Optimal. Leaf size=186 \[ \frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \left (8 a^2 (c e-2 a g)-6 a b^2 e-4 a b (3 c d-2 a f)+5 b^3 d\right )}{16 a^{7/2}}-\frac{\sqrt{a+b x+c x^2} \left (24 a^2 f-18 a b e-16 a c d+15 b^2 d\right )}{24 a^3 x}+\frac{\sqrt{a+b x+c x^2} (5 b d-6 a e)}{12 a^2 x^2}-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3} \]
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Rubi [A] time = 0.319922, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {1650, 806, 724, 206} \[ \frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right ) \left (8 a^2 (c e-2 a g)-6 a b^2 e-4 a b (3 c d-2 a f)+5 b^3 d\right )}{16 a^{7/2}}-\frac{\sqrt{a+b x+c x^2} \left (24 a^2 f-18 a b e-16 a c d+15 b^2 d\right )}{24 a^3 x}+\frac{\sqrt{a+b x+c x^2} (5 b d-6 a e)}{12 a^2 x^2}-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2+g x^3}{x^4 \sqrt{a+b x+c x^2}} \, dx &=-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3}-\frac{\int \frac{\frac{1}{2} (5 b d-6 a e)+(2 c d-3 a f) x-3 a g x^2}{x^3 \sqrt{a+b x+c x^2}} \, dx}{3 a}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 b d-6 a e) \sqrt{a+b x+c x^2}}{12 a^2 x^2}+\frac{\int \frac{\frac{1}{4} \left (15 b^2 d-16 a c d-18 a b e+24 a^2 f\right )+\frac{1}{2} \left (5 b c d-6 a c e+12 a^2 g\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{6 a^2}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 b d-6 a e) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{\left (15 b^2 d-16 a c d-18 a b e+24 a^2 f\right ) \sqrt{a+b x+c x^2}}{24 a^3 x}-\frac{\left (5 b^3 d-6 a b^2 e-4 a b (3 c d-2 a f)+8 a^2 (c e-2 a g)\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{16 a^3}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 b d-6 a e) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{\left (15 b^2 d-16 a c d-18 a b e+24 a^2 f\right ) \sqrt{a+b x+c x^2}}{24 a^3 x}+\frac{\left (5 b^3 d-6 a b^2 e-4 a b (3 c d-2 a f)+8 a^2 (c e-2 a g)\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 )}{8 a^3}\\ &=-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 b d-6 a e) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{\left (15 b^2 d-16 a c d-18 a b e+24 a^2 f\right ) \sqrt{a+b x+c x^2}}{24 a^3 x}+\frac{\left (5 b^3 d-6 a b^2 e-4 a b (3 c d-2 a f)+8 a^2 (c e-2 a g)\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.313395, size = 150, normalized size = 0.81 \[ \frac{\tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right ) \left (8 a^2 (c e-2 a g)-6 a b^2 e+4 a b (2 a f-3 c d)+5 b^3 d\right )}{16 a^{7/2}}-\frac{\sqrt{a+x (b+c x)} \left (4 a^2 (2 d+3 x (e+2 f x))-2 a x (5 b d+9 b e x+8 c d x)+15 b^2 d x^2\right )}{24 a^3 x^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 375, normalized size = 2. \begin{align*} -{g\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}{3\,a{x}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,bd}{12\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{b}^{2}d}{8\,x{a}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{3}d}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{3\,bcd}{4}\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{2\,cd}{3\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{e}{2\,a{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,be}{4\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}e}{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{ce}{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{f}{ax}\sqrt{c{x}^{2}+bx+a}}+{\frac{bf}{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 = 21.0011, size = 848, normalized size = 4.56 \begin{align*} \left [-\frac{3 \,{\left (8 \, a^{2} b f - 16 \, a^{3} g +{\left (5 \, b^{3} - 12 \, a b c\right )} d - 2 \,{\left (3 \, a b^{2} - 4 \, a^{2} c\right )} e\right )} \sqrt{a} x^{3} \log \left (-\frac{8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{3} d -{\left (18 \, a^{2} b e - 24 \, a^{3} f -{\left (15 \, a b^{2} - 16 \, a^{2} c\right )} d\right )} x^{2} - 2 \,{\left (5 \, a^{2} b d - 6 \, a^{3} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{96 \, a^{4} x^{3}}, -\frac{3 \,{\left (8 \, a^{2} b f - 16 \, a^{3} g +{\left (5 \, b^{3} - 12 \, a b c\right )} d - 2 \,{\left (3 \, a b^{2} - 4 \, a^{2} c\right )} e\right )} \sqrt{-a} x^{3} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \,{\left (8 \, a^{3} d -{\left (18 \, a^{2} b e - 24 \, a^{3} f -{\left (15 \, a b^{2} - 16 \, a^{2} c\right )} d\right )} x^{2} - 2 \,{\left (5 \, a^{2} b d - 6 \, a^{3} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{48 \, a^{4} x^{3}}\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}}{x^{4} \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 = 1.18562, size = 930, normalized size = 5. \begin{align*} -\frac{{\left (5 \, b^{3} d - 12 \, a b c d + 8 \, a^{2} b f - 16 \, a^{3} g - 6 \, a b^{2} e + 8 \, a^{2} c e\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{3}} + \frac{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} b^{3} d - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} a b c d + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} a^{2} b f - 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} a b^{2} e + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} a^{2} c e + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a^{3} \sqrt{c} f - 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a b^{3} d + 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a^{2} b c d - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a^{3} b f + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a^{2} b^{2} e + 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{3} c^{\frac{3}{2}} d - 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{4} \sqrt{c} f + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{3} b \sqrt{c} e + 33 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{2} b^{3} d + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{3} b c d + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{4} b f - 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{3} b^{2} e - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{4} c e + 48 \, a^{3} b^{2} \sqrt{c} d - 32 \, a^{4} c^{\frac{3}{2}} d + 48 \, a^{5} \sqrt{c} f - 48 \, a^{4} b \sqrt{c} e}{24 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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