Optimal. Leaf size=173 \[ -\frac{\sqrt{a x^2+b x^3+c x^4}}{x^2}-\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}+\frac{\sqrt{c} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.124074, 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 = {1920, 1933, 843, 621, 206, 724} \[ -\frac{\sqrt{a x^2+b x^3+c x^4}}{x^2}-\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}+\frac{\sqrt{c} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1920
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^3} \, dx &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{x^2}+\frac{1}{2} \int \frac{b+2 c x}{\sqrt{a x^2+b x^3+c x^4}} \, dx\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{x^2}+\frac{\left (x \sqrt{a+b x+c x^2}\right ) \int \frac{b+2 c 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^2}+\frac{\left (b x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{2 \sqrt{a x^2+b x^3+c x^4}}+\frac{\left (c x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{\sqrt{a x^2+b x^3+c x^4}}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{x^2}-\frac{\left (b 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 (2 c 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^2}-\frac{b 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 )}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}+\frac{\sqrt{c} 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 )}{\sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.127059, size = 131, normalized size = 0.76 \[ -\frac{\sqrt{a+x (b+c x)} \left (b x \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )+2 \sqrt{a} \left (\sqrt{a+x (b+c x)}-\sqrt{c} x \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )}{2 \sqrt{a} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 174, normalized size = 1. \begin{align*}{\frac{1}{2\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 2\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}{x}^{2}-{c}^{{\frac{3}{2}}}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) xb-2\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{3/2}+2\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}xb+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) xa{c}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\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 \frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96342, size = 1489, normalized size = 8.61 \begin{align*} \left [\frac{2 \, a \sqrt{c} x^{2} \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 ) + \sqrt{a} b x^{2} \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}} a}{4 \, a x^{2}}, -\frac{4 \, a \sqrt{-c} x^{2} \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} b x^{2} \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}} a}{4 \, a x^{2}}, \frac{\sqrt{-a} b x^{2} \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 ) + a \sqrt{c} x^{2} \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{c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}, \frac{\sqrt{-a} b x^{2} \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 ) - 2 \, a \sqrt{-c} x^{2} \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}} a}{2 \, a x^{2}}\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^{3}}\, 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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