Optimal. Leaf size=177 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right )}{16 c^{7/2}}+\frac{\sqrt{a+b x+c x^2} \left (-16 a c g+15 b^2 g-18 b c f+24 c^2 e\right )}{24 c^3}+\frac{x \sqrt{a+b x+c x^2} (6 c f-5 b g)}{12 c^2}+\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c} \]
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Rubi [A] time = 0.235167, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1661, 640, 621, 206} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right )}{16 c^{7/2}}+\frac{\sqrt{a+b x+c x^2} \left (-16 a c g+15 b^2 g-18 b c f+24 c^2 e\right )}{24 c^3}+\frac{x \sqrt{a+b x+c x^2} (6 c f-5 b g)}{12 c^2}+\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3}{\sqrt{a+b x+c x^2}} \, dx &=\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c}+\frac{\int \frac{3 c d+(3 c e-2 a g) x+\frac{1}{2} (6 c f-5 b g) x^2}{\sqrt{a+b x+c x^2}} \, dx}{3 c}\\ &=\frac{(6 c f-5 b g) x \sqrt{a+b x+c x^2}}{12 c^2}+\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c}+\frac{\int \frac{\frac{1}{2} \left (12 c^2 d-6 a c f+5 a b g\right )+\frac{1}{4} \left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) x}{\sqrt{a+b x+c x^2}} \, dx}{6 c^2}\\ &=\frac{\left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) \sqrt{a+b x+c x^2}}{24 c^3}+\frac{(6 c f-5 b g) x \sqrt{a+b x+c x^2}}{12 c^2}+\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c}+\frac{\left (16 c^3 d-8 c^2 (b e+a f)-5 b^3 g+6 b c (b f+2 a g)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c^3}\\ &=\frac{\left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) \sqrt{a+b x+c x^2}}{24 c^3}+\frac{(6 c f-5 b g) x \sqrt{a+b x+c x^2}}{12 c^2}+\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c}+\frac{\left (16 c^3 d-8 c^2 (b e+a f)-5 b^3 g+6 b c (b f+2 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 )}{8 c^3}\\ &=\frac{\left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) \sqrt{a+b x+c x^2}}{24 c^3}+\frac{(6 c f-5 b g) x \sqrt{a+b x+c x^2}}{12 c^2}+\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c}+\frac{\left (16 c^3 d-8 c^2 (b e+a f)-5 b^3 g+6 b c (b f+2 a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.271913, size = 141, normalized size = 0.8 \[ \frac{3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right )+2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-2 c (8 a g+9 b f+5 b g x)+15 b^2 g+4 c^2 (6 e+x (3 f+2 g x))\right )}{48 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.086, size = 333, normalized size = 1.9 \begin{align*}{\frac{g{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,bgx}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{2}g}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{b}^{3}g}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,bga}{4}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{2\,ag}{3\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{fx}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,bf}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}f}{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{af}{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{e}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{be}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{d\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.53496, size = 803, normalized size = 4.54 \begin{align*} \left [\frac{3 \,{\left (16 \, c^{3} d - 8 \, b c^{2} e + 2 \,{\left (3 \, b^{2} c - 4 \, a c^{2}\right )} f -{\left (5 \, b^{3} - 12 \, a b c\right )} g\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) + 4 \,{\left (8 \, c^{3} g x^{2} + 24 \, c^{3} e - 18 \, b c^{2} f +{\left (15 \, b^{2} c - 16 \, a c^{2}\right )} g + 2 \,{\left (6 \, c^{3} f - 5 \, b c^{2} g\right )} x\right )} \sqrt{c x^{2} + b x + a}}{96 \, c^{4}}, -\frac{3 \,{\left (16 \, c^{3} d - 8 \, b c^{2} e + 2 \,{\left (3 \, b^{2} c - 4 \, a c^{2}\right )} f -{\left (5 \, b^{3} - 12 \, a b c\right )} g\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \,{\left (8 \, c^{3} g x^{2} + 24 \, c^{3} e - 18 \, b c^{2} f +{\left (15 \, b^{2} c - 16 \, a c^{2}\right )} g + 2 \,{\left (6 \, c^{3} f - 5 \, b c^{2} g\right )} x\right )} \sqrt{c x^{2} + b x + a}}{48 \, c^{4}}\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{d + e x + f x^{2} + g x^{3}}{\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 [A] time = 1.18691, size = 201, normalized size = 1.14 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (\frac{4 \, g x}{c} + \frac{6 \, c^{2} f - 5 \, b c g}{c^{3}}\right )} x - \frac{18 \, b c f - 15 \, b^{2} g + 16 \, a c g - 24 \, c^{2} e}{c^{3}}\right )} - \frac{{\left (16 \, c^{3} d + 6 \, b^{2} c f - 8 \, a c^{2} f - 5 \, b^{3} g + 12 \, a b c g - 8 \, b c^{2} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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