Integrand size = 15, antiderivative size = 220 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{12 a x^{12}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{69 a^2 x^{23/2}}-\frac {7 b^2 \left (a+b \sqrt {x}\right )^{16}}{759 a^3 x^{11}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{759 a^4 x^{21/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{1518 a^5 x^{10}}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{14421 a^6 x^{19/2}}-\frac {b^6 \left (a+b \sqrt {x}\right )^{16}}{43263 a^7 x^9}+\frac {2 b^7 \left (a+b \sqrt {x}\right )^{16}}{735471 a^8 x^{17/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{5883768 a^9 x^8} \] Output:
-1/12*(a+b*x^(1/2))^16/a/x^12+2/69*b*(a+b*x^(1/2))^16/a^2/x^(23/2)-7/759*b ^2*(a+b*x^(1/2))^16/a^3/x^11+2/759*b^3*(a+b*x^(1/2))^16/a^4/x^(21/2)-1/151 8*b^4*(a+b*x^(1/2))^16/a^5/x^10+2/14421*b^5*(a+b*x^(1/2))^16/a^6/x^(19/2)- 1/43263*b^6*(a+b*x^(1/2))^16/a^7/x^9+2/735471*b^7*(a+b*x^(1/2))^16/a^8/x^( 17/2)-1/5883768*b^8*(a+b*x^(1/2))^16/a^9/x^8
Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=\frac {-490314 a^{15}-7674480 a^{14} b \sqrt {x}-56163240 a^{13} b^2 x-254963280 a^{12} b^3 x^{3/2}-803134332 a^{11} b^4 x^2-1859890032 a^{10} b^5 x^{5/2}-3272028760 a^9 b^6 x^3-4454358480 a^8 b^7 x^{7/2}-4732755885 a^7 b^8 x^4-3926434512 a^6 b^9 x^{9/2}-2524136472 a^5 b^{10} x^5-1235591280 a^4 b^{11} x^{11/2}-446185740 a^3 b^{12} x^6-112326480 a^2 b^{13} x^{13/2}-17651304 a b^{14} x^7-1307504 b^{15} x^{15/2}}{5883768 x^{12}} \] Input:
Integrate[(a + b*Sqrt[x])^15/x^13,x]
Output:
(-490314*a^15 - 7674480*a^14*b*Sqrt[x] - 56163240*a^13*b^2*x - 254963280*a ^12*b^3*x^(3/2) - 803134332*a^11*b^4*x^2 - 1859890032*a^10*b^5*x^(5/2) - 3 272028760*a^9*b^6*x^3 - 4454358480*a^8*b^7*x^(7/2) - 4732755885*a^7*b^8*x^ 4 - 3926434512*a^6*b^9*x^(9/2) - 2524136472*a^5*b^10*x^5 - 1235591280*a^4* b^11*x^(11/2) - 446185740*a^3*b^12*x^6 - 112326480*a^2*b^13*x^(13/2) - 176 51304*a*b^14*x^7 - 1307504*b^15*x^(15/2))/(5883768*x^12)
Time = 0.44 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {798, 55, 55, 55, 55, 55, 55, 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.
| \(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{25/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}}d\sqrt {x}}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{23/2}}d\sqrt {x}}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}}d\sqrt {x}}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{21/2}}d\sqrt {x}}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}}d\sqrt {x}}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{19/2}}d\sqrt {x}}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9}d\sqrt {x}}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17/2}}d\sqrt {x}}{17 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle 2 \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (\frac {b \left (a+b \sqrt {x}\right )^{16}}{272 a^2 x^8}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )\) |
Input:
Int[(a + b*Sqrt[x])^15/x^13,x]
Output:
2*(-1/3*(b*((-7*b*((-3*b*((-5*b*(-1/5*(b*((-3*b*(-1/9*(b*(-1/17*(a + b*Sqr t[x])^16/(a*x^(17/2)) + (b*(a + b*Sqrt[x])^16)/(272*a^2*x^8)))/a - (a + b* Sqrt[x])^16/(18*a*x^9)))/(19*a) - (a + b*Sqrt[x])^16/(19*a*x^(19/2))))/a - (a + b*Sqrt[x])^16/(20*a*x^10)))/(21*a) - (a + b*Sqrt[x])^16/(21*a*x^(21/ 2))))/(11*a) - (a + b*Sqrt[x])^16/(22*a*x^11)))/(23*a) - (a + b*Sqrt[x])^1 6/(23*a*x^(23/2))))/a - (a + b*Sqrt[x])^16/(24*a*x^12))
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 23.90 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(-\frac {30 a^{14} b}{23 x^{\frac {23}{2}}}-\frac {130 a^{12} b^{3}}{3 x^{\frac {21}{2}}}-\frac {2002 a^{6} b^{9}}{3 x^{\frac {15}{2}}}-\frac {2 b^{15}}{9 x^{\frac {9}{2}}}-\frac {273 a^{11} b^{4}}{2 x^{10}}-\frac {5005 a^{9} b^{6}}{9 x^{9}}-\frac {a^{15}}{12 x^{12}}-\frac {3 a \,b^{14}}{x^{5}}-\frac {105 a^{13} b^{2}}{11 x^{11}}-\frac {210 a^{4} b^{11}}{x^{\frac {13}{2}}}-\frac {210 a^{2} b^{13}}{11 x^{\frac {11}{2}}}-\frac {455 a^{3} b^{12}}{6 x^{6}}-\frac {429 a^{5} b^{10}}{x^{7}}-\frac {12870 a^{8} b^{7}}{17 x^{\frac {17}{2}}}-\frac {6006 a^{10} b^{5}}{19 x^{\frac {19}{2}}}-\frac {6435 a^{7} b^{8}}{8 x^{8}}\) | \(168\) |
| default | \(-\frac {30 a^{14} b}{23 x^{\frac {23}{2}}}-\frac {130 a^{12} b^{3}}{3 x^{\frac {21}{2}}}-\frac {2002 a^{6} b^{9}}{3 x^{\frac {15}{2}}}-\frac {2 b^{15}}{9 x^{\frac {9}{2}}}-\frac {273 a^{11} b^{4}}{2 x^{10}}-\frac {5005 a^{9} b^{6}}{9 x^{9}}-\frac {a^{15}}{12 x^{12}}-\frac {3 a \,b^{14}}{x^{5}}-\frac {105 a^{13} b^{2}}{11 x^{11}}-\frac {210 a^{4} b^{11}}{x^{\frac {13}{2}}}-\frac {210 a^{2} b^{13}}{11 x^{\frac {11}{2}}}-\frac {455 a^{3} b^{12}}{6 x^{6}}-\frac {429 a^{5} b^{10}}{x^{7}}-\frac {12870 a^{8} b^{7}}{17 x^{\frac {17}{2}}}-\frac {6006 a^{10} b^{5}}{19 x^{\frac {19}{2}}}-\frac {6435 a^{7} b^{8}}{8 x^{8}}\) | \(168\) |
| orering | \(-\frac {\left (41186376 b^{28} x^{14}-454506220 a^{2} b^{26} x^{13}+2529574500 a^{4} b^{24} x^{12}-9035261235 a^{6} b^{22} x^{11}+22770783035 a^{8} b^{20} x^{10}-42473774343 a^{10} b^{18} x^{9}+60167436615 a^{12} b^{16} x^{8}-65546319475 a^{14} b^{14} x^{7}+55062474075 a^{16} b^{12} x^{6}-35440923375 a^{18} b^{10} x^{5}+17185959791 a^{20} b^{8} x^{4}-6083267190 a^{22} b^{6} x^{3}+1484887950 a^{24} b^{4} x^{2}-223616130 a^{26} b^{2} x +15668730 a^{28}\right ) \left (a +b \sqrt {x}\right )^{15}}{88256520 x^{12} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (1961256 b^{28} x^{14}-19761140 a^{2} b^{26} x^{13}+101182980 a^{4} b^{24} x^{12}-334639305 a^{6} b^{22} x^{11}+785199415 a^{8} b^{20} x^{10}-1370121753 a^{10} b^{18} x^{9}+1823255655 a^{12} b^{16} x^{8}-1872751985 a^{14} b^{14} x^{7}+1488174975 a^{16} b^{12} x^{6}-908741625 a^{18} b^{10} x^{5}+419169751 a^{20} b^{8} x^{4}-141471330 a^{22} b^{6} x^{3}+32997510 a^{24} b^{4} x^{2}-4757790 a^{26} b^{2} x +319770 a^{28}\right ) x^{2} \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} b}{2 x^{\frac {27}{2}}}-\frac {13 \left (a +b \sqrt {x}\right )^{15}}{x^{14}}\right )}{44128260 \left (-b^{2} x +a^{2}\right )^{14}}\) | \(385\) |
| trager | \(\frac {\left (-1+x \right ) \left (66 a^{14} x^{11}+7560 a^{12} b^{2} x^{11}+108108 a^{10} b^{4} x^{11}+440440 a^{8} b^{6} x^{11}+637065 a^{6} b^{8} x^{11}+339768 a^{4} b^{10} x^{11}+60060 a^{2} b^{12} x^{11}+2376 b^{14} x^{11}+66 a^{14} x^{10}+7560 a^{12} b^{2} x^{10}+108108 a^{10} b^{4} x^{10}+440440 a^{8} b^{6} x^{10}+637065 a^{6} b^{8} x^{10}+339768 a^{4} b^{10} x^{10}+60060 a^{2} b^{12} x^{10}+2376 b^{14} x^{10}+66 a^{14} x^{9}+7560 a^{12} b^{2} x^{9}+108108 a^{10} b^{4} x^{9}+440440 a^{8} b^{6} x^{9}+637065 a^{6} b^{8} x^{9}+339768 a^{4} b^{10} x^{9}+60060 a^{2} b^{12} x^{9}+2376 b^{14} x^{9}+66 a^{14} x^{8}+7560 a^{12} b^{2} x^{8}+108108 a^{10} b^{4} x^{8}+440440 a^{8} b^{6} x^{8}+637065 a^{6} b^{8} x^{8}+339768 a^{4} b^{10} x^{8}+60060 a^{2} b^{12} x^{8}+2376 b^{14} x^{8}+66 a^{14} x^{7}+7560 a^{12} b^{2} x^{7}+108108 a^{10} b^{4} x^{7}+440440 a^{8} b^{6} x^{7}+637065 a^{6} b^{8} x^{7}+339768 b^{10} x^{7} a^{4}+60060 a^{2} b^{12} x^{7}+2376 x^{7} b^{14}+66 a^{14} x^{6}+7560 a^{12} b^{2} x^{6}+108108 a^{10} b^{4} x^{6}+440440 a^{8} b^{6} x^{6}+637065 b^{8} x^{6} a^{6}+339768 a^{4} b^{10} x^{6}+60060 a^{2} b^{12} x^{6}+66 a^{14} x^{5}+7560 a^{12} b^{2} x^{5}+108108 a^{10} b^{4} x^{5}+440440 a^{8} b^{6} x^{5}+637065 a^{6} b^{8} x^{5}+339768 a^{4} b^{10} x^{5}+66 a^{14} x^{4}+7560 a^{12} b^{2} x^{4}+108108 b^{4} x^{4} a^{10}+440440 x^{4} b^{6} a^{8}+637065 a^{6} b^{8} x^{4}+66 a^{14} x^{3}+7560 a^{12} b^{2} x^{3}+108108 a^{10} b^{4} x^{3}+440440 a^{8} b^{6} x^{3}+66 a^{14} x^{2}+7560 a^{12} b^{2} x^{2}+108108 a^{10} b^{4} x^{2}+66 a^{14} x +7560 a^{12} b^{2} x +66 a^{14}\right ) a}{792 x^{12}}-\frac {2 \left (81719 x^{7} b^{14}+7020405 a^{2} b^{12} x^{6}+77224455 a^{4} b^{10} x^{5}+245402157 a^{6} b^{8} x^{4}+278397405 a^{8} b^{6} x^{3}+116243127 a^{10} b^{4} x^{2}+15935205 a^{12} b^{2} x +479655 a^{14}\right ) b}{735471 x^{\frac {23}{2}}}\) | \(786\) |
Input:
int((a+b*x^(1/2))^15/x^13,x,method=_RETURNVERBOSE)
Output:
-30/23*a^14*b/x^(23/2)-130/3*a^12*b^3/x^(21/2)-2002/3*a^6*b^9/x^(15/2)-2/9 *b^15/x^(9/2)-273/2*a^11*b^4/x^10-5005/9*a^9*b^6/x^9-1/12*a^15/x^12-3*a*b^ 14/x^5-105/11*a^13*b^2/x^11-210*a^4*b^11/x^(13/2)-210/11*a^2*b^13/x^(11/2) -455/6*a^3*b^12/x^6-429*a^5*b^10/x^7-12870/17*a^8*b^7/x^(17/2)-6006/19*a^1 0*b^5/x^(19/2)-6435/8*a^7*b^8/x^8
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {17651304 \, a b^{14} x^{7} + 446185740 \, a^{3} b^{12} x^{6} + 2524136472 \, a^{5} b^{10} x^{5} + 4732755885 \, a^{7} b^{8} x^{4} + 3272028760 \, a^{9} b^{6} x^{3} + 803134332 \, a^{11} b^{4} x^{2} + 56163240 \, a^{13} b^{2} x + 490314 \, a^{15} + 16 \, {\left (81719 \, b^{15} x^{7} + 7020405 \, a^{2} b^{13} x^{6} + 77224455 \, a^{4} b^{11} x^{5} + 245402157 \, a^{6} b^{9} x^{4} + 278397405 \, a^{8} b^{7} x^{3} + 116243127 \, a^{10} b^{5} x^{2} + 15935205 \, a^{12} b^{3} x + 479655 \, a^{14} b\right )} \sqrt {x}}{5883768 \, x^{12}} \] Input:
integrate((a+b*x^(1/2))^15/x^13,x, algorithm="fricas")
Output:
-1/5883768*(17651304*a*b^14*x^7 + 446185740*a^3*b^12*x^6 + 2524136472*a^5* b^10*x^5 + 4732755885*a^7*b^8*x^4 + 3272028760*a^9*b^6*x^3 + 803134332*a^1 1*b^4*x^2 + 56163240*a^13*b^2*x + 490314*a^15 + 16*(81719*b^15*x^7 + 70204 05*a^2*b^13*x^6 + 77224455*a^4*b^11*x^5 + 245402157*a^6*b^9*x^4 + 27839740 5*a^8*b^7*x^3 + 116243127*a^10*b^5*x^2 + 15935205*a^12*b^3*x + 479655*a^14 *b)*sqrt(x))/x^12
Time = 1.92 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=- \frac {a^{15}}{12 x^{12}} - \frac {30 a^{14} b}{23 x^{\frac {23}{2}}} - \frac {105 a^{13} b^{2}}{11 x^{11}} - \frac {130 a^{12} b^{3}}{3 x^{\frac {21}{2}}} - \frac {273 a^{11} b^{4}}{2 x^{10}} - \frac {6006 a^{10} b^{5}}{19 x^{\frac {19}{2}}} - \frac {5005 a^{9} b^{6}}{9 x^{9}} - \frac {12870 a^{8} b^{7}}{17 x^{\frac {17}{2}}} - \frac {6435 a^{7} b^{8}}{8 x^{8}} - \frac {2002 a^{6} b^{9}}{3 x^{\frac {15}{2}}} - \frac {429 a^{5} b^{10}}{x^{7}} - \frac {210 a^{4} b^{11}}{x^{\frac {13}{2}}} - \frac {455 a^{3} b^{12}}{6 x^{6}} - \frac {210 a^{2} b^{13}}{11 x^{\frac {11}{2}}} - \frac {3 a b^{14}}{x^{5}} - \frac {2 b^{15}}{9 x^{\frac {9}{2}}} \] Input:
integrate((a+b*x**(1/2))**15/x**13,x)
Output:
-a**15/(12*x**12) - 30*a**14*b/(23*x**(23/2)) - 105*a**13*b**2/(11*x**11) - 130*a**12*b**3/(3*x**(21/2)) - 273*a**11*b**4/(2*x**10) - 6006*a**10*b** 5/(19*x**(19/2)) - 5005*a**9*b**6/(9*x**9) - 12870*a**8*b**7/(17*x**(17/2) ) - 6435*a**7*b**8/(8*x**8) - 2002*a**6*b**9/(3*x**(15/2)) - 429*a**5*b**1 0/x**7 - 210*a**4*b**11/x**(13/2) - 455*a**3*b**12/(6*x**6) - 210*a**2*b** 13/(11*x**(11/2)) - 3*a*b**14/x**5 - 2*b**15/(9*x**(9/2))
Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {1307504 \, b^{15} x^{\frac {15}{2}} + 17651304 \, a b^{14} x^{7} + 112326480 \, a^{2} b^{13} x^{\frac {13}{2}} + 446185740 \, a^{3} b^{12} x^{6} + 1235591280 \, a^{4} b^{11} x^{\frac {11}{2}} + 2524136472 \, a^{5} b^{10} x^{5} + 3926434512 \, a^{6} b^{9} x^{\frac {9}{2}} + 4732755885 \, a^{7} b^{8} x^{4} + 4454358480 \, a^{8} b^{7} x^{\frac {7}{2}} + 3272028760 \, a^{9} b^{6} x^{3} + 1859890032 \, a^{10} b^{5} x^{\frac {5}{2}} + 803134332 \, a^{11} b^{4} x^{2} + 254963280 \, a^{12} b^{3} x^{\frac {3}{2}} + 56163240 \, a^{13} b^{2} x + 7674480 \, a^{14} b \sqrt {x} + 490314 \, a^{15}}{5883768 \, x^{12}} \] Input:
integrate((a+b*x^(1/2))^15/x^13,x, algorithm="maxima")
Output:
-1/5883768*(1307504*b^15*x^(15/2) + 17651304*a*b^14*x^7 + 112326480*a^2*b^ 13*x^(13/2) + 446185740*a^3*b^12*x^6 + 1235591280*a^4*b^11*x^(11/2) + 2524 136472*a^5*b^10*x^5 + 3926434512*a^6*b^9*x^(9/2) + 4732755885*a^7*b^8*x^4 + 4454358480*a^8*b^7*x^(7/2) + 3272028760*a^9*b^6*x^3 + 1859890032*a^10*b^ 5*x^(5/2) + 803134332*a^11*b^4*x^2 + 254963280*a^12*b^3*x^(3/2) + 56163240 *a^13*b^2*x + 7674480*a^14*b*sqrt(x) + 490314*a^15)/x^12
Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {1307504 \, b^{15} x^{\frac {15}{2}} + 17651304 \, a b^{14} x^{7} + 112326480 \, a^{2} b^{13} x^{\frac {13}{2}} + 446185740 \, a^{3} b^{12} x^{6} + 1235591280 \, a^{4} b^{11} x^{\frac {11}{2}} + 2524136472 \, a^{5} b^{10} x^{5} + 3926434512 \, a^{6} b^{9} x^{\frac {9}{2}} + 4732755885 \, a^{7} b^{8} x^{4} + 4454358480 \, a^{8} b^{7} x^{\frac {7}{2}} + 3272028760 \, a^{9} b^{6} x^{3} + 1859890032 \, a^{10} b^{5} x^{\frac {5}{2}} + 803134332 \, a^{11} b^{4} x^{2} + 254963280 \, a^{12} b^{3} x^{\frac {3}{2}} + 56163240 \, a^{13} b^{2} x + 7674480 \, a^{14} b \sqrt {x} + 490314 \, a^{15}}{5883768 \, x^{12}} \] Input:
integrate((a+b*x^(1/2))^15/x^13,x, algorithm="giac")
Output:
-1/5883768*(1307504*b^15*x^(15/2) + 17651304*a*b^14*x^7 + 112326480*a^2*b^ 13*x^(13/2) + 446185740*a^3*b^12*x^6 + 1235591280*a^4*b^11*x^(11/2) + 2524 136472*a^5*b^10*x^5 + 3926434512*a^6*b^9*x^(9/2) + 4732755885*a^7*b^8*x^4 + 4454358480*a^8*b^7*x^(7/2) + 3272028760*a^9*b^6*x^3 + 1859890032*a^10*b^ 5*x^(5/2) + 803134332*a^11*b^4*x^2 + 254963280*a^12*b^3*x^(3/2) + 56163240 *a^13*b^2*x + 7674480*a^14*b*sqrt(x) + 490314*a^15)/x^12
Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=-\frac {\frac {a^{15}}{12}+\frac {2\,b^{15}\,x^{15/2}}{9}+\frac {105\,a^{13}\,b^2\,x}{11}+\frac {30\,a^{14}\,b\,\sqrt {x}}{23}+3\,a\,b^{14}\,x^7+\frac {273\,a^{11}\,b^4\,x^2}{2}+\frac {5005\,a^9\,b^6\,x^3}{9}+\frac {6435\,a^7\,b^8\,x^4}{8}+429\,a^5\,b^{10}\,x^5+\frac {130\,a^{12}\,b^3\,x^{3/2}}{3}+\frac {455\,a^3\,b^{12}\,x^6}{6}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{19}+\frac {12870\,a^8\,b^7\,x^{7/2}}{17}+\frac {2002\,a^6\,b^9\,x^{9/2}}{3}+210\,a^4\,b^{11}\,x^{11/2}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{11}}{x^{12}} \] Input:
int((a + b*x^(1/2))^15/x^13,x)
Output:
-(a^15/12 + (2*b^15*x^(15/2))/9 + (105*a^13*b^2*x)/11 + (30*a^14*b*x^(1/2) )/23 + 3*a*b^14*x^7 + (273*a^11*b^4*x^2)/2 + (5005*a^9*b^6*x^3)/9 + (6435* a^7*b^8*x^4)/8 + 429*a^5*b^10*x^5 + (130*a^12*b^3*x^(3/2))/3 + (455*a^3*b^ 12*x^6)/6 + (6006*a^10*b^5*x^(5/2))/19 + (12870*a^8*b^7*x^(7/2))/17 + (200 2*a^6*b^9*x^(9/2))/3 + 210*a^4*b^11*x^(11/2) + (210*a^2*b^13*x^(13/2))/11) /x^12
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}} \, dx=\frac {-490314 \sqrt {x}\, a^{15}-56163240 \sqrt {x}\, a^{13} b^{2} x -803134332 \sqrt {x}\, a^{11} b^{4} x^{2}-3272028760 \sqrt {x}\, a^{9} b^{6} x^{3}-4732755885 \sqrt {x}\, a^{7} b^{8} x^{4}-2524136472 \sqrt {x}\, a^{5} b^{10} x^{5}-446185740 \sqrt {x}\, a^{3} b^{12} x^{6}-17651304 \sqrt {x}\, a \,b^{14} x^{7}-7674480 a^{14} b x -254963280 a^{12} b^{3} x^{2}-1859890032 a^{10} b^{5} x^{3}-4454358480 a^{8} b^{7} x^{4}-3926434512 a^{6} b^{9} x^{5}-1235591280 a^{4} b^{11} x^{6}-112326480 a^{2} b^{13} x^{7}-1307504 b^{15} x^{8}}{5883768 \sqrt {x}\, x^{12}} \] Input:
int((a+b*x^(1/2))^15/x^13,x)
Output:
( - 490314*sqrt(x)*a**15 - 56163240*sqrt(x)*a**13*b**2*x - 803134332*sqrt( x)*a**11*b**4*x**2 - 3272028760*sqrt(x)*a**9*b**6*x**3 - 4732755885*sqrt(x )*a**7*b**8*x**4 - 2524136472*sqrt(x)*a**5*b**10*x**5 - 446185740*sqrt(x)* a**3*b**12*x**6 - 17651304*sqrt(x)*a*b**14*x**7 - 7674480*a**14*b*x - 2549 63280*a**12*b**3*x**2 - 1859890032*a**10*b**5*x**3 - 4454358480*a**8*b**7* x**4 - 3926434512*a**6*b**9*x**5 - 1235591280*a**4*b**11*x**6 - 112326480* a**2*b**13*x**7 - 1307504*b**15*x**8)/(5883768*sqrt(x)*x**12)