Optimal. Leaf size=146 \[ \frac{a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac{a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac{5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac{5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac{5 a (A b-3 a B)}{b^7 (a+b x)}+\frac{(A b-6 a B) \log (a+b x)}{b^7}+\frac{B x}{b^6} \]
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Rubi [A] time = 0.163201, antiderivative size = 146, 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} \[ \frac{a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac{a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac{5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac{5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac{5 a (A b-3 a B)}{b^7 (a+b x)}+\frac{(A b-6 a B) \log (a+b x)}{b^7}+\frac{B x}{b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{x^5 (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (\frac{B}{b^6}+\frac{a^5 (-A b+a B)}{b^6 (a+b x)^6}-\frac{a^4 (-5 A b+6 a B)}{b^6 (a+b x)^5}+\frac{5 a^3 (-2 A b+3 a B)}{b^6 (a+b x)^4}-\frac{10 a^2 (-A b+2 a B)}{b^6 (a+b x)^3}+\frac{5 a (-A b+3 a B)}{b^6 (a+b x)^2}+\frac{A b-6 a B}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{B x}{b^6}+\frac{a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac{a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac{5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac{5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac{5 a (A b-3 a B)}{b^7 (a+b x)}+\frac{(A b-6 a B) \log (a+b x)}{b^7}\\ \end{align*}
Mathematica [A] time = 0.0838994, size = 130, normalized size = 0.89 \[ \frac{\frac{12 a^5 (A b-a B)}{(a+b x)^5}+\frac{15 a^4 (6 a B-5 A b)}{(a+b x)^4}+\frac{100 a^3 (2 A b-3 a B)}{(a+b x)^3}+\frac{300 a^2 (2 a B-A b)}{(a+b x)^2}+\frac{300 a (A b-3 a B)}{a+b x}+60 (A b-6 a B) \log (a+b x)+60 b B x}{60 b^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 190, normalized size = 1.3 \begin{align*}{\frac{Bx}{{b}^{6}}}+5\,{\frac{aA}{{b}^{6} \left ( bx+a \right ) }}-15\,{\frac{B{a}^{2}}{{b}^{7} \left ( bx+a \right ) }}-5\,{\frac{A{a}^{2}}{{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{B{a}^{3}}{{b}^{7} \left ( bx+a \right ) ^{2}}}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{6}}}-6\,{\frac{\ln \left ( bx+a \right ) aB}{{b}^{7}}}+{\frac{{a}^{5}A}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}-{\frac{B{a}^{6}}{5\,{b}^{7} \left ( bx+a \right ) ^{5}}}-{\frac{5\,{a}^{4}A}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}+{\frac{3\,B{a}^{5}}{2\,{b}^{7} \left ( bx+a \right ) ^{4}}}+{\frac{10\,A{a}^{3}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-5\,{\frac{B{a}^{4}}{{b}^{7} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04954, size = 257, normalized size = 1.76 \begin{align*} -\frac{522 \, B a^{6} - 137 \, A a^{5} b + 300 \,{\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \,{\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \,{\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \,{\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac{B x}{b^{6}} - \frac{{\left (6 \, B a - A b\right )} \log \left (b x + a\right )}{b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31446, size = 663, normalized size = 4.54 \begin{align*} \frac{60 \, B b^{6} x^{6} + 300 \, B a b^{5} x^{5} - 522 \, B a^{6} + 137 \, A a^{5} b - 300 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} x^{4} - 300 \,{\left (8 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 100 \,{\left (36 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} - 125 \,{\left (18 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x - 60 \,{\left (6 \, B a^{6} - A a^{5} b +{\left (6 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \,{\left (6 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \,{\left (6 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (6 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (6 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{60 \,{\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.45064, size = 190, normalized size = 1.3 \begin{align*} \frac{B x}{b^{6}} - \frac{- 137 A a^{5} b + 522 B a^{6} + x^{4} \left (- 300 A a b^{5} + 900 B a^{2} b^{4}\right ) + x^{3} \left (- 900 A a^{2} b^{4} + 3000 B a^{3} b^{3}\right ) + x^{2} \left (- 1100 A a^{3} b^{3} + 3900 B a^{4} b^{2}\right ) + x \left (- 625 A a^{4} b^{2} + 2310 B a^{5} b\right )}{60 a^{5} b^{7} + 300 a^{4} b^{8} x + 600 a^{3} b^{9} x^{2} + 600 a^{2} b^{10} x^{3} + 300 a b^{11} x^{4} + 60 b^{12} x^{5}} - \frac{\left (- A b + 6 B a\right ) \log{\left (a + b x \right )}}{b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17584, size = 194, normalized size = 1.33 \begin{align*} \frac{B x}{b^{6}} - \frac{{\left (6 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{522 \, B a^{6} - 137 \, A a^{5} b + 300 \,{\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \,{\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \,{\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \,{\left (b x + a\right )}^{5} b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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