Optimal. Leaf size=134 \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)-\frac{a^5 (a B+6 A b)}{3 x^3}-\frac{3 a^4 b (2 a B+5 A b)}{2 x^2}-\frac{a^6 A}{4 x^4}+\frac{1}{2} b^5 x^2 (6 a B+A b)+3 a b^4 x (5 a B+2 A b)+\frac{1}{3} b^6 B x^3 \]
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Rubi [A] time = 0.0820024, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)-\frac{a^5 (a B+6 A b)}{3 x^3}-\frac{3 a^4 b (2 a B+5 A b)}{2 x^2}-\frac{a^6 A}{4 x^4}+\frac{1}{2} b^5 x^2 (6 a B+A b)+3 a b^4 x (5 a B+2 A b)+\frac{1}{3} b^6 B x^3 \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^5} \, dx\\ &=\int \left (3 a b^4 (2 A b+5 a B)+\frac{a^6 A}{x^5}+\frac{a^5 (6 A b+a B)}{x^4}+\frac{3 a^4 b (5 A b+2 a B)}{x^3}+\frac{5 a^3 b^2 (4 A b+3 a B)}{x^2}+\frac{5 a^2 b^3 (3 A b+4 a B)}{x}+b^5 (A b+6 a B) x+b^6 B x^2\right ) \, dx\\ &=-\frac{a^6 A}{4 x^4}-\frac{a^5 (6 A b+a B)}{3 x^3}-\frac{3 a^4 b (5 A b+2 a B)}{2 x^2}-\frac{5 a^3 b^2 (4 A b+3 a B)}{x}+3 a b^4 (2 A b+5 a B) x+\frac{1}{2} b^5 (A b+6 a B) x^2+\frac{1}{3} b^6 B x^3+5 a^2 b^3 (3 A b+4 a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0506153, size = 128, normalized size = 0.96 \[ -\frac{15 a^4 b^2 (A+2 B x)}{2 x^2}+5 a^2 b^3 \log (x) (4 a B+3 A b)-\frac{20 a^3 A b^3}{x}-\frac{a^5 b (2 A+3 B x)}{x^3}-\frac{a^6 (3 A+4 B x)}{12 x^4}+15 a^2 b^4 B x+3 a b^5 x (2 A+B x)+\frac{1}{6} b^6 x^2 (3 A+2 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 144, normalized size = 1.1 \begin{align*}{\frac{{b}^{6}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{6}}{2}}+3\,B{x}^{2}a{b}^{5}+6\,Aa{b}^{5}x+15\,B{a}^{2}{b}^{4}x+15\,A\ln \left ( x \right ){a}^{2}{b}^{4}+20\,B\ln \left ( x \right ){a}^{3}{b}^{3}-2\,{\frac{A{a}^{5}b}{{x}^{3}}}-{\frac{B{a}^{6}}{3\,{x}^{3}}}-{\frac{A{a}^{6}}{4\,{x}^{4}}}-{\frac{15\,A{a}^{4}{b}^{2}}{2\,{x}^{2}}}-3\,{\frac{B{a}^{5}b}{{x}^{2}}}-20\,{\frac{A{a}^{3}{b}^{3}}{x}}-15\,{\frac{B{a}^{4}{b}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987675, size = 196, normalized size = 1.46 \begin{align*} \frac{1}{3} \, B b^{6} x^{3} + \frac{1}{2} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{2} + 3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \left (x\right ) - \frac{3 \, A a^{6} + 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41241, size = 323, normalized size = 2.41 \begin{align*} \frac{4 \, B b^{6} x^{7} - 3 \, A a^{6} + 6 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 36 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} \log \left (x\right ) - 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 4 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.83099, size = 144, normalized size = 1.07 \begin{align*} \frac{B b^{6} x^{3}}{3} + 5 a^{2} b^{3} \left (3 A b + 4 B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{6}}{2} + 3 B a b^{5}\right ) + x \left (6 A a b^{5} + 15 B a^{2} b^{4}\right ) - \frac{3 A a^{6} + x^{3} \left (240 A a^{3} b^{3} + 180 B a^{4} b^{2}\right ) + x^{2} \left (90 A a^{4} b^{2} + 36 B a^{5} b\right ) + x \left (24 A a^{5} b + 4 B a^{6}\right )}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16283, size = 196, normalized size = 1.46 \begin{align*} \frac{1}{3} \, B b^{6} x^{3} + 3 \, B a b^{5} x^{2} + \frac{1}{2} \, A b^{6} x^{2} + 15 \, B a^{2} b^{4} x + 6 \, A a b^{5} x + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac{3 \, A a^{6} + 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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