Optimal. Leaf size=171 \[ -\frac{a^6 (A b-a B)}{5 b^8 (a+b x)^5}+\frac{a^5 (6 A b-7 a B)}{4 b^8 (a+b x)^4}-\frac{a^4 (5 A b-7 a B)}{b^8 (a+b x)^3}+\frac{5 a^3 (4 A b-7 a B)}{2 b^8 (a+b x)^2}-\frac{5 a^2 (3 A b-7 a B)}{b^8 (a+b x)}+\frac{x (A b-6 a B)}{b^7}-\frac{3 a (2 A b-7 a B) \log (a+b x)}{b^8}+\frac{B x^2}{2 b^6} \]
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Rubi [A] time = 0.216397, antiderivative size = 171, 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} \[ -\frac{a^6 (A b-a B)}{5 b^8 (a+b x)^5}+\frac{a^5 (6 A b-7 a B)}{4 b^8 (a+b x)^4}-\frac{a^4 (5 A b-7 a B)}{b^8 (a+b x)^3}+\frac{5 a^3 (4 A b-7 a B)}{2 b^8 (a+b x)^2}-\frac{5 a^2 (3 A b-7 a B)}{b^8 (a+b x)}+\frac{x (A b-6 a B)}{b^7}-\frac{3 a (2 A b-7 a B) \log (a+b x)}{b^8}+\frac{B x^2}{2 b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{x^6 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{x^6 (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (\frac{A b-6 a B}{b^7}+\frac{B x}{b^6}-\frac{a^6 (-A b+a B)}{b^7 (a+b x)^6}+\frac{a^5 (-6 A b+7 a B)}{b^7 (a+b x)^5}-\frac{3 a^4 (-5 A b+7 a B)}{b^7 (a+b x)^4}+\frac{5 a^3 (-4 A b+7 a B)}{b^7 (a+b x)^3}-\frac{5 a^2 (-3 A b+7 a B)}{b^7 (a+b x)^2}+\frac{3 a (-2 A b+7 a B)}{b^7 (a+b x)}\right ) \, dx\\ &=\frac{(A b-6 a B) x}{b^7}+\frac{B x^2}{2 b^6}-\frac{a^6 (A b-a B)}{5 b^8 (a+b x)^5}+\frac{a^5 (6 A b-7 a B)}{4 b^8 (a+b x)^4}-\frac{a^4 (5 A b-7 a B)}{b^8 (a+b x)^3}+\frac{5 a^3 (4 A b-7 a B)}{2 b^8 (a+b x)^2}-\frac{5 a^2 (3 A b-7 a B)}{b^8 (a+b x)}-\frac{3 a (2 A b-7 a B) \log (a+b x)}{b^8}\\ \end{align*}
Mathematica [A] time = 0.100685, size = 151, normalized size = 0.88 \[ \frac{\frac{4 a^6 (a B-A b)}{(a+b x)^5}+\frac{5 a^5 (6 A b-7 a B)}{(a+b x)^4}+\frac{20 a^4 (7 a B-5 A b)}{(a+b x)^3}-\frac{50 a^3 (7 a B-4 A b)}{(a+b x)^2}+\frac{100 a^2 (7 a B-3 A b)}{a+b x}+20 b x (A b-6 a B)+60 a (7 a B-2 A b) \log (a+b x)+10 b^2 B x^2}{20 b^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 213, normalized size = 1.3 \begin{align*}{\frac{B{x}^{2}}{2\,{b}^{6}}}+{\frac{Ax}{{b}^{6}}}-6\,{\frac{aBx}{{b}^{7}}}-15\,{\frac{A{a}^{2}}{{b}^{7} \left ( bx+a \right ) }}+35\,{\frac{B{a}^{3}}{{b}^{8} \left ( bx+a \right ) }}+10\,{\frac{A{a}^{3}}{{b}^{7} \left ( bx+a \right ) ^{2}}}-{\frac{35\,B{a}^{4}}{2\,{b}^{8} \left ( bx+a \right ) ^{2}}}+{\frac{3\,A{a}^{5}}{2\,{b}^{7} \left ( bx+a \right ) ^{4}}}-{\frac{7\,B{a}^{6}}{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}-6\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{7}}}+21\,{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{8}}}-{\frac{{a}^{6}A}{5\,{b}^{7} \left ( bx+a \right ) ^{5}}}+{\frac{B{a}^{7}}{5\,{b}^{8} \left ( bx+a \right ) ^{5}}}-5\,{\frac{A{a}^{4}}{{b}^{7} \left ( bx+a \right ) ^{3}}}+7\,{\frac{B{a}^{5}}{{b}^{8} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07207, size = 288, normalized size = 1.68 \begin{align*} \frac{459 \, B a^{7} - 174 \, A a^{6} b + 100 \,{\left (7 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 50 \,{\left (49 \, B a^{4} b^{3} - 20 \, A a^{3} b^{4}\right )} x^{3} + 10 \,{\left (329 \, B a^{5} b^{2} - 130 \, A a^{4} b^{3}\right )} x^{2} + 35 \,{\left (57 \, B a^{6} b - 22 \, A a^{5} b^{2}\right )} x}{20 \,{\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac{B b x^{2} - 2 \,{\left (6 \, B a - A b\right )} x}{2 \, b^{7}} + \frac{3 \,{\left (7 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.32106, size = 755, normalized size = 4.42 \begin{align*} \frac{10 \, B b^{7} x^{7} + 459 \, B a^{7} - 174 \, A a^{6} b - 10 \,{\left (7 \, B a b^{6} - 2 \, A b^{7}\right )} x^{6} - 100 \,{\left (5 \, B a^{2} b^{5} - A a b^{6}\right )} x^{5} - 100 \,{\left (4 \, B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{4} + 100 \,{\left (13 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{3} + 300 \,{\left (9 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3}\right )} x^{2} + 375 \,{\left (5 \, B a^{6} b - 2 \, A a^{5} b^{2}\right )} x + 60 \,{\left (7 \, B a^{7} - 2 \, A a^{6} b +{\left (7 \, B a^{2} b^{5} - 2 \, A a b^{6}\right )} x^{5} + 5 \,{\left (7 \, B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} x^{4} + 10 \,{\left (7 \, B a^{4} b^{3} - 2 \, A a^{3} b^{4}\right )} x^{3} + 10 \,{\left (7 \, B a^{5} b^{2} - 2 \, A a^{4} b^{3}\right )} x^{2} + 5 \,{\left (7 \, B a^{6} b - 2 \, A a^{5} b^{2}\right )} x\right )} \log \left (b x + a\right )}{20 \,{\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.01449, size = 214, normalized size = 1.25 \begin{align*} \frac{B x^{2}}{2 b^{6}} + \frac{3 a \left (- 2 A b + 7 B a\right ) \log{\left (a + b x \right )}}{b^{8}} + \frac{- 174 A a^{6} b + 459 B a^{7} + x^{4} \left (- 300 A a^{2} b^{5} + 700 B a^{3} b^{4}\right ) + x^{3} \left (- 1000 A a^{3} b^{4} + 2450 B a^{4} b^{3}\right ) + x^{2} \left (- 1300 A a^{4} b^{3} + 3290 B a^{5} b^{2}\right ) + x \left (- 770 A a^{5} b^{2} + 1995 B a^{6} b\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} - \frac{x \left (- A b + 6 B a\right )}{b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17538, size = 232, normalized size = 1.36 \begin{align*} \frac{3 \,{\left (7 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac{B b^{6} x^{2} - 12 \, B a b^{5} x + 2 \, A b^{6} x}{2 \, b^{12}} + \frac{459 \, B a^{7} - 174 \, A a^{6} b + 100 \,{\left (7 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 50 \,{\left (49 \, B a^{4} b^{3} - 20 \, A a^{3} b^{4}\right )} x^{3} + 10 \,{\left (329 \, B a^{5} b^{2} - 130 \, A a^{4} b^{3}\right )} x^{2} + 35 \,{\left (57 \, B a^{6} b - 22 \, A a^{5} b^{2}\right )} x}{20 \,{\left (b x + a\right )}^{5} b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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