Optimal. Leaf size=113 \[ -\frac{a^4 (A b-a B)}{b^6 (a+b x)}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a^3 (4 A b-5 a B) \log (a+b x)}{b^6}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^4}{4 b^2} \]
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Rubi [A] time = 0.10855, antiderivative size = 113, 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, 77} \[ -\frac{a^4 (A b-a B)}{b^6 (a+b x)}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a^3 (4 A b-5 a B) \log (a+b x)}{b^6}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{x^4 (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{x^4 (A+B x)}{(a+b x)^2} \, dx\\ &=\int \left (-\frac{a^2 (-3 A b+4 a B)}{b^5}+\frac{a (-2 A b+3 a B) x}{b^4}+\frac{(A b-2 a B) x^2}{b^3}+\frac{B x^3}{b^2}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)^2}+\frac{a^3 (-4 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{a^2 (3 A b-4 a B) x}{b^5}-\frac{a (2 A b-3 a B) x^2}{2 b^4}+\frac{(A b-2 a B) x^3}{3 b^3}+\frac{B x^4}{4 b^2}-\frac{a^4 (A b-a B)}{b^6 (a+b x)}-\frac{a^3 (4 A b-5 a B) \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.053374, size = 107, normalized size = 0.95 \[ \frac{\frac{12 a^4 (a B-A b)}{a+b x}-12 a^2 b x (4 a B-3 A b)+12 a^3 (5 a B-4 A b) \log (a+b x)+6 a b^2 x^2 (3 a B-2 A b)+4 b^3 x^3 (A b-2 a B)+3 b^4 B x^4}{12 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 133, normalized size = 1.2 \begin{align*}{\frac{B{x}^{4}}{4\,{b}^{2}}}+{\frac{A{x}^{3}}{3\,{b}^{2}}}-{\frac{2\,aB{x}^{3}}{3\,{b}^{3}}}-{\frac{aA{x}^{2}}{{b}^{3}}}+{\frac{3\,{a}^{2}B{x}^{2}}{2\,{b}^{4}}}+3\,{\frac{{a}^{2}Ax}{{b}^{4}}}-4\,{\frac{{a}^{3}Bx}{{b}^{5}}}-{\frac{{a}^{4}A}{{b}^{5} \left ( bx+a \right ) }}+{\frac{B{a}^{5}}{{b}^{6} \left ( bx+a \right ) }}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) A}{{b}^{5}}}+5\,{\frac{{a}^{4}\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 = 1.02119, size = 166, normalized size = 1.47 \begin{align*} \frac{B a^{5} - A a^{4} b}{b^{7} x + a b^{6}} + \frac{3 \, B b^{3} x^{4} - 4 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{3} + 6 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2} - 12 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} x}{12 \, b^{5}} + \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} 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 = 1.18285, size = 352, normalized size = 3.12 \begin{align*} \frac{3 \, B b^{5} x^{5} + 12 \, B a^{5} - 12 \, A a^{4} b -{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} x^{4} + 2 \,{\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{3} - 6 \,{\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{2} - 12 \,{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x + 12 \,{\left (5 \, B a^{5} - 4 \, A a^{4} b +{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.633376, size = 114, normalized size = 1.01 \begin{align*} \frac{B x^{4}}{4 b^{2}} + \frac{a^{3} \left (- 4 A b + 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{- A a^{4} b + B a^{5}}{a b^{6} + b^{7} x} - \frac{x^{3} \left (- A b + 2 B a\right )}{3 b^{3}} + \frac{x^{2} \left (- 2 A a b + 3 B a^{2}\right )}{2 b^{4}} - \frac{x \left (- 3 A a^{2} b + 4 B a^{3}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11476, size = 170, normalized size = 1.5 \begin{align*} \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{B a^{5} - A a^{4} b}{{\left (b x + a\right )} b^{6}} + \frac{3 \, B b^{6} x^{4} - 8 \, B a b^{5} x^{3} + 4 \, A b^{6} x^{3} + 18 \, B a^{2} b^{4} x^{2} - 12 \, A a b^{5} x^{2} - 48 \, B a^{3} b^{3} x + 36 \, A a^{2} b^{4} x}{12 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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