Integrand size = 16, antiderivative size = 101 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {A (a+b x)^{11}}{14 a x^{14}}+\frac {(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b (3 A b-14 a B) (a+b x)^{11}}{1092 a^3 x^{12}}+\frac {b^2 (3 A b-14 a B) (a+b x)^{11}}{12012 a^4 x^{11}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=\frac {b^2 (a+b x)^{11} (3 A b-14 a B)}{12012 a^4 x^{11}}-\frac {b (a+b x)^{11} (3 A b-14 a B)}{1092 a^3 x^{12}}+\frac {(a+b x)^{11} (3 A b-14 a B)}{182 a^2 x^{13}}-\frac {A (a+b x)^{11}}{14 a x^{14}} \]
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^{11}}{14 a x^{14}}+\frac {(-3 A b+14 a B) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{14 a} \\ & = -\frac {A (a+b x)^{11}}{14 a x^{14}}+\frac {(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}+\frac {(b (3 A b-14 a B)) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{91 a^2} \\ & = -\frac {A (a+b x)^{11}}{14 a x^{14}}+\frac {(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b (3 A b-14 a B) (a+b x)^{11}}{1092 a^3 x^{12}}-\frac {\left (b^2 (3 A b-14 a B)\right ) \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{1092 a^3} \\ & = -\frac {A (a+b x)^{11}}{14 a x^{14}}+\frac {(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b (3 A b-14 a B) (a+b x)^{11}}{1092 a^3 x^{12}}+\frac {b^2 (3 A b-14 a B) (a+b x)^{11}}{12012 a^4 x^{11}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {1001 b^{10} x^{10} (3 A+4 B x)+6006 a b^9 x^9 (4 A+5 B x)+18018 a^2 b^8 x^8 (5 A+6 B x)+34320 a^3 b^7 x^7 (6 A+7 B x)+45045 a^4 b^6 x^6 (7 A+8 B x)+42042 a^5 b^5 x^5 (8 A+9 B x)+28028 a^6 b^4 x^4 (9 A+10 B x)+13104 a^7 b^3 x^3 (10 A+11 B x)+4095 a^8 b^2 x^2 (11 A+12 B x)+770 a^9 b x (12 A+13 B x)+66 a^{10} (13 A+14 B x)}{12012 x^{14}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(93)=186\).
Time = 0.40 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.06
method | result | size |
default | \(-\frac {5 a^{2} b^{7} \left (3 A b +8 B a \right )}{2 x^{6}}-\frac {a^{10} A}{14 x^{14}}-\frac {30 a^{3} b^{6} \left (4 A b +7 B a \right )}{7 x^{7}}-\frac {21 a^{4} b^{5} \left (5 A b +6 B a \right )}{4 x^{8}}-\frac {3 a^{6} b^{3} \left (7 A b +4 B a \right )}{x^{10}}-\frac {a^{9} \left (10 A b +B a \right )}{13 x^{13}}-\frac {b^{10} B}{3 x^{3}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{12 x^{12}}-\frac {b^{9} \left (A b +10 B a \right )}{4 x^{4}}-\frac {a \,b^{8} \left (2 A b +9 B a \right )}{x^{5}}-\frac {14 a^{5} b^{4} \left (6 A b +5 B a \right )}{3 x^{9}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{11 x^{11}}\) | \(208\) |
norman | \(\frac {-\frac {a^{10} A}{14}+\left (-\frac {10}{13} a^{9} b A -\frac {1}{13} a^{10} B \right ) x +\left (-\frac {15}{4} a^{8} b^{2} A -\frac {5}{6} a^{9} b B \right ) x^{2}+\left (-\frac {120}{11} a^{7} b^{3} A -\frac {45}{11} a^{8} b^{2} B \right ) x^{3}+\left (-21 a^{6} b^{4} A -12 a^{7} b^{3} B \right ) x^{4}+\left (-28 a^{5} b^{5} A -\frac {70}{3} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {105}{4} a^{4} b^{6} A -\frac {63}{2} a^{5} b^{5} B \right ) x^{6}+\left (-\frac {120}{7} a^{3} b^{7} A -30 a^{4} b^{6} B \right ) x^{7}+\left (-\frac {15}{2} a^{2} b^{8} A -20 a^{3} b^{7} B \right ) x^{8}+\left (-2 a \,b^{9} A -9 a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{4} b^{10} A -\frac {5}{2} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{3}}{x^{14}}\) | \(235\) |
risch | \(\frac {-\frac {a^{10} A}{14}+\left (-\frac {10}{13} a^{9} b A -\frac {1}{13} a^{10} B \right ) x +\left (-\frac {15}{4} a^{8} b^{2} A -\frac {5}{6} a^{9} b B \right ) x^{2}+\left (-\frac {120}{11} a^{7} b^{3} A -\frac {45}{11} a^{8} b^{2} B \right ) x^{3}+\left (-21 a^{6} b^{4} A -12 a^{7} b^{3} B \right ) x^{4}+\left (-28 a^{5} b^{5} A -\frac {70}{3} a^{6} b^{4} B \right ) x^{5}+\left (-\frac {105}{4} a^{4} b^{6} A -\frac {63}{2} a^{5} b^{5} B \right ) x^{6}+\left (-\frac {120}{7} a^{3} b^{7} A -30 a^{4} b^{6} B \right ) x^{7}+\left (-\frac {15}{2} a^{2} b^{8} A -20 a^{3} b^{7} B \right ) x^{8}+\left (-2 a \,b^{9} A -9 a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{4} b^{10} A -\frac {5}{2} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{3}}{x^{14}}\) | \(235\) |
gosper | \(-\frac {4004 b^{10} B \,x^{11}+3003 A \,b^{10} x^{10}+30030 B a \,b^{9} x^{10}+24024 a A \,b^{9} x^{9}+108108 B \,a^{2} b^{8} x^{9}+90090 a^{2} A \,b^{8} x^{8}+240240 B \,a^{3} b^{7} x^{8}+205920 a^{3} A \,b^{7} x^{7}+360360 B \,a^{4} b^{6} x^{7}+315315 a^{4} A \,b^{6} x^{6}+378378 B \,a^{5} b^{5} x^{6}+336336 a^{5} A \,b^{5} x^{5}+280280 B \,a^{6} b^{4} x^{5}+252252 a^{6} A \,b^{4} x^{4}+144144 B \,a^{7} b^{3} x^{4}+131040 a^{7} A \,b^{3} x^{3}+49140 B \,a^{8} b^{2} x^{3}+45045 a^{8} A \,b^{2} x^{2}+10010 B \,a^{9} b \,x^{2}+9240 a^{9} A b x +924 a^{10} B x +858 a^{10} A}{12012 x^{14}}\) | \(244\) |
parallelrisch | \(-\frac {4004 b^{10} B \,x^{11}+3003 A \,b^{10} x^{10}+30030 B a \,b^{9} x^{10}+24024 a A \,b^{9} x^{9}+108108 B \,a^{2} b^{8} x^{9}+90090 a^{2} A \,b^{8} x^{8}+240240 B \,a^{3} b^{7} x^{8}+205920 a^{3} A \,b^{7} x^{7}+360360 B \,a^{4} b^{6} x^{7}+315315 a^{4} A \,b^{6} x^{6}+378378 B \,a^{5} b^{5} x^{6}+336336 a^{5} A \,b^{5} x^{5}+280280 B \,a^{6} b^{4} x^{5}+252252 a^{6} A \,b^{4} x^{4}+144144 B \,a^{7} b^{3} x^{4}+131040 a^{7} A \,b^{3} x^{3}+49140 B \,a^{8} b^{2} x^{3}+45045 a^{8} A \,b^{2} x^{2}+10010 B \,a^{9} b \,x^{2}+9240 a^{9} A b x +924 a^{10} B x +858 a^{10} A}{12012 x^{14}}\) | \(244\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (93) = 186\).
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (93) = 186\).
Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (93) = 186\).
Time = 0.31 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {4004 \, B b^{10} x^{11} + 30030 \, B a b^{9} x^{10} + 3003 \, A b^{10} x^{10} + 108108 \, B a^{2} b^{8} x^{9} + 24024 \, A a b^{9} x^{9} + 240240 \, B a^{3} b^{7} x^{8} + 90090 \, A a^{2} b^{8} x^{8} + 360360 \, B a^{4} b^{6} x^{7} + 205920 \, A a^{3} b^{7} x^{7} + 378378 \, B a^{5} b^{5} x^{6} + 315315 \, A a^{4} b^{6} x^{6} + 280280 \, B a^{6} b^{4} x^{5} + 336336 \, A a^{5} b^{5} x^{5} + 144144 \, B a^{7} b^{3} x^{4} + 252252 \, A a^{6} b^{4} x^{4} + 49140 \, B a^{8} b^{2} x^{3} + 131040 \, A a^{7} b^{3} x^{3} + 10010 \, B a^{9} b x^{2} + 45045 \, A a^{8} b^{2} x^{2} + 924 \, B a^{10} x + 9240 \, A a^{9} b x + 858 \, A a^{10}}{12012 \, x^{14}} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.33 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {x\,\left (\frac {B\,a^{10}}{13}+\frac {10\,A\,b\,a^9}{13}\right )+\frac {A\,a^{10}}{14}+x^9\,\left (9\,B\,a^2\,b^8+2\,A\,a\,b^9\right )+x^2\,\left (\frac {5\,B\,a^9\,b}{6}+\frac {15\,A\,a^8\,b^2}{4}\right )+x^{10}\,\left (\frac {A\,b^{10}}{4}+\frac {5\,B\,a\,b^9}{2}\right )+x^4\,\left (12\,B\,a^7\,b^3+21\,A\,a^6\,b^4\right )+x^8\,\left (20\,B\,a^3\,b^7+\frac {15\,A\,a^2\,b^8}{2}\right )+x^5\,\left (\frac {70\,B\,a^6\,b^4}{3}+28\,A\,a^5\,b^5\right )+x^7\,\left (30\,B\,a^4\,b^6+\frac {120\,A\,a^3\,b^7}{7}\right )+x^6\,\left (\frac {63\,B\,a^5\,b^5}{2}+\frac {105\,A\,a^4\,b^6}{4}\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{11}+\frac {120\,A\,a^7\,b^3}{11}\right )+\frac {B\,b^{10}\,x^{11}}{3}}{x^{14}} \]
[In]
[Out]