Integrand size = 16, antiderivative size = 101 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {A (a+b x)^{11}}{14 a x^{14}}+\frac {(3 A b-14 a B) (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b (3 A b-14 a B) (a+b x)^{11}}{1092 a^3 x^{12}}+\frac {b^2 (3 A b-14 a B) (a+b x)^{11}}{12012 a^4 x^{11}} \]
-1/14*A*(b*x+a)^11/a/x^14+1/182*(3*A*b-14*B*a)*(b*x+a)^11/a^2/x^13-1/1092* b*(3*A*b-14*B*a)*(b*x+a)^11/a^3/x^12+1/12012*b^2*(3*A*b-14*B*a)*(b*x+a)^11 /a^4/x^11
Time = 0.04 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {1001 b^{10} x^{10} (3 A+4 B x)+6006 a b^9 x^9 (4 A+5 B x)+18018 a^2 b^8 x^8 (5 A+6 B x)+34320 a^3 b^7 x^7 (6 A+7 B x)+45045 a^4 b^6 x^6 (7 A+8 B x)+42042 a^5 b^5 x^5 (8 A+9 B x)+28028 a^6 b^4 x^4 (9 A+10 B x)+13104 a^7 b^3 x^3 (10 A+11 B x)+4095 a^8 b^2 x^2 (11 A+12 B x)+770 a^9 b x (12 A+13 B x)+66 a^{10} (13 A+14 B x)}{12012 x^{14}} \]
-1/12012*(1001*b^10*x^10*(3*A + 4*B*x) + 6006*a*b^9*x^9*(4*A + 5*B*x) + 18 018*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))/x ^14
Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 55, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(3 A b-14 a B) \int \frac {(a+b x)^{10}}{x^{14}}dx}{14 a}-\frac {A (a+b x)^{11}}{14 a x^{14}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(3 A b-14 a B) \left (-\frac {2 b \int \frac {(a+b x)^{10}}{x^{13}}dx}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {A (a+b x)^{11}}{14 a x^{14}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(3 A b-14 a B) \left (-\frac {2 b \left (-\frac {b \int \frac {(a+b x)^{10}}{x^{12}}dx}{12 a}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {A (a+b x)^{11}}{14 a x^{14}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (-\frac {2 b \left (\frac {b (a+b x)^{11}}{132 a^2 x^{11}}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right ) (3 A b-14 a B)}{14 a}-\frac {A (a+b x)^{11}}{14 a x^{14}}\) |
-1/14*(A*(a + b*x)^11)/(a*x^14) - ((3*A*b - 14*a*B)*(-1/13*(a + b*x)^11/(a *x^13) - (2*b*(-1/12*(a + b*x)^11/(a*x^12) + (b*(a + b*x)^11)/(132*a^2*x^1 1)))/(13*a)))/(14*a)
3.2.62.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 | \(-\frac {4004 b^{10} B \,x^{11}+3003 A \,b^{10} x^{10}+30030 B a \,b^{9} x^{10}+24024 a A \,b^{9} x^{9}+108108 B \,a^{2} b^{8} x^{9}+90090 a^{2} A \,b^{8} x^{8}+240240 B \,a^{3} b^{7} x^{8}+205920 a^{3} A \,b^{7} x^{7}+360360 B \,a^{4} b^{6} x^{7}+315315 a^{4} A \,b^{6} x^{6}+378378 B \,a^{5} b^{5} x^{6}+336336 a^{5} A \,b^{5} x^{5}+280280 B \,a^{6} b^{4} x^{5}+252252 a^{6} A \,b^{4} x^{4}+144144 B \,a^{7} b^{3} x^{4}+131040 a^{7} A \,b^{3} x^{3}+49140 B \,a^{8} b^{2} x^{3}+45045 a^{8} A \,b^{2} x^{2}+10010 B \,a^{9} b \,x^{2}+9240 a^{9} A b x +924 a^{10} B x +858 a^{10} A}{12012 x^{14}}\) | \(244\) |
-5/2*a^2*b^7*(3*A*b+8*B*a)/x^6-1/14*a^10*A/x^14-30/7*a^3*b^6*(4*A*b+7*B*a) /x^7-21/4*a^4*b^5*(5*A*b+6*B*a)/x^8-3*a^6*b^3*(7*A*b+4*B*a)/x^10-1/13*a^9* (10*A*b+B*a)/x^13-1/3*b^10*B/x^3-5/12*a^8*b*(9*A*b+2*B*a)/x^12-1/4*b^9*(A* b+10*B*a)/x^4-a*b^8*(2*A*b+9*B*a)/x^5-14/3*a^5*b^4*(6*A*b+5*B*a)/x^9-15/11 *a^7*b^2*(8*A*b+3*B*a)/x^11
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 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \]
-1/12012*(4004*B*b^10*x^11 + 858*A*a^10 + 3003*(10*B*a*b^9 + A*b^10)*x^10 + 12012*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 30030*(8*B*a^3*b^7 + 3*A*a^2*b^8)* x^8 + 51480*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 63063*(6*B*a^5*b^5 + 5*A*a^4 *b^6)*x^6 + 56056*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 36036*(4*B*a^7*b^3 + 7 *A*a^6*b^4)*x^4 + 16380*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 924*(B*a^10 + 10*A*a^9*b)*x)/x^14
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=\text {Timed out} \]
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 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {4004 \, B b^{10} x^{11} + 858 \, A a^{10} + 3003 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 12012 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 30030 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 51480 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 63063 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 56056 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 36036 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 16380 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 5005 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 924 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12012 \, x^{14}} \]
-1/12012*(4004*B*b^10*x^11 + 858*A*a^10 + 3003*(10*B*a*b^9 + A*b^10)*x^10 + 12012*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 30030*(8*B*a^3*b^7 + 3*A*a^2*b^8)* x^8 + 51480*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 63063*(6*B*a^5*b^5 + 5*A*a^4 *b^6)*x^6 + 56056*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 36036*(4*B*a^7*b^3 + 7 *A*a^6*b^4)*x^4 + 16380*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 5005*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 924*(B*a^10 + 10*A*a^9*b)*x)/x^14
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 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {4004 \, B b^{10} x^{11} + 30030 \, B a b^{9} x^{10} + 3003 \, A b^{10} x^{10} + 108108 \, B a^{2} b^{8} x^{9} + 24024 \, A a b^{9} x^{9} + 240240 \, B a^{3} b^{7} x^{8} + 90090 \, A a^{2} b^{8} x^{8} + 360360 \, B a^{4} b^{6} x^{7} + 205920 \, A a^{3} b^{7} x^{7} + 378378 \, B a^{5} b^{5} x^{6} + 315315 \, A a^{4} b^{6} x^{6} + 280280 \, B a^{6} b^{4} x^{5} + 336336 \, A a^{5} b^{5} x^{5} + 144144 \, B a^{7} b^{3} x^{4} + 252252 \, A a^{6} b^{4} x^{4} + 49140 \, B a^{8} b^{2} x^{3} + 131040 \, A a^{7} b^{3} x^{3} + 10010 \, B a^{9} b x^{2} + 45045 \, A a^{8} b^{2} x^{2} + 924 \, B a^{10} x + 9240 \, A a^{9} b x + 858 \, A a^{10}}{12012 \, x^{14}} \]
-1/12012*(4004*B*b^10*x^11 + 30030*B*a*b^9*x^10 + 3003*A*b^10*x^10 + 10810 8*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 + 1 44144*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 + 9 240*A*a^9*b*x + 858*A*a^10)/x^14
Time = 0.18 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.33 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{15}} \, dx=-\frac {x\,\left (\frac {B\,a^{10}}{13}+\frac {10\,A\,b\,a^9}{13}\right )+\frac {A\,a^{10}}{14}+x^9\,\left (9\,B\,a^2\,b^8+2\,A\,a\,b^9\right )+x^2\,\left (\frac {5\,B\,a^9\,b}{6}+\frac {15\,A\,a^8\,b^2}{4}\right )+x^{10}\,\left (\frac {A\,b^{10}}{4}+\frac {5\,B\,a\,b^9}{2}\right )+x^4\,\left (12\,B\,a^7\,b^3+21\,A\,a^6\,b^4\right )+x^8\,\left (20\,B\,a^3\,b^7+\frac {15\,A\,a^2\,b^8}{2}\right )+x^5\,\left (\frac {70\,B\,a^6\,b^4}{3}+28\,A\,a^5\,b^5\right )+x^7\,\left (30\,B\,a^4\,b^6+\frac {120\,A\,a^3\,b^7}{7}\right )+x^6\,\left (\frac {63\,B\,a^5\,b^5}{2}+\frac {105\,A\,a^4\,b^6}{4}\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{11}+\frac {120\,A\,a^7\,b^3}{11}\right )+\frac {B\,b^{10}\,x^{11}}{3}}{x^{14}} \]
-(x*((B*a^10)/13 + (10*A*a^9*b)/13) + (A*a^10)/14 + x^9*(9*B*a^2*b^8 + 2*A *a*b^9) + x^2*((15*A*a^8*b^2)/4 + (5*B*a^9*b)/6) + x^10*((A*b^10)/4 + (5*B *a*b^9)/2) + x^4*(21*A*a^6*b^4 + 12*B*a^7*b^3) + x^8*((15*A*a^2*b^8)/2 + 2 0*B*a^3*b^7) + x^5*(28*A*a^5*b^5 + (70*B*a^6*b^4)/3) + x^7*((120*A*a^3*b^7 )/7 + 30*B*a^4*b^6) + x^6*((105*A*a^4*b^6)/4 + (63*B*a^5*b^5)/2) + x^3*((1 20*A*a^7*b^3)/11 + (45*B*a^8*b^2)/11) + (B*b^10*x^11)/3)/x^14