Optimal. Leaf size=108 \[ 10 a^2 b^2 \log (x) (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}-\frac{5 a^3 b (a B+2 A b)}{x}-\frac{a^5 A}{3 x^3}+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{3} b^5 B x^3 \]
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Rubi [A] time = 0.0645475, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ 10 a^2 b^2 \log (x) (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}-\frac{5 a^3 b (a B+2 A b)}{x}-\frac{a^5 A}{3 x^3}+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{3} b^5 B x^3 \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^5 (A+B x)}{x^4} \, dx &=\int \left (5 a b^3 (A b+2 a B)+\frac{a^5 A}{x^4}+\frac{a^4 (5 A b+a B)}{x^3}+\frac{5 a^3 b (2 A b+a B)}{x^2}+\frac{10 a^2 b^2 (A b+a B)}{x}+b^4 (A b+5 a B) x+b^5 B x^2\right ) \, dx\\ &=-\frac{a^5 A}{3 x^3}-\frac{a^4 (5 A b+a B)}{2 x^2}-\frac{5 a^3 b (2 A b+a B)}{x}+5 a b^3 (A b+2 a B) x+\frac{1}{2} b^4 (A b+5 a B) x^2+\frac{1}{3} b^5 B x^3+10 a^2 b^2 (A b+a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0281629, size = 109, normalized size = 1.01 \[ \frac{60 a^2 b^2 x^3 \log (x) (a B+A b)-60 a^3 A b^2 x^2-15 a^4 b x (A+2 B x)+a^5 (-(2 A+3 B x))+60 a^2 b^3 B x^4+15 a b^4 x^4 (2 A+B x)+b^5 x^5 (3 A+2 B x)}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 120, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{5}}{2}}+{\frac{5\,B{x}^{2}a{b}^{4}}{2}}+5\,a{b}^{4}Ax+10\,{a}^{2}{b}^{3}Bx+10\,A\ln \left ( x \right ){a}^{2}{b}^{3}+10\,B\ln \left ( x \right ){a}^{3}{b}^{2}-{\frac{A{a}^{5}}{3\,{x}^{3}}}-{\frac{5\,{a}^{4}bA}{2\,{x}^{2}}}-{\frac{{a}^{5}B}{2\,{x}^{2}}}-10\,{\frac{{a}^{3}{b}^{2}A}{x}}-5\,{\frac{{a}^{4}bB}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01424, size = 158, normalized size = 1.46 \begin{align*} \frac{1}{3} \, B b^{5} x^{3} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{2} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{5} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64932, size = 261, normalized size = 2.42 \begin{align*} \frac{2 \, B b^{5} x^{6} - 2 \, A a^{5} + 3 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 60 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} \log \left (x\right ) - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00922, size = 119, normalized size = 1.1 \begin{align*} \frac{B b^{5} x^{3}}{3} + 10 a^{2} b^{2} \left (A b + B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{5}}{2} + \frac{5 B a b^{4}}{2}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) - \frac{2 A a^{5} + x^{2} \left (60 A a^{3} b^{2} + 30 B a^{4} b\right ) + x \left (15 A a^{4} b + 3 B a^{5}\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18152, size = 159, normalized size = 1.47 \begin{align*} \frac{1}{3} \, B b^{5} x^{3} + \frac{5}{2} \, B a b^{4} x^{2} + \frac{1}{2} \, A b^{5} x^{2} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, A a^{5} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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