c} \sqrt{x} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c^{2} x^{2} + b c\right )} \sqrt{x}}{32 \, c^{2} x}, \frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c x^{2} + b\right )} \sqrt{-c} \sqrt{x}}{2 \,{\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right ) + 2 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c^{2} x^{2} + b c\right )} \sqrt{x}}{16 \, c^{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 \sqrt{x} \sqrt{x \left (a + b x^{2} + c x^{4}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19448, size = 171, normalized size = 1.33 \begin{align*} \frac{1}{8} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, x^{2} + \frac{b}{c}\right )} + \frac{{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{3}{2}}} - \frac{b^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 4 \, a c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2 \, \sqrt{a} b \sqrt{c}}{16 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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