Optimal. Leaf size=159 \[ \frac {b^4 (a+b x)^{11} (5 A b-16 a B)}{240240 a^6 x^{11}}-\frac {b^3 (a+b x)^{11} (5 A b-16 a B)}{21840 a^5 x^{12}}+\frac {b^2 (a+b x)^{11} (5 A b-16 a B)}{3640 a^4 x^{13}}-\frac {b (a+b x)^{11} (5 A b-16 a B)}{840 a^3 x^{14}}+\frac {(a+b x)^{11} (5 A b-16 a B)}{240 a^2 x^{15}}-\frac {A (a+b x)^{11}}{16 a x^{16}} \]
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Rubi [A] time = 0.06, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ \frac {b^4 (a+b x)^{11} (5 A b-16 a B)}{240240 a^6 x^{11}}-\frac {b^3 (a+b x)^{11} (5 A b-16 a B)}{21840 a^5 x^{12}}+\frac {b^2 (a+b x)^{11} (5 A b-16 a B)}{3640 a^4 x^{13}}-\frac {b (a+b x)^{11} (5 A b-16 a B)}{840 a^3 x^{14}}+\frac {(a+b x)^{11} (5 A b-16 a B)}{240 a^2 x^{15}}-\frac {A (a+b x)^{11}}{16 a x^{16}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^{17}} \, dx &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(-5 A b+16 a B) \int \frac {(a+b x)^{10}}{x^{16}} \, dx}{16 a}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}+\frac {(b (5 A b-16 a B)) \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{60 a^2}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}-\frac {\left (b^2 (5 A b-16 a B)\right ) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{280 a^3}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}+\frac {b^2 (5 A b-16 a B) (a+b x)^{11}}{3640 a^4 x^{13}}+\frac {\left (b^3 (5 A b-16 a B)\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{1820 a^4}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}+\frac {b^2 (5 A b-16 a B) (a+b x)^{11}}{3640 a^4 x^{13}}-\frac {b^3 (5 A b-16 a B) (a+b x)^{11}}{21840 a^5 x^{12}}-\frac {\left (b^4 (5 A b-16 a B)\right ) \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{21840 a^5}\\ &=-\frac {A (a+b x)^{11}}{16 a x^{16}}+\frac {(5 A b-16 a B) (a+b x)^{11}}{240 a^2 x^{15}}-\frac {b (5 A b-16 a B) (a+b x)^{11}}{840 a^3 x^{14}}+\frac {b^2 (5 A b-16 a B) (a+b x)^{11}}{3640 a^4 x^{13}}-\frac {b^3 (5 A b-16 a B) (a+b x)^{11}}{21840 a^5 x^{12}}+\frac {b^4 (5 A b-16 a B) (a+b x)^{11}}{240240 a^6 x^{11}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 222, normalized size = 1.40 \[ -\frac {a^{10} (15 A+16 B x)}{240 x^{16}}-\frac {a^9 b (14 A+15 B x)}{21 x^{15}}-\frac {45 a^8 b^2 (13 A+14 B x)}{182 x^{14}}-\frac {10 a^7 b^3 (12 A+13 B x)}{13 x^{13}}-\frac {35 a^6 b^4 (11 A+12 B x)}{22 x^{12}}-\frac {126 a^5 b^5 (10 A+11 B x)}{55 x^{11}}-\frac {7 a^4 b^6 (9 A+10 B x)}{3 x^{10}}-\frac {5 a^3 b^7 (8 A+9 B x)}{3 x^9}-\frac {45 a^2 b^8 (7 A+8 B x)}{56 x^8}-\frac {5 a b^9 (6 A+7 B x)}{21 x^7}-\frac {b^{10} (5 A+6 B x)}{30 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 243, normalized size = 1.53 \[ -\frac {48048 \, B b^{10} x^{11} + 15015 \, A a^{10} + 40040 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 171600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 450450 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 800800 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 1009008 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 917280 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 600600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 277200 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 85800 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 16016 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{240240 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 243, normalized size = 1.53 \[ -\frac {48048 \, B b^{10} x^{11} + 400400 \, B a b^{9} x^{10} + 40040 \, A b^{10} x^{10} + 1544400 \, B a^{2} b^{8} x^{9} + 343200 \, A a b^{9} x^{9} + 3603600 \, B a^{3} b^{7} x^{8} + 1351350 \, A a^{2} b^{8} x^{8} + 5605600 \, B a^{4} b^{6} x^{7} + 3203200 \, A a^{3} b^{7} x^{7} + 6054048 \, B a^{5} b^{5} x^{6} + 5045040 \, A a^{4} b^{6} x^{6} + 4586400 \, B a^{6} b^{4} x^{5} + 5503680 \, A a^{5} b^{5} x^{5} + 2402400 \, B a^{7} b^{3} x^{4} + 4204200 \, A a^{6} b^{4} x^{4} + 831600 \, B a^{8} b^{2} x^{3} + 2217600 \, A a^{7} b^{3} x^{3} + 171600 \, B a^{9} b x^{2} + 772200 \, A a^{8} b^{2} x^{2} + 16016 \, B a^{10} x + 160160 \, A a^{9} b x + 15015 \, A a^{10}}{240240 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 208, normalized size = 1.31 \[ -\frac {B \,b^{10}}{5 x^{5}}-\frac {\left (A b +10 B a \right ) b^{9}}{6 x^{6}}-\frac {5 \left (2 A b +9 B a \right ) a \,b^{8}}{7 x^{7}}-\frac {15 \left (3 A b +8 B a \right ) a^{2} b^{7}}{8 x^{8}}-\frac {10 \left (4 A b +7 B a \right ) a^{3} b^{6}}{3 x^{9}}-\frac {21 \left (5 A b +6 B a \right ) a^{4} b^{5}}{5 x^{10}}-\frac {42 \left (6 A b +5 B a \right ) a^{5} b^{4}}{11 x^{11}}-\frac {5 \left (7 A b +4 B a \right ) a^{6} b^{3}}{2 x^{12}}-\frac {15 \left (8 A b +3 B a \right ) a^{7} b^{2}}{13 x^{13}}-\frac {A \,a^{10}}{16 x^{16}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{14 x^{14}}-\frac {\left (10 A b +B a \right ) a^{9}}{15 x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 243, normalized size = 1.53 \[ -\frac {48048 \, B b^{10} x^{11} + 15015 \, A a^{10} + 40040 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 171600 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 450450 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 800800 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 1009008 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 917280 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 600600 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 277200 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 85800 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 16016 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{240240 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 235, normalized size = 1.48 \[ -\frac {x\,\left (\frac {B\,a^{10}}{15}+\frac {2\,A\,b\,a^9}{3}\right )+\frac {A\,a^{10}}{16}+x^2\,\left (\frac {5\,B\,a^9\,b}{7}+\frac {45\,A\,a^8\,b^2}{14}\right )+x^9\,\left (\frac {45\,B\,a^2\,b^8}{7}+\frac {10\,A\,a\,b^9}{7}\right )+x^{10}\,\left (\frac {A\,b^{10}}{6}+\frac {5\,B\,a\,b^9}{3}\right )+x^4\,\left (10\,B\,a^7\,b^3+\frac {35\,A\,a^6\,b^4}{2}\right )+x^8\,\left (15\,B\,a^3\,b^7+\frac {45\,A\,a^2\,b^8}{8}\right )+x^7\,\left (\frac {70\,B\,a^4\,b^6}{3}+\frac {40\,A\,a^3\,b^7}{3}\right )+x^6\,\left (\frac {126\,B\,a^5\,b^5}{5}+21\,A\,a^4\,b^6\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{13}+\frac {120\,A\,a^7\,b^3}{13}\right )+x^5\,\left (\frac {210\,B\,a^6\,b^4}{11}+\frac {252\,A\,a^5\,b^5}{11}\right )+\frac {B\,b^{10}\,x^{11}}{5}}{x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 86.74, size = 260, normalized size = 1.64 \[ \frac {- 15015 A a^{10} - 48048 B b^{10} x^{11} + x^{10} \left (- 40040 A b^{10} - 400400 B a b^{9}\right ) + x^{9} \left (- 343200 A a b^{9} - 1544400 B a^{2} b^{8}\right ) + x^{8} \left (- 1351350 A a^{2} b^{8} - 3603600 B a^{3} b^{7}\right ) + x^{7} \left (- 3203200 A a^{3} b^{7} - 5605600 B a^{4} b^{6}\right ) + x^{6} \left (- 5045040 A a^{4} b^{6} - 6054048 B a^{5} b^{5}\right ) + x^{5} \left (- 5503680 A a^{5} b^{5} - 4586400 B a^{6} b^{4}\right ) + x^{4} \left (- 4204200 A a^{6} b^{4} - 2402400 B a^{7} b^{3}\right ) + x^{3} \left (- 2217600 A a^{7} b^{3} - 831600 B a^{8} b^{2}\right ) + x^{2} \left (- 772200 A a^{8} b^{2} - 171600 B a^{9} b\right ) + x \left (- 160160 A a^{9} b - 16016 B a^{10}\right )}{240240 x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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