Optimal. Leaf size=106 \[ -\frac{3 a^2 B}{b^6 (a+b x)^2}+\frac{4 a^3 B}{3 b^6 (a+b x)^3}-\frac{a^4 B}{4 b^6 (a+b x)^4}+\frac{x^5 (A b-a B)}{5 a b (a+b x)^5}+\frac{4 a B}{b^6 (a+b x)}+\frac{B \log (a+b x)}{b^6} \]
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Rubi [A] time = 0.0617757, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 78, 43} \[ -\frac{3 a^2 B}{b^6 (a+b x)^2}+\frac{4 a^3 B}{3 b^6 (a+b x)^3}-\frac{a^4 B}{4 b^6 (a+b x)^4}+\frac{x^5 (A b-a B)}{5 a b (a+b x)^5}+\frac{4 a B}{b^6 (a+b x)}+\frac{B \log (a+b x)}{b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 43
Rubi steps
\begin{align*} \int \frac{x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{x^4 (A+B x)}{(a+b x)^6} \, dx\\ &=\frac{(A b-a B) x^5}{5 a b (a+b x)^5}+\frac{B \int \frac{x^4}{(a+b x)^5} \, dx}{b}\\ &=\frac{(A b-a B) x^5}{5 a b (a+b x)^5}+\frac{B \int \left (\frac{a^4}{b^4 (a+b x)^5}-\frac{4 a^3}{b^4 (a+b x)^4}+\frac{6 a^2}{b^4 (a+b x)^3}-\frac{4 a}{b^4 (a+b x)^2}+\frac{1}{b^4 (a+b x)}\right ) \, dx}{b}\\ &=\frac{(A b-a B) x^5}{5 a b (a+b x)^5}-\frac{a^4 B}{4 b^6 (a+b x)^4}+\frac{4 a^3 B}{3 b^6 (a+b x)^3}-\frac{3 a^2 B}{b^6 (a+b x)^2}+\frac{4 a B}{b^6 (a+b x)}+\frac{B \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0426029, size = 113, normalized size = 1.07 \[ \frac{60 a^2 b^3 x^2 (15 B x-2 A)+20 a^3 b^2 x (55 B x-3 A)+a^4 (625 b B x-12 A b)+137 a^5 B+60 a b^4 x^3 (5 B x-2 A)+60 B (a+b x)^5 \log (a+b x)-60 A b^5 x^4}{60 b^6 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 165, normalized size = 1.6 \begin{align*} -{\frac{A}{{b}^{5} \left ( bx+a \right ) }}+5\,{\frac{aB}{{b}^{6} \left ( bx+a \right ) }}+{\frac{B\ln \left ( bx+a \right ) }{{b}^{6}}}-{\frac{A{a}^{4}}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}}+{\frac{B{a}^{5}}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}+{\frac{A{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{5\,B{a}^{4}}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}+2\,{\frac{aA}{{b}^{5} \left ( bx+a \right ) ^{2}}}-5\,{\frac{B{a}^{2}}{{b}^{6} \left ( bx+a \right ) ^{2}}}-2\,{\frac{A{a}^{2}}{{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{10\,B{a}^{3}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07605, size = 230, normalized size = 2.17 \begin{align*} \frac{137 \, B a^{5} - 12 \, A a^{4} b + 60 \,{\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 60 \,{\left (15 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 20 \,{\left (55 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{2} + 5 \,{\left (125 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac{B \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 [B] time = 1.34417, size = 485, normalized size = 4.58 \begin{align*} \frac{137 \, B a^{5} - 12 \, A a^{4} b + 60 \,{\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 60 \,{\left (15 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 20 \,{\left (55 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{2} + 5 \,{\left (125 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x + 60 \,{\left (B b^{5} x^{5} + 5 \, B a b^{4} x^{4} + 10 \, B a^{2} b^{3} x^{3} + 10 \, B a^{3} b^{2} x^{2} + 5 \, B a^{4} b x + B a^{5}\right )} \log \left (b x + a\right )}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.68125, size = 172, normalized size = 1.62 \begin{align*} \frac{B \log{\left (a + b x \right )}}{b^{6}} + \frac{- 12 A a^{4} b + 137 B a^{5} + x^{4} \left (- 60 A b^{5} + 300 B a b^{4}\right ) + x^{3} \left (- 120 A a b^{4} + 900 B a^{2} b^{3}\right ) + x^{2} \left (- 120 A a^{2} b^{3} + 1100 B a^{3} b^{2}\right ) + x \left (- 60 A a^{3} b^{2} + 625 B a^{4} b\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14564, size = 167, normalized size = 1.58 \begin{align*} \frac{B \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{60 \,{\left (5 \, B a b^{3} - A b^{4}\right )} x^{4} + 60 \,{\left (15 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 20 \,{\left (55 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x^{2} + 5 \,{\left (125 \, B a^{4} - 12 \, A a^{3} b\right )} x + \frac{137 \, B a^{5} - 12 \, A a^{4} b}{b}}{60 \,{\left (b x + a\right )}^{5} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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