Optimal. Leaf size=231 \[ \frac{\left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{9/2}}+\frac{\sqrt{a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{192 a^4 x}-\frac{\sqrt{a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{96 a^3 x^2}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4} \]
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Rubi [A] time = 0.276561, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {834, 806, 724, 206} \[ \frac{\left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{9/2}}+\frac{\sqrt{a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{192 a^4 x}-\frac{\sqrt{a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{96 a^3 x^2}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4} \]
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^5 \sqrt{a+b x+c x^2}} \, dx &=-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4}-\frac{\int \frac{\frac{1}{2} (7 A b-8 a B)+3 A c x}{x^4 \sqrt{a+b x+c x^2}} \, dx}{4 a}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}+\frac{\int \frac{\frac{1}{4} \left (35 A b^2-40 a b B-36 a A c\right )+(7 A b-8 a B) c x}{x^3 \sqrt{a+b x+c x^2}} \, dx}{12 a^2}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}-\frac{\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt{a+b x+c x^2}}{96 a^3 x^2}-\frac{\int \frac{\frac{1}{8} \left (-8 a B \left (15 b^2-16 a c\right )+5 A \left (21 b^3-44 a b c\right )\right )+\frac{1}{4} c \left (35 A b^2-40 a b B-36 a A c\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{24 a^3}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}-\frac{\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt{a+b x+c x^2}}{96 a^3 x^2}+\frac{\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt{a+b x+c x^2}}{192 a^4 x}-\frac{\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{128 a^4}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}-\frac{\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt{a+b x+c x^2}}{96 a^3 x^2}+\frac{\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt{a+b x+c x^2}}{192 a^4 x}+\frac{\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )\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 )}{64 a^4}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}-\frac{\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt{a+b x+c x^2}}{96 a^3 x^2}+\frac{\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt{a+b x+c x^2}}{192 a^4 x}+\frac{\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.235602, size = 176, normalized size = 0.76 \[ \frac{\sqrt{a+x (b+c x)} \left (8 a^2 x (A (7 b+9 c x)+2 B x (5 b+8 c x))-16 a^3 (3 A+4 B x)-10 a b x^2 (7 A b+22 A c x+12 b B x)+105 A b^3 x^3\right )}{192 a^4 x^4}-\frac{\left (A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )+8 a b B \left (12 a c-5 b^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{128 a^{9/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 417, normalized size = 1.8 \begin{align*} -{\frac{B}{3\,a{x}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,bB}{12\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{b}^{2}B}{8\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{3}B}{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\,Bcb}{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\,Bc}{3\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{A}{4\,a{x}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{7\,Ab}{24\,{a}^{2}{x}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,A{b}^{2}}{96\,{a}^{3}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,A{b}^{3}}{64\,{a}^{4}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{15\,A{b}^{2}c}{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{55\,Abc}{48\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Ac}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,A{c}^{2}}{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}}}} \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 = 6.06718, size = 1007, normalized size = 4.36 \begin{align*} \left [-\frac{3 \,{\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \,{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} c\right )} \sqrt{a} x^{4} \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 (48 \, A a^{4} +{\left (120 \, B a^{2} b^{2} - 105 \, A a b^{3} - 4 \,{\left (32 \, B a^{3} - 55 \, A a^{2} b\right )} c\right )} x^{3} - 2 \,{\left (40 \, B a^{3} b - 35 \, A a^{2} b^{2} + 36 \, A a^{3} c\right )} x^{2} + 8 \,{\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \, a^{5} x^{4}}, -\frac{3 \,{\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \,{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} c\right )} \sqrt{-a} x^{4} \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 (48 \, A a^{4} +{\left (120 \, B a^{2} b^{2} - 105 \, A a b^{3} - 4 \,{\left (32 \, B a^{3} - 55 \, A a^{2} b\right )} c\right )} x^{3} - 2 \,{\left (40 \, B a^{3} b - 35 \, A a^{2} b^{2} + 36 \, A a^{3} c\right )} x^{2} + 8 \,{\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt{c x^{2} + b x + a}}{384 \, a^{5} x^{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{A + B x}{x^{5} \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 = 1.34663, size = 1193, normalized size = 5.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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