Optimal. Leaf size=194 \[ \frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}-\frac{\sqrt{a} \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a x+b x^3+c x^5}}+\frac{b \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} \sqrt{a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.208964, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1921, 1953, 1251, 843, 621, 206, 724} \[ \frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}-\frac{\sqrt{a} \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a x+b x^3+c x^5}}+\frac{b \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} \sqrt{a x+b x^3+c x^5}} \]
Antiderivative was successfully verified.
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Rule 1921
Rule 1953
Rule 1251
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\sqrt{a x+b x^3+c x^5}}{x^{3/2}} \, dx &=\frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}+\frac{1}{2} \int \frac{2 a+b x^2}{\sqrt{x} \sqrt{a x+b x^3+c x^5}} \, dx\\ &=\frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}+\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{2 a+b x^2}{x \sqrt{a+b x^2+c x^4}} \, dx}{2 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}+\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{2 a+b x}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{4 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}+\frac{\left (a \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 \sqrt{a x+b x^3+c x^5}}+\frac{\left (b \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 )}{4 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}-\frac{\left (a \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{\sqrt{a x+b x^3+c x^5}}+\frac{\left (b \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 )}{2 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}-\frac{\sqrt{a} \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a x+b x^3+c x^5}}+\frac{b \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 )}{4 \sqrt{c} \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [A] time = 0.0663108, size = 155, normalized size = 0.8 \[ \frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}-2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )+b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )\right )}{4 \sqrt{c} \sqrt{x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 136, normalized size = 0.7 \begin{align*} -{\frac{1}{4}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( 2\,\sqrt{a}\ln \left ({\frac{2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a}}{{x}^{2}}} \right ) \sqrt{c}-b\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 ) -2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}{\frac{1}{\sqrt{c}}}} \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^{5} + b x^{3} + a x}}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64065, size = 1575, normalized size = 8.12 \begin{align*} \left [\frac{b \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 ) + 2 \, \sqrt{a} c x \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} \sqrt{x}}{x^{5}}\right ) + 4 \, \sqrt{c x^{5} + b x^{3} + a x} c \sqrt{x}}{8 \, c x}, -\frac{b \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 ) - \sqrt{a} c x \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} \sqrt{x}}{x^{5}}\right ) - 2 \, \sqrt{c x^{5} + b x^{3} + a x} c \sqrt{x}}{4 \, c x}, \frac{4 \, \sqrt{-a} c x \arctan \left (\frac{\sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a} \sqrt{x}}{2 \,{\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right ) + b \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} c \sqrt{x}}{8 \, c x}, \frac{2 \, \sqrt{-a} c x \arctan \left (\frac{\sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a} \sqrt{x}}{2 \,{\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right ) - b \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} c \sqrt{x}}{4 \, c 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 \left (a + b x^{2} + c x^{4}\right )}}{x^{\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{\sqrt{c x^{5} + b x^{3} + a x}}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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