Optimal. Leaf size=143 \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{8 x^8}-\frac{5 a^2 b^3 (4 a B+3 A b)}{7 x^7}-\frac{a^5 (a B+6 A b)}{10 x^{10}}-\frac{a^4 b (2 a B+5 A b)}{3 x^9}-\frac{a^6 A}{11 x^{11}}-\frac{a b^4 (5 a B+2 A b)}{2 x^6}-\frac{b^5 (6 a B+A b)}{5 x^5}-\frac{b^6 B}{4 x^4} \]
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Rubi [A] time = 0.0716047, antiderivative size = 143, 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, 76} \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{8 x^8}-\frac{5 a^2 b^3 (4 a B+3 A b)}{7 x^7}-\frac{a^5 (a B+6 A b)}{10 x^{10}}-\frac{a^4 b (2 a B+5 A b)}{3 x^9}-\frac{a^6 A}{11 x^{11}}-\frac{a b^4 (5 a B+2 A b)}{2 x^6}-\frac{b^5 (6 a B+A b)}{5 x^5}-\frac{b^6 B}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^{12}} \, dx\\ &=\int \left (\frac{a^6 A}{x^{12}}+\frac{a^5 (6 A b+a B)}{x^{11}}+\frac{3 a^4 b (5 A b+2 a B)}{x^{10}}+\frac{5 a^3 b^2 (4 A b+3 a B)}{x^9}+\frac{5 a^2 b^3 (3 A b+4 a B)}{x^8}+\frac{3 a b^4 (2 A b+5 a B)}{x^7}+\frac{b^5 (A b+6 a B)}{x^6}+\frac{b^6 B}{x^5}\right ) \, dx\\ &=-\frac{a^6 A}{11 x^{11}}-\frac{a^5 (6 A b+a B)}{10 x^{10}}-\frac{a^4 b (5 A b+2 a B)}{3 x^9}-\frac{5 a^3 b^2 (4 A b+3 a B)}{8 x^8}-\frac{5 a^2 b^3 (3 A b+4 a B)}{7 x^7}-\frac{a b^4 (2 A b+5 a B)}{2 x^6}-\frac{b^5 (A b+6 a B)}{5 x^5}-\frac{b^6 B}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0337319, size = 126, normalized size = 0.88 \[ -\frac{1925 a^4 b^2 x^2 (8 A+9 B x)+3300 a^3 b^3 x^3 (7 A+8 B x)+3300 a^2 b^4 x^4 (6 A+7 B x)+616 a^5 b x (9 A+10 B x)+84 a^6 (10 A+11 B x)+1848 a b^5 x^5 (5 A+6 B x)+462 b^6 x^6 (4 A+5 B x)}{9240 x^{11}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 128, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{6}}{11\,{x}^{11}}}-{\frac{{a}^{5} \left ( 6\,Ab+aB \right ) }{10\,{x}^{10}}}-{\frac{{a}^{4}b \left ( 5\,Ab+2\,aB \right ) }{3\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2} \left ( 4\,Ab+3\,aB \right ) }{8\,{x}^{8}}}-{\frac{5\,{a}^{2}{b}^{3} \left ( 3\,Ab+4\,aB \right ) }{7\,{x}^{7}}}-{\frac{a{b}^{4} \left ( 2\,Ab+5\,aB \right ) }{2\,{x}^{6}}}-{\frac{{b}^{5} \left ( Ab+6\,aB \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{6}}{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980805, size = 198, normalized size = 1.38 \begin{align*} -\frac{2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19436, size = 343, normalized size = 2.4 \begin{align*} -\frac{2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 33.9449, size = 150, normalized size = 1.05 \begin{align*} - \frac{840 A a^{6} + 2310 B b^{6} x^{7} + x^{6} \left (1848 A b^{6} + 11088 B a b^{5}\right ) + x^{5} \left (9240 A a b^{5} + 23100 B a^{2} b^{4}\right ) + x^{4} \left (19800 A a^{2} b^{4} + 26400 B a^{3} b^{3}\right ) + x^{3} \left (23100 A a^{3} b^{3} + 17325 B a^{4} b^{2}\right ) + x^{2} \left (15400 A a^{4} b^{2} + 6160 B a^{5} b\right ) + x \left (5544 A a^{5} b + 924 B a^{6}\right )}{9240 x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09211, size = 198, normalized size = 1.38 \begin{align*} -\frac{2310 \, B b^{6} x^{7} + 11088 \, B a b^{5} x^{6} + 1848 \, A b^{6} x^{6} + 23100 \, B a^{2} b^{4} x^{5} + 9240 \, A a b^{5} x^{5} + 26400 \, B a^{3} b^{3} x^{4} + 19800 \, A a^{2} b^{4} x^{4} + 17325 \, B a^{4} b^{2} x^{3} + 23100 \, A a^{3} b^{3} x^{3} + 6160 \, B a^{5} b x^{2} + 15400 \, A a^{4} b^{2} x^{2} + 924 \, B a^{6} x + 5544 \, A a^{5} b x + 840 \, A a^{6}}{9240 \, x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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