Optimal. Leaf size=167 \[ -\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{24 a^3 x}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{7/2}}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3} \]
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Rubi [A] time = 0.161842, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {834, 806, 724, 206} \[ -\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{24 a^3 x}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{7/2}}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x}{x^4 \sqrt{a+b x+c x^2}} \, dx &=-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3}-\frac{\int \frac{\frac{1}{2} (5 A b-6 a B)+2 A c x}{x^3 \sqrt{a+b x+c x^2}} \, dx}{3 a}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}+\frac{\int \frac{\frac{1}{4} \left (15 A b^2-18 a b B-16 a A c\right )+\frac{1}{2} (5 A b-6 a B) c x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{6 a^2}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{24 a^3 x}-\frac{\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{16 a^3}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{24 a^3 x}+\frac{\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\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{A \sqrt{a+b x+c x^2}}{3 a x^3}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}-\frac{\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{24 a^3 x}+\frac{\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\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 = 0.144477, size = 132, normalized size = 0.79 \[ \frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{16 a^{7/2}}-\frac{\sqrt{a+x (b+c x)} \left (4 a^2 (2 A+3 B x)-2 a x (5 A b+8 A c x+9 b B x)+15 A b^2 x^2\right )}{24 a^3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 283, normalized size = 1.7 \begin{align*} -{\frac{A}{3\,a{x}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,Ab}{12\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,A{b}^{2}}{8\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,A{b}^{3}}{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\,Abc}{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\,Ac}{3\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{B}{2\,a{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,bB}{4\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}B}{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{Bc}{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 = 3.72652, size = 751, normalized size = 4.5 \begin{align*} \left [\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c\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 a^{3} -{\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{2} + 2 \,{\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt{c x^{2} + b x + a}}{96 \, a^{4} x^{3}}, \frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c\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 a^{3} -{\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{2} + 2 \,{\left (6 \, B a^{3} - 5 \, A a^{2} b\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{A + B x}{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 = 1.2184, size = 690, normalized size = 4.13 \begin{align*} \frac{{\left (6 \, B a b^{2} - 5 \, A b^{3} - 8 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{3}} - \frac{18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} B a b^{2} - 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} A b^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} B a^{2} c + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} A a b c - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A a b^{3} - 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A a^{2} b c - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt{c} - 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac{3}{2}} + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} B a^{3} b^{2} - 33 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a^{2} b^{3} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} B a^{4} c - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt{c} - 48 \, A a^{3} b^{2} \sqrt{c} + 32 \, A a^{4} c^{\frac{3}{2}}}{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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