Optimal. Leaf size=130 \[ -\frac {b^3 (a+b x)^{11} (4 A b-15 a B)}{60060 a^5 x^{11}}+\frac {b^2 (a+b x)^{11} (4 A b-15 a B)}{5460 a^4 x^{12}}-\frac {b (a+b x)^{11} (4 A b-15 a B)}{910 a^3 x^{13}}+\frac {(a+b x)^{11} (4 A b-15 a B)}{210 a^2 x^{14}}-\frac {A (a+b x)^{11}}{15 a x^{15}} \]
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Rubi [A] time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ -\frac {b^3 (a+b x)^{11} (4 A b-15 a B)}{60060 a^5 x^{11}}+\frac {b^2 (a+b x)^{11} (4 A b-15 a B)}{5460 a^4 x^{12}}-\frac {b (a+b x)^{11} (4 A b-15 a B)}{910 a^3 x^{13}}+\frac {(a+b x)^{11} (4 A b-15 a B)}{210 a^2 x^{14}}-\frac {A (a+b x)^{11}}{15 a x^{15}} \]
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^{16}} \, dx &=-\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(-4 A b+15 a B) \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{15 a}\\ &=-\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}+\frac {(b (4 A b-15 a B)) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{70 a^2}\\ &=-\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac {b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}-\frac {\left (b^2 (4 A b-15 a B)\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{455 a^3}\\ &=-\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac {b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac {b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}+\frac {\left (b^3 (4 A b-15 a B)\right ) \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{5460 a^4}\\ &=-\frac {A (a+b x)^{11}}{15 a x^{15}}+\frac {(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac {b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac {b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}-\frac {b^3 (4 A b-15 a B) (a+b x)^{11}}{60060 a^5 x^{11}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 202, normalized size = 1.55 \[ -\frac {286 a^{10} (14 A+15 B x)+3300 a^9 b x (13 A+14 B x)+17325 a^8 b^2 x^2 (12 A+13 B x)+54600 a^7 b^3 x^3 (11 A+12 B x)+114660 a^6 b^4 x^4 (10 A+11 B x)+168168 a^5 b^5 x^5 (9 A+10 B x)+175175 a^4 b^6 x^6 (8 A+9 B x)+128700 a^3 b^7 x^7 (7 A+8 B x)+64350 a^2 b^8 x^8 (6 A+7 B x)+20020 a b^9 x^9 (5 A+6 B x)+3003 b^{10} x^{10} (4 A+5 B x)}{60060 x^{15}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 243, normalized size = 1.87 \[ -\frac {15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 243, normalized size = 1.87 \[ -\frac {15015 \, B b^{10} x^{11} + 120120 \, B a b^{9} x^{10} + 12012 \, A b^{10} x^{10} + 450450 \, B a^{2} b^{8} x^{9} + 100100 \, A a b^{9} x^{9} + 1029600 \, B a^{3} b^{7} x^{8} + 386100 \, A a^{2} b^{8} x^{8} + 1576575 \, B a^{4} b^{6} x^{7} + 900900 \, A a^{3} b^{7} x^{7} + 1681680 \, B a^{5} b^{5} x^{6} + 1401400 \, A a^{4} b^{6} x^{6} + 1261260 \, B a^{6} b^{4} x^{5} + 1513512 \, A a^{5} b^{5} x^{5} + 655200 \, B a^{7} b^{3} x^{4} + 1146600 \, A a^{6} b^{4} x^{4} + 225225 \, B a^{8} b^{2} x^{3} + 600600 \, A a^{7} b^{3} x^{3} + 46200 \, B a^{9} b x^{2} + 207900 \, A a^{8} b^{2} x^{2} + 4290 \, B a^{10} x + 42900 \, A a^{9} b x + 4004 \, A a^{10}}{60060 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 208, normalized size = 1.60 \[ -\frac {B \,b^{10}}{4 x^{4}}-\frac {\left (A b +10 B a \right ) b^{9}}{5 x^{5}}-\frac {5 \left (2 A b +9 B a \right ) a \,b^{8}}{6 x^{6}}-\frac {15 \left (3 A b +8 B a \right ) a^{2} b^{7}}{7 x^{7}}-\frac {15 \left (4 A b +7 B a \right ) a^{3} b^{6}}{4 x^{8}}-\frac {14 \left (5 A b +6 B a \right ) a^{4} b^{5}}{3 x^{9}}-\frac {21 \left (6 A b +5 B a \right ) a^{5} b^{4}}{5 x^{10}}-\frac {30 \left (7 A b +4 B a \right ) a^{6} b^{3}}{11 x^{11}}-\frac {5 \left (8 A b +3 B a \right ) a^{7} b^{2}}{4 x^{12}}-\frac {A \,a^{10}}{15 x^{15}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{13 x^{13}}-\frac {\left (10 A b +B a \right ) a^{9}}{14 x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.07, size = 243, normalized size = 1.87 \[ -\frac {15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 235, normalized size = 1.81 \[ -\frac {x\,\left (\frac {B\,a^{10}}{14}+\frac {5\,A\,b\,a^9}{7}\right )+\frac {A\,a^{10}}{15}+x^9\,\left (\frac {15\,B\,a^2\,b^8}{2}+\frac {5\,A\,a\,b^9}{3}\right )+x^2\,\left (\frac {10\,B\,a^9\,b}{13}+\frac {45\,A\,a^8\,b^2}{13}\right )+x^{10}\,\left (\frac {A\,b^{10}}{5}+2\,B\,a\,b^9\right )+x^3\,\left (\frac {15\,B\,a^8\,b^2}{4}+10\,A\,a^7\,b^3\right )+x^6\,\left (28\,B\,a^5\,b^5+\frac {70\,A\,a^4\,b^6}{3}\right )+x^7\,\left (\frac {105\,B\,a^4\,b^6}{4}+15\,A\,a^3\,b^7\right )+x^5\,\left (21\,B\,a^6\,b^4+\frac {126\,A\,a^5\,b^5}{5}\right )+x^8\,\left (\frac {120\,B\,a^3\,b^7}{7}+\frac {45\,A\,a^2\,b^8}{7}\right )+x^4\,\left (\frac {120\,B\,a^7\,b^3}{11}+\frac {210\,A\,a^6\,b^4}{11}\right )+\frac {B\,b^{10}\,x^{11}}{4}}{x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 64.37, size = 260, normalized size = 2.00 \[ \frac {- 4004 A a^{10} - 15015 B b^{10} x^{11} + x^{10} \left (- 12012 A b^{10} - 120120 B a b^{9}\right ) + x^{9} \left (- 100100 A a b^{9} - 450450 B a^{2} b^{8}\right ) + x^{8} \left (- 386100 A a^{2} b^{8} - 1029600 B a^{3} b^{7}\right ) + x^{7} \left (- 900900 A a^{3} b^{7} - 1576575 B a^{4} b^{6}\right ) + x^{6} \left (- 1401400 A a^{4} b^{6} - 1681680 B a^{5} b^{5}\right ) + x^{5} \left (- 1513512 A a^{5} b^{5} - 1261260 B a^{6} b^{4}\right ) + x^{4} \left (- 1146600 A a^{6} b^{4} - 655200 B a^{7} b^{3}\right ) + x^{3} \left (- 600600 A a^{7} b^{3} - 225225 B a^{8} b^{2}\right ) + x^{2} \left (- 207900 A a^{8} b^{2} - 46200 B a^{9} b\right ) + x \left (- 42900 A a^{9} b - 4290 B a^{10}\right )}{60060 x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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