Optimal. Leaf size=85 \[ -\frac{5 a^2 b^3 B}{x^2}-\frac{10 a^3 b^2 B}{3 x^3}-\frac{5 a^4 b B}{4 x^4}-\frac{a^5 B}{5 x^5}-\frac{A (a+b x)^6}{6 a x^6}-\frac{5 a b^4 B}{x}+b^5 B \log (x) \]
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Rubi [A] time = 0.0345449, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 43} \[ -\frac{5 a^2 b^3 B}{x^2}-\frac{10 a^3 b^2 B}{3 x^3}-\frac{5 a^4 b B}{4 x^4}-\frac{a^5 B}{5 x^5}-\frac{A (a+b x)^6}{6 a x^6}-\frac{5 a b^4 B}{x}+b^5 B \log (x) \]
Antiderivative was successfully verified.
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Rule 78
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5 (A+B x)}{x^7} \, dx &=-\frac{A (a+b x)^6}{6 a x^6}+B \int \frac{(a+b x)^5}{x^6} \, dx\\ &=-\frac{A (a+b x)^6}{6 a x^6}+B \int \left (\frac{a^5}{x^6}+\frac{5 a^4 b}{x^5}+\frac{10 a^3 b^2}{x^4}+\frac{10 a^2 b^3}{x^3}+\frac{5 a b^4}{x^2}+\frac{b^5}{x}\right ) \, dx\\ &=-\frac{a^5 B}{5 x^5}-\frac{5 a^4 b B}{4 x^4}-\frac{10 a^3 b^2 B}{3 x^3}-\frac{5 a^2 b^3 B}{x^2}-\frac{5 a b^4 B}{x}-\frac{A (a+b x)^6}{6 a x^6}+b^5 B \log (x)\\ \end{align*}
Mathematica [A] time = 0.037397, size = 109, normalized size = 1.28 \[ -\frac{100 a^2 b^3 x^3 (2 A+3 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+15 a^4 b x (4 A+5 B x)+2 a^5 (5 A+6 B x)+150 a b^4 x^4 (A+2 B x)+60 A b^5 x^5-60 b^5 B x^6 \log (x)}{60 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 124, normalized size = 1.5 \begin{align*}{b}^{5}B\ln \left ( x \right ) -{\frac{10\,{a}^{2}{b}^{3}A}{3\,{x}^{3}}}-{\frac{10\,{a}^{3}{b}^{2}B}{3\,{x}^{3}}}-{\frac{{a}^{4}bA}{{x}^{5}}}-{\frac{{a}^{5}B}{5\,{x}^{5}}}-{\frac{5\,{a}^{3}{b}^{2}A}{2\,{x}^{4}}}-{\frac{5\,{a}^{4}bB}{4\,{x}^{4}}}-{\frac{5\,a{b}^{4}A}{2\,{x}^{2}}}-5\,{\frac{{a}^{2}{b}^{3}B}{{x}^{2}}}-{\frac{A{a}^{5}}{6\,{x}^{6}}}-{\frac{{b}^{5}A}{x}}-5\,{\frac{a{b}^{4}B}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02222, size = 159, normalized size = 1.87 \begin{align*} B b^{5} \log \left (x\right ) - \frac{10 \, A a^{5} + 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \,{\left (B a^{5} + 5 \, A a^{4} 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 = 1.76978, size = 270, normalized size = 3.18 \begin{align*} \frac{60 \, B b^{5} x^{6} \log \left (x\right ) - 10 \, A a^{5} - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \,{\left (B a^{5} + 5 \, A a^{4} 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 = 3.6827, size = 122, normalized size = 1.44 \begin{align*} B b^{5} \log{\left (x \right )} - \frac{10 A a^{5} + x^{5} \left (60 A b^{5} + 300 B a b^{4}\right ) + x^{4} \left (150 A a b^{4} + 300 B a^{2} b^{3}\right ) + x^{3} \left (200 A a^{2} b^{3} + 200 B a^{3} b^{2}\right ) + x^{2} \left (150 A a^{3} b^{2} + 75 B a^{4} b\right ) + x \left (60 A a^{4} b + 12 B a^{5}\right )}{60 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26332, size = 161, normalized size = 1.89 \begin{align*} B b^{5} \log \left ({\left | x \right |}\right ) - \frac{10 \, A a^{5} + 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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