Optimal. Leaf size=155 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a g+b f)+3 b^2 g+8 c^2 e\right )}{8 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} (4 c f-3 b g)}{4 c^2}-\frac{d \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a}}+\frac{g x \sqrt{a+b x+c x^2}}{2 c} \]
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Rubi [A] time = 0.256308, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1653, 843, 621, 206, 724} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a g+b f)+3 b^2 g+8 c^2 e\right )}{8 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} (4 c f-3 b g)}{4 c^2}-\frac{d \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a}}+\frac{g x \sqrt{a+b x+c x^2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3}{x \sqrt{a+b x+c x^2}} \, dx &=\frac{g x \sqrt{a+b x+c x^2}}{2 c}+\frac{\int \frac{2 c d+(2 c e-a g) x+\frac{1}{2} (4 c f-3 b g) x^2}{x \sqrt{a+b x+c x^2}} \, dx}{2 c}\\ &=\frac{(4 c f-3 b g) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{g x \sqrt{a+b x+c x^2}}{2 c}+\frac{\int \frac{2 c^2 d+\frac{1}{4} \left (8 c^2 e+3 b^2 g-4 c (b f+a g)\right ) x}{x \sqrt{a+b x+c x^2}} \, dx}{2 c^2}\\ &=\frac{(4 c f-3 b g) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{g x \sqrt{a+b x+c x^2}}{2 c}+d \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx+\frac{\left (8 c^2 e+3 b^2 g-4 c (b f+a g)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c^2}\\ &=\frac{(4 c f-3 b g) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{g x \sqrt{a+b x+c x^2}}{2 c}-(2 d) \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 )+\frac{\left (8 c^2 e+3 b^2 g-4 c (b f+a g)\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 )}{4 c^2}\\ &=\frac{(4 c f-3 b g) \sqrt{a+b x+c x^2}}{4 c^2}+\frac{g x \sqrt{a+b x+c x^2}}{2 c}-\frac{d \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a}}+\frac{\left (8 c^2 e+3 b^2 g-4 c (b f+a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.384614, size = 134, normalized size = 0.86 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (-4 c (a g+b f)+3 b^2 g+8 c^2 e\right )}{8 c^{5/2}}+\frac{\sqrt{a+x (b+c x)} (-3 b g+4 c f+2 c g x)}{4 c^2}-\frac{d \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 220, normalized size = 1.4 \begin{align*}{\frac{gx}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,bg}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}g}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{ag}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{f}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{bf}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{e\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{d\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 19.8931, size = 1759, normalized size = 11.35 \begin{align*} \text{result too large to display} \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{d + e x + f x^{2} + g x^{3}}{x \sqrt{a + b x + c 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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