Optimal. Leaf size=143 \[ -\frac {a^6 A}{11 x^{11}}-\frac {a^5 (a B+6 A b)}{10 x^{10}}-\frac {a^4 b (2 a B+5 A b)}{3 x^9}-\frac {5 a^3 b^2 (3 a B+4 A b)}{8 x^8}-\frac {5 a^2 b^3 (4 a B+3 A b)}{7 x^7}-\frac {b^5 (6 a B+A b)}{5 x^5}-\frac {a b^4 (5 a B+2 A b)}{2 x^6}-\frac {b^6 B}{4 x^4} \]
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Rubi [A] time = 0.07, antiderivative size = 143, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}
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*}
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Mathematica [A] time = 0.03, size = 126, normalized size = 0.88 \begin {gather*} -\frac {84 a^6 (10 A+11 B x)+616 a^5 b x (9 A+10 B x)+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)+1848 a b^5 x^5 (5 A+6 B x)+462 b^6 x^6 (4 A+5 B x)}{9240 x^{11}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 147, normalized size = 1.03 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 147, normalized size = 1.03 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 128, normalized size = 0.90 \begin {gather*} -\frac {B \,b^{6}}{4 x^{4}}-\frac {\left (A b +6 B a \right ) b^{5}}{5 x^{5}}-\frac {\left (2 A b +5 B a \right ) a \,b^{4}}{2 x^{6}}-\frac {5 \left (3 A b +4 B a \right ) a^{2} b^{3}}{7 x^{7}}-\frac {5 \left (4 A b +3 B a \right ) a^{3} b^{2}}{8 x^{8}}-\frac {A \,a^{6}}{11 x^{11}}-\frac {\left (5 A b +2 B a \right ) a^{4} b}{3 x^{9}}-\frac {\left (6 A b +B a \right ) a^{5}}{10 x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 147, normalized size = 1.03 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 142, normalized size = 0.99 \begin {gather*} -\frac {x\,\left (\frac {B\,a^6}{10}+\frac {3\,A\,b\,a^5}{5}\right )+\frac {A\,a^6}{11}+x^5\,\left (\frac {5\,B\,a^2\,b^4}{2}+A\,a\,b^5\right )+x^2\,\left (\frac {2\,B\,a^5\,b}{3}+\frac {5\,A\,a^4\,b^2}{3}\right )+x^6\,\left (\frac {A\,b^6}{5}+\frac {6\,B\,a\,b^5}{5}\right )+x^3\,\left (\frac {15\,B\,a^4\,b^2}{8}+\frac {5\,A\,a^3\,b^3}{2}\right )+x^4\,\left (\frac {20\,B\,a^3\,b^3}{7}+\frac {15\,A\,a^2\,b^4}{7}\right )+\frac {B\,b^6\,x^7}{4}}{x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.11, size = 158, normalized size = 1.10 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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