Optimal. Leaf size=108 \[ \frac{a^2 x^2 (A b-a B)}{2 b^4}-\frac{a^3 x (A b-a B)}{b^5}+\frac{a^4 (A b-a B) \log (a+b x)}{b^6}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{B x^5}{5 b} \]
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Rubi [A] time = 0.0909027, 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 = {77} \[ \frac{a^2 x^2 (A b-a B)}{2 b^4}-\frac{a^3 x (A b-a B)}{b^5}+\frac{a^4 (A b-a B) \log (a+b x)}{b^6}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{B x^5}{5 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x^4 (A+B x)}{a+b x} \, dx &=\int \left (\frac{a^3 (-A b+a B)}{b^5}-\frac{a^2 (-A b+a B) x}{b^4}+\frac{a (-A b+a B) x^2}{b^3}+\frac{(A b-a B) x^3}{b^2}+\frac{B x^4}{b}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac{a^3 (A b-a B) x}{b^5}+\frac{a^2 (A b-a B) x^2}{2 b^4}-\frac{a (A b-a B) x^3}{3 b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^5}{5 b}+\frac{a^4 (A b-a B) \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0365787, size = 100, normalized size = 0.93 \[ \frac{b x \left (10 a^2 b^2 x (3 A+2 B x)-30 a^3 b (2 A+B x)+60 a^4 B-5 a b^3 x^2 (4 A+3 B x)+3 b^4 x^3 (5 A+4 B x)\right )-60 a^4 (a B-A b) \log (a+b x)}{60 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 124, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{5\,b}}+{\frac{A{x}^{4}}{4\,b}}-{\frac{B{x}^{4}a}{4\,{b}^{2}}}-{\frac{aA{x}^{3}}{3\,{b}^{2}}}+{\frac{B{x}^{3}{a}^{2}}{3\,{b}^{3}}}+{\frac{{a}^{2}A{x}^{2}}{2\,{b}^{3}}}-{\frac{B{x}^{2}{a}^{3}}{2\,{b}^{4}}}-{\frac{{a}^{3}Ax}{{b}^{4}}}+{\frac{B{a}^{4}x}{{b}^{5}}}+{\frac{{a}^{4}\ln \left ( bx+a \right ) A}{{b}^{5}}}-{\frac{{a}^{5}\ln \left ( bx+a \right ) B}{{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0333, size = 157, normalized size = 1.45 \begin{align*} \frac{12 \, B b^{4} x^{5} - 15 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 20 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 30 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 60 \,{\left (B a^{4} - A a^{3} b\right )} x}{60 \, b^{5}} - \frac{{\left (B a^{5} - A a^{4} b\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41963, size = 244, normalized size = 2.26 \begin{align*} \frac{12 \, B b^{5} x^{5} - 15 \,{\left (B a b^{4} - A b^{5}\right )} x^{4} + 20 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 30 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 60 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} x - 60 \,{\left (B a^{5} - A a^{4} b\right )} \log \left (b x + a\right )}{60 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.478746, size = 99, normalized size = 0.92 \begin{align*} \frac{B x^{5}}{5 b} - \frac{a^{4} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{x^{4} \left (- A b + B a\right )}{4 b^{2}} + \frac{x^{3} \left (- A a b + B a^{2}\right )}{3 b^{3}} - \frac{x^{2} \left (- A a^{2} b + B a^{3}\right )}{2 b^{4}} + \frac{x \left (- A a^{3} b + B a^{4}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19186, size = 161, normalized size = 1.49 \begin{align*} \frac{12 \, B b^{4} x^{5} - 15 \, B a b^{3} x^{4} + 15 \, A b^{4} x^{4} + 20 \, B a^{2} b^{2} x^{3} - 20 \, A a b^{3} x^{3} - 30 \, B a^{3} b x^{2} + 30 \, A a^{2} b^{2} x^{2} + 60 \, B a^{4} x - 60 \, A a^{3} b x}{60 \, b^{5}} - \frac{{\left (B a^{5} - A a^{4} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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