Optimal. Leaf size=155 \[ -\frac{a^2 (a B+3 A b)}{5 x^5}-\frac{a^3 A}{6 x^6}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{2 x^2}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{4 x^4}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{3 x^3}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{x}+c^2 \log (x) (A c+3 b B)+B c^3 x \]
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Rubi [A] time = 0.109169, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ -\frac{a^2 (a B+3 A b)}{5 x^5}-\frac{a^3 A}{6 x^6}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{2 x^2}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{4 x^4}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{3 x^3}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{x}+c^2 \log (x) (A c+3 b B)+B c^3 x \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x^7} \, dx &=\int \left (B c^3+\frac{a^3 A}{x^7}+\frac{a^2 (3 A b+a B)}{x^6}+\frac{3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^5}+\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^4}+\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^3}+\frac{3 c \left (b^2 B+A b c+a B c\right )}{x^2}+\frac{c^2 (3 b B+A c)}{x}\right ) \, dx\\ &=-\frac{a^3 A}{6 x^6}-\frac{a^2 (3 A b+a B)}{5 x^5}-\frac{3 a \left (a b B+A \left (b^2+a c\right )\right )}{4 x^4}-\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{3 x^3}-\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{2 x^2}-\frac{3 c \left (b^2 B+A b c+a B c\right )}{x}+B c^3 x+c^2 (3 b B+A c) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0909048, size = 169, normalized size = 1.09 \[ -\frac{3 a^2 x (3 A (4 b+5 c x)+5 B x (3 b+4 c x))+2 a^3 (5 A+6 B x)+15 a x^2 \left (A \left (3 b^2+8 b c x+6 c^2 x^2\right )+4 B x \left (b^2+3 b c x+3 c^2 x^2\right )\right )+10 x^3 \left (A b \left (2 b^2+9 b c x+18 c^2 x^2\right )+3 B x \left (6 b^2 c x+b^3-2 c^3 x^3\right )\right )-60 c^2 x^6 \log (x) (A c+3 b B)}{60 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 188, normalized size = 1.2 \begin{align*} B{c}^{3}x+A\ln \left ( x \right ){c}^{3}+3\,B\ln \left ( x \right ) b{c}^{2}-2\,{\frac{Aabc}{{x}^{3}}}-{\frac{A{b}^{3}}{3\,{x}^{3}}}-{\frac{B{a}^{2}c}{{x}^{3}}}-{\frac{Ba{b}^{2}}{{x}^{3}}}-{\frac{3\,aA{c}^{2}}{2\,{x}^{2}}}-{\frac{3\,A{b}^{2}c}{2\,{x}^{2}}}-3\,{\frac{abBc}{{x}^{2}}}-{\frac{{b}^{3}B}{2\,{x}^{2}}}-3\,{\frac{Ab{c}^{2}}{x}}-3\,{\frac{Ba{c}^{2}}{x}}-3\,{\frac{B{b}^{2}c}{x}}-{\frac{3\,Ab{a}^{2}}{5\,{x}^{5}}}-{\frac{B{a}^{3}}{5\,{x}^{5}}}-{\frac{3\,A{a}^{2}c}{4\,{x}^{4}}}-{\frac{3\,Aa{b}^{2}}{4\,{x}^{4}}}-{\frac{3\,B{a}^{2}b}{4\,{x}^{4}}}-{\frac{A{a}^{3}}{6\,{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09071, size = 219, normalized size = 1.41 \begin{align*} B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x\right ) - \frac{180 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 30 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30756, size = 381, normalized size = 2.46 \begin{align*} \frac{60 \, B c^{3} x^{7} + 60 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} \log \left (x\right ) - 180 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} - 30 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 10 \, A a^{3} - 20 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 39.2824, size = 178, normalized size = 1.15 \begin{align*} B c^{3} x + c^{2} \left (A c + 3 B b\right ) \log{\left (x \right )} - \frac{10 A a^{3} + x^{5} \left (180 A b c^{2} + 180 B a c^{2} + 180 B b^{2} c\right ) + x^{4} \left (90 A a c^{2} + 90 A b^{2} c + 180 B a b c + 30 B b^{3}\right ) + x^{3} \left (120 A a b c + 20 A b^{3} + 60 B a^{2} c + 60 B a b^{2}\right ) + x^{2} \left (45 A a^{2} c + 45 A a b^{2} + 45 B a^{2} b\right ) + x \left (36 A a^{2} b + 12 B a^{3}\right )}{60 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37135, size = 219, normalized size = 1.41 \begin{align*} B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac{180 \,{\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} x^{5} + 30 \,{\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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