Optimal. Leaf size=173 \[ \frac{\sqrt{a x^2+b x^3+c x^4}}{x}-\frac{\sqrt{a} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}+\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.126323, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1921, 1933, 843, 621, 206, 724} \[ \frac{\sqrt{a x^2+b x^3+c x^4}}{x}-\frac{\sqrt{a} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}+\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1921
Rule 1933
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^2} \, dx &=\frac{\sqrt{a x^2+b x^3+c x^4}}{x}+\frac{1}{2} \int \frac{2 a+b x}{\sqrt{a x^2+b x^3+c x^4}} \, dx\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{x}+\frac{\left (x \sqrt{a+b x+c x^2}\right ) \int \frac{2 a+b x}{x \sqrt{a+b x+c x^2}} \, dx}{2 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{x}+\frac{\left (a x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{\sqrt{a x^2+b x^3+c x^4}}+\frac{\left (b x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{x}-\frac{\left (2 a x \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}+\frac{\left (b x \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\sqrt{a x^2+b x^3+c x^4}}{x}-\frac{\sqrt{a} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}+\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.110016, size = 134, normalized size = 0.77 \[ \frac{x \sqrt{a+x (b+c x)} \left (2 \sqrt{c} \sqrt{a+x (b+c x)}-2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )+b \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{2 \sqrt{c} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 126, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 2\,\sqrt{a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c}-2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}-b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b \right ){\frac{1}{\sqrt{c}}}} \right ) \right ){\frac{1}{\sqrt{c{x}^{2}+bx+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^{4} + b x^{3} + a x^{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84208, size = 1457, normalized size = 8.42 \begin{align*} \left [\frac{b \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 2 \, \sqrt{a} c x \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{4 \, c x}, -\frac{b \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - \sqrt{a} c x \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{2 \, c x}, \frac{4 \, \sqrt{-a} c x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + b \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{4 \, c x}, \frac{2 \, \sqrt{-a} c x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - b \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{2 \, 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^{2} \left (a + b x + c x^{2}\right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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