Optimal. Leaf size=159 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \left (8 a^2 f-4 a b e-4 a c d+3 b^2 d\right )}{8 a^{5/2}}+\frac{\sqrt{a+b x+c x^2} (3 b d-4 a e)}{4 a^2 x}-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}+\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.244307, antiderivative size = 159, 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 = {1650, 843, 621, 206, 724} \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \left (8 a^2 f-4 a b e-4 a c d+3 b^2 d\right )}{8 a^{5/2}}+\frac{\sqrt{a+b x+c x^2} (3 b d-4 a e)}{4 a^2 x}-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}+\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3}{x^3 \sqrt{a+b x+c x^2}} \, dx &=-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}-\frac{\int \frac{\frac{1}{2} (3 b d-4 a e)+(c d-2 a f) x-2 a g x^2}{x^2 \sqrt{a+b x+c x^2}} \, dx}{2 a}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}+\frac{(3 b d-4 a e) \sqrt{a+b x+c x^2}}{4 a^2 x}+\frac{\int \frac{\frac{1}{4} \left (3 b^2 d-4 a b e-4 a (c d-2 a f)\right )+2 a^2 g x}{x \sqrt{a+b x+c x^2}} \, dx}{2 a^2}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}+\frac{(3 b d-4 a e) \sqrt{a+b x+c x^2}}{4 a^2 x}+\frac{\left (3 b^2 d-4 a c d-4 a b e+8 a^2 f\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{8 a^2}+g \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}+\frac{(3 b d-4 a e) \sqrt{a+b x+c x^2}}{4 a^2 x}-\frac{\left (3 b^2 d-4 a c d-4 a b e+8 a^2 f\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 )}{4 a^2}+(2 g) \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 )\\ &=-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}+\frac{(3 b d-4 a e) \sqrt{a+b x+c x^2}}{4 a^2 x}-\frac{\left (3 b^2 d-4 a c d-4 a b e+8 a^2 f\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{5/2}}+\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.371441, size = 137, normalized size = 0.86 \[ \frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right ) \left (4 a b e+4 a (c d-2 a f)-3 b^2 d\right )}{8 a^{5/2}}+\frac{\sqrt{a+x (b+c x)} (3 b d x-2 a (d+2 e x))}{4 a^2 x^2}+\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 241, normalized size = 1.5 \begin{align*}{g\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{f\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{d}{2\,a{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,bd}{4\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}d}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{cd}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{e}{ax}\sqrt{c{x}^{2}+bx+a}}+{\frac{be}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \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 = 18.345, size = 1837, normalized size = 11.55 \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^{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 [B] time = 1.29618, size = 475, normalized size = 2.99 \begin{align*} -\frac{g \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c - b \sqrt{c} \right |}\right )}{\sqrt{c}} + \frac{{\left (3 \, b^{2} d - 4 \, a c d + 8 \, a^{2} f - 4 \, a b e\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{2}} - \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} b^{2} d - 4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a c d - 4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a b e - 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{2} \sqrt{c} e - 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a b^{2} d - 4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{2} c d + 4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{2} b e - 8 \, a^{2} b \sqrt{c} d + 8 \, a^{3} \sqrt{c} e}{4 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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