Optimal. Leaf size=116 \[ -\frac{a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac{a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac{2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}+\frac{x^2 (A b-3 a B)}{2 b^4}-\frac{3 a x (A b-2 a B)}{b^5}+\frac{B x^3}{3 b^3} \]
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Rubi [A] time = 0.113754, antiderivative size = 116, 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} \[ -\frac{a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac{a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac{2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}+\frac{x^2 (A b-3 a B)}{2 b^4}-\frac{3 a x (A b-2 a B)}{b^5}+\frac{B x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x^4 (A+B x)}{(a+b x)^3} \, dx &=\int \left (\frac{3 a (-A b+2 a B)}{b^5}+\frac{(A b-3 a B) x}{b^4}+\frac{B x^2}{b^3}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)^3}+\frac{a^3 (-4 A b+5 a B)}{b^5 (a+b x)^2}-\frac{2 a^2 (-3 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac{3 a (A b-2 a B) x}{b^5}+\frac{(A b-3 a B) x^2}{2 b^4}+\frac{B x^3}{3 b^3}-\frac{a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac{a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac{2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0653461, size = 108, normalized size = 0.93 \[ \frac{\frac{3 a^4 (a B-A b)}{(a+b x)^2}+\frac{6 a^3 (4 A b-5 a B)}{a+b x}-12 a^2 (5 a B-3 A b) \log (a+b x)+3 b^2 x^2 (A b-3 a B)+18 a b x (2 a B-A b)+2 b^3 B x^3}{6 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 142, normalized size = 1.2 \begin{align*}{\frac{B{x}^{3}}{3\,{b}^{3}}}+{\frac{A{x}^{2}}{2\,{b}^{3}}}-{\frac{3\,B{x}^{2}a}{2\,{b}^{4}}}-3\,{\frac{aAx}{{b}^{4}}}+6\,{\frac{{a}^{2}Bx}{{b}^{5}}}+4\,{\frac{{a}^{3}A}{{b}^{5} \left ( bx+a \right ) }}-5\,{\frac{{a}^{4}B}{{b}^{6} \left ( bx+a \right ) }}-{\frac{{a}^{4}A}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{B{a}^{5}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+6\,{\frac{{a}^{2}\ln \left ( bx+a \right ) A}{{b}^{5}}}-10\,{\frac{{a}^{3}\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 = 1.07015, size = 180, normalized size = 1.55 \begin{align*} -\frac{9 \, B a^{5} - 7 \, A a^{4} b + 2 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac{2 \, B b^{2} x^{3} - 3 \,{\left (3 \, B a b - A b^{2}\right )} x^{2} + 18 \,{\left (2 \, B a^{2} - A a b\right )} x}{6 \, b^{5}} - \frac{2 \,{\left (5 \, B a^{3} - 3 \, A a^{2} 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 = 1.99657, size = 417, normalized size = 3.59 \begin{align*} \frac{2 \, B b^{5} x^{5} - 27 \, B a^{5} + 21 \, A a^{4} b -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 3 \,{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 6 \,{\left (B a^{4} b + A a^{3} b^{2}\right )} x - 12 \,{\left (5 \, B a^{5} - 3 \, A a^{4} b +{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.15259, size = 131, normalized size = 1.13 \begin{align*} \frac{B x^{3}}{3 b^{3}} - \frac{2 a^{2} \left (- 3 A b + 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{- 7 A a^{4} b + 9 B a^{5} + x \left (- 8 A a^{3} b^{2} + 10 B a^{4} b\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac{x^{2} \left (- A b + 3 B a\right )}{2 b^{4}} + \frac{x \left (- 3 A a b + 6 B a^{2}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16045, size = 169, normalized size = 1.46 \begin{align*} -\frac{2 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{9 \, B a^{5} - 7 \, A a^{4} b + 2 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, B b^{6} x^{3} - 9 \, B a b^{5} x^{2} + 3 \, A b^{6} x^{2} + 36 \, B a^{2} b^{4} x - 18 \, A a b^{5} x}{6 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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