Integrand size = 15, antiderivative size = 70 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{9 a x^9}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{153 a^2 x^{17/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{1224 a^3 x^8} \]
-1/9*(a+b*x^(1/2))^16/a/x^9+2/153*b*(a+b*x^(1/2))^16/a^2/x^(17/2)-1/1224*b ^2*(a+b*x^(1/2))^16/a^3/x^8
Leaf count is larger than twice the leaf count of optimal. \(185\) vs. \(2(70)=140\).
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx=\frac {-136 a^{15}-2160 a^{14} b \sqrt {x}-16065 a^{13} b^2 x-74256 a^{12} b^3 x^{3/2}-238680 a^{11} b^4 x^2-565488 a^{10} b^5 x^{5/2}-1021020 a^9 b^6 x^3-1432080 a^8 b^7 x^{7/2}-1575288 a^7 b^8 x^4-1361360 a^6 b^9 x^{9/2}-918918 a^5 b^{10} x^5-477360 a^4 b^{11} x^{11/2}-185640 a^3 b^{12} x^6-51408 a^2 b^{13} x^{13/2}-9180 a b^{14} x^7-816 b^{15} x^{15/2}}{1224 x^9} \]
(-136*a^15 - 2160*a^14*b*Sqrt[x] - 16065*a^13*b^2*x - 74256*a^12*b^3*x^(3/ 2) - 238680*a^11*b^4*x^2 - 565488*a^10*b^5*x^(5/2) - 1021020*a^9*b^6*x^3 - 1432080*a^8*b^7*x^(7/2) - 1575288*a^7*b^8*x^4 - 1361360*a^6*b^9*x^(9/2) - 918918*a^5*b^10*x^5 - 477360*a^4*b^11*x^(11/2) - 185640*a^3*b^12*x^6 - 51 408*a^2*b^13*x^(13/2) - 9180*a*b^14*x^7 - 816*b^15*x^(15/2))/(1224*x^9)
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {798, 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.
\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{19/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9}d\sqrt {x}}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (-\frac {b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17/2}}d\sqrt {x}}{17 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle 2 \left (-\frac {b \left (\frac {b \left (a+b \sqrt {x}\right )^{16}}{272 a^2 x^8}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )\) |
2*(-1/9*(b*(-1/17*(a + b*Sqrt[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))
3.22.83.3.1 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[(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]]
Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(56)=112\).
Time = 3.65 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.40
method | result | size |
derivativedivides | \(-\frac {42 a^{2} b^{13}}{x^{\frac {5}{2}}}-\frac {2 b^{15}}{3 x^{\frac {3}{2}}}-\frac {3003 a^{5} b^{10}}{4 x^{4}}-\frac {105 a^{13} b^{2}}{8 x^{8}}-\frac {390 a^{4} b^{11}}{x^{\frac {7}{2}}}-\frac {1170 a^{8} b^{7}}{x^{\frac {11}{2}}}-\frac {182 a^{12} b^{3}}{3 x^{\frac {15}{2}}}-\frac {15 a \,b^{14}}{2 x^{2}}-\frac {455 a^{3} b^{12}}{3 x^{3}}-\frac {30 a^{14} b}{17 x^{\frac {17}{2}}}-\frac {10010 a^{6} b^{9}}{9 x^{\frac {9}{2}}}-\frac {462 a^{10} b^{5}}{x^{\frac {13}{2}}}-\frac {1287 a^{7} b^{8}}{x^{5}}-\frac {5005 a^{9} b^{6}}{6 x^{6}}-\frac {195 a^{11} b^{4}}{x^{7}}-\frac {a^{15}}{9 x^{9}}\) | \(168\) |
default | \(-\frac {42 a^{2} b^{13}}{x^{\frac {5}{2}}}-\frac {2 b^{15}}{3 x^{\frac {3}{2}}}-\frac {3003 a^{5} b^{10}}{4 x^{4}}-\frac {105 a^{13} b^{2}}{8 x^{8}}-\frac {390 a^{4} b^{11}}{x^{\frac {7}{2}}}-\frac {1170 a^{8} b^{7}}{x^{\frac {11}{2}}}-\frac {182 a^{12} b^{3}}{3 x^{\frac {15}{2}}}-\frac {15 a \,b^{14}}{2 x^{2}}-\frac {455 a^{3} b^{12}}{3 x^{3}}-\frac {30 a^{14} b}{17 x^{\frac {17}{2}}}-\frac {10010 a^{6} b^{9}}{9 x^{\frac {9}{2}}}-\frac {462 a^{10} b^{5}}{x^{\frac {13}{2}}}-\frac {1287 a^{7} b^{8}}{x^{5}}-\frac {5005 a^{9} b^{6}}{6 x^{6}}-\frac {195 a^{11} b^{4}}{x^{7}}-\frac {a^{15}}{9 x^{9}}\) | \(168\) |
trager | \(\frac {\left (-1+x \right ) \left (8 a^{14} x^{8}+945 a^{12} b^{2} x^{8}+14040 a^{10} b^{4} x^{8}+60060 a^{8} b^{6} x^{8}+92664 a^{6} b^{8} x^{8}+54054 a^{4} b^{10} x^{8}+10920 a^{2} b^{12} x^{8}+540 b^{14} x^{8}+8 a^{14} x^{7}+945 a^{12} b^{2} x^{7}+14040 a^{10} b^{4} x^{7}+60060 a^{8} b^{6} x^{7}+92664 a^{6} b^{8} x^{7}+54054 a^{4} b^{10} x^{7}+10920 a^{2} b^{12} x^{7}+540 x^{7} b^{14}+8 a^{14} x^{6}+945 a^{12} b^{2} x^{6}+14040 a^{10} b^{4} x^{6}+60060 a^{8} b^{6} x^{6}+92664 a^{6} b^{8} x^{6}+54054 a^{4} b^{10} x^{6}+10920 a^{2} b^{12} x^{6}+8 a^{14} x^{5}+945 a^{12} b^{2} x^{5}+14040 a^{10} b^{4} x^{5}+60060 a^{8} b^{6} x^{5}+92664 a^{6} b^{8} x^{5}+54054 a^{4} b^{10} x^{5}+8 a^{14} x^{4}+945 x^{4} a^{12} b^{2}+14040 x^{4} a^{10} b^{4}+60060 a^{8} b^{6} x^{4}+92664 a^{6} b^{8} x^{4}+8 a^{14} x^{3}+945 a^{12} b^{2} x^{3}+14040 a^{10} b^{4} x^{3}+60060 a^{8} b^{6} x^{3}+8 x^{2} a^{14}+945 a^{12} b^{2} x^{2}+14040 a^{10} b^{4} x^{2}+8 a^{14} x +945 a^{12} b^{2} x +8 a^{14}\right ) a}{72 x^{9}}-\frac {2 \left (51 x^{7} b^{14}+3213 a^{2} b^{12} x^{6}+29835 a^{4} b^{10} x^{5}+85085 a^{6} b^{8} x^{4}+89505 a^{8} b^{6} x^{3}+35343 a^{10} b^{4} x^{2}+4641 a^{12} b^{2} x +135 a^{14}\right ) b}{153 x^{\frac {17}{2}}}\) | \(540\) |
-42*a^2*b^13/x^(5/2)-2/3*b^15/x^(3/2)-3003/4*a^5*b^10/x^4-105/8*a^13*b^2/x ^8-390*a^4*b^11/x^(7/2)-1170*a^8*b^7/x^(11/2)-182/3*a^12*b^3/x^(15/2)-15/2 *a*b^14/x^2-455/3*a^3*b^12/x^3-30/17*a^14*b/x^(17/2)-10010/9*a^6*b^9/x^(9/ 2)-462*a^10*b^5/x^(13/2)-1287*a^7*b^8/x^5-5005/6*a^9*b^6/x^6-195*a^11*b^4/ x^7-1/9*a^15/x^9
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (56) = 112\).
Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx=-\frac {9180 \, a b^{14} x^{7} + 185640 \, a^{3} b^{12} x^{6} + 918918 \, a^{5} b^{10} x^{5} + 1575288 \, a^{7} b^{8} x^{4} + 1021020 \, a^{9} b^{6} x^{3} + 238680 \, a^{11} b^{4} x^{2} + 16065 \, a^{13} b^{2} x + 136 \, a^{15} + 16 \, {\left (51 \, b^{15} x^{7} + 3213 \, a^{2} b^{13} x^{6} + 29835 \, a^{4} b^{11} x^{5} + 85085 \, a^{6} b^{9} x^{4} + 89505 \, a^{8} b^{7} x^{3} + 35343 \, a^{10} b^{5} x^{2} + 4641 \, a^{12} b^{3} x + 135 \, a^{14} b\right )} \sqrt {x}}{1224 \, x^{9}} \]
-1/1224*(9180*a*b^14*x^7 + 185640*a^3*b^12*x^6 + 918918*a^5*b^10*x^5 + 157 5288*a^7*b^8*x^4 + 1021020*a^9*b^6*x^3 + 238680*a^11*b^4*x^2 + 16065*a^13* b^2*x + 136*a^15 + 16*(51*b^15*x^7 + 3213*a^2*b^13*x^6 + 29835*a^4*b^11*x^ 5 + 85085*a^6*b^9*x^4 + 89505*a^8*b^7*x^3 + 35343*a^10*b^5*x^2 + 4641*a^12 *b^3*x + 135*a^14*b)*sqrt(x))/x^9
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (61) = 122\).
Time = 0.79 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.99 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx=- \frac {a^{15}}{9 x^{9}} - \frac {30 a^{14} b}{17 x^{\frac {17}{2}}} - \frac {105 a^{13} b^{2}}{8 x^{8}} - \frac {182 a^{12} b^{3}}{3 x^{\frac {15}{2}}} - \frac {195 a^{11} b^{4}}{x^{7}} - \frac {462 a^{10} b^{5}}{x^{\frac {13}{2}}} - \frac {5005 a^{9} b^{6}}{6 x^{6}} - \frac {1170 a^{8} b^{7}}{x^{\frac {11}{2}}} - \frac {1287 a^{7} b^{8}}{x^{5}} - \frac {10010 a^{6} b^{9}}{9 x^{\frac {9}{2}}} - \frac {3003 a^{5} b^{10}}{4 x^{4}} - \frac {390 a^{4} b^{11}}{x^{\frac {7}{2}}} - \frac {455 a^{3} b^{12}}{3 x^{3}} - \frac {42 a^{2} b^{13}}{x^{\frac {5}{2}}} - \frac {15 a b^{14}}{2 x^{2}} - \frac {2 b^{15}}{3 x^{\frac {3}{2}}} \]
-a**15/(9*x**9) - 30*a**14*b/(17*x**(17/2)) - 105*a**13*b**2/(8*x**8) - 18 2*a**12*b**3/(3*x**(15/2)) - 195*a**11*b**4/x**7 - 462*a**10*b**5/x**(13/2 ) - 5005*a**9*b**6/(6*x**6) - 1170*a**8*b**7/x**(11/2) - 1287*a**7*b**8/x* *5 - 10010*a**6*b**9/(9*x**(9/2)) - 3003*a**5*b**10/(4*x**4) - 390*a**4*b* *11/x**(7/2) - 455*a**3*b**12/(3*x**3) - 42*a**2*b**13/x**(5/2) - 15*a*b** 14/(2*x**2) - 2*b**15/(3*x**(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (56) = 112\).
Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.39 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx=-\frac {816 \, b^{15} x^{\frac {15}{2}} + 9180 \, a b^{14} x^{7} + 51408 \, a^{2} b^{13} x^{\frac {13}{2}} + 185640 \, a^{3} b^{12} x^{6} + 477360 \, a^{4} b^{11} x^{\frac {11}{2}} + 918918 \, a^{5} b^{10} x^{5} + 1361360 \, a^{6} b^{9} x^{\frac {9}{2}} + 1575288 \, a^{7} b^{8} x^{4} + 1432080 \, a^{8} b^{7} x^{\frac {7}{2}} + 1021020 \, a^{9} b^{6} x^{3} + 565488 \, a^{10} b^{5} x^{\frac {5}{2}} + 238680 \, a^{11} b^{4} x^{2} + 74256 \, a^{12} b^{3} x^{\frac {3}{2}} + 16065 \, a^{13} b^{2} x + 2160 \, a^{14} b \sqrt {x} + 136 \, a^{15}}{1224 \, x^{9}} \]
-1/1224*(816*b^15*x^(15/2) + 9180*a*b^14*x^7 + 51408*a^2*b^13*x^(13/2) + 1 85640*a^3*b^12*x^6 + 477360*a^4*b^11*x^(11/2) + 918918*a^5*b^10*x^5 + 1361 360*a^6*b^9*x^(9/2) + 1575288*a^7*b^8*x^4 + 1432080*a^8*b^7*x^(7/2) + 1021 020*a^9*b^6*x^3 + 565488*a^10*b^5*x^(5/2) + 238680*a^11*b^4*x^2 + 74256*a^ 12*b^3*x^(3/2) + 16065*a^13*b^2*x + 2160*a^14*b*sqrt(x) + 136*a^15)/x^9
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.39 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx=-\frac {816 \, b^{15} x^{\frac {15}{2}} + 9180 \, a b^{14} x^{7} + 51408 \, a^{2} b^{13} x^{\frac {13}{2}} + 185640 \, a^{3} b^{12} x^{6} + 477360 \, a^{4} b^{11} x^{\frac {11}{2}} + 918918 \, a^{5} b^{10} x^{5} + 1361360 \, a^{6} b^{9} x^{\frac {9}{2}} + 1575288 \, a^{7} b^{8} x^{4} + 1432080 \, a^{8} b^{7} x^{\frac {7}{2}} + 1021020 \, a^{9} b^{6} x^{3} + 565488 \, a^{10} b^{5} x^{\frac {5}{2}} + 238680 \, a^{11} b^{4} x^{2} + 74256 \, a^{12} b^{3} x^{\frac {3}{2}} + 16065 \, a^{13} b^{2} x + 2160 \, a^{14} b \sqrt {x} + 136 \, a^{15}}{1224 \, x^{9}} \]
-1/1224*(816*b^15*x^(15/2) + 9180*a*b^14*x^7 + 51408*a^2*b^13*x^(13/2) + 1 85640*a^3*b^12*x^6 + 477360*a^4*b^11*x^(11/2) + 918918*a^5*b^10*x^5 + 1361 360*a^6*b^9*x^(9/2) + 1575288*a^7*b^8*x^4 + 1432080*a^8*b^7*x^(7/2) + 1021 020*a^9*b^6*x^3 + 565488*a^10*b^5*x^(5/2) + 238680*a^11*b^4*x^2 + 74256*a^ 12*b^3*x^(3/2) + 16065*a^13*b^2*x + 2160*a^14*b*sqrt(x) + 136*a^15)/x^9
Time = 5.93 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.39 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}} \, dx=-\frac {\frac {a^{15}}{9}+\frac {2\,b^{15}\,x^{15/2}}{3}+\frac {105\,a^{13}\,b^2\,x}{8}+\frac {30\,a^{14}\,b\,\sqrt {x}}{17}+\frac {15\,a\,b^{14}\,x^7}{2}+195\,a^{11}\,b^4\,x^2+\frac {5005\,a^9\,b^6\,x^3}{6}+1287\,a^7\,b^8\,x^4+\frac {3003\,a^5\,b^{10}\,x^5}{4}+\frac {182\,a^{12}\,b^3\,x^{3/2}}{3}+\frac {455\,a^3\,b^{12}\,x^6}{3}+462\,a^{10}\,b^5\,x^{5/2}+1170\,a^8\,b^7\,x^{7/2}+\frac {10010\,a^6\,b^9\,x^{9/2}}{9}+390\,a^4\,b^{11}\,x^{11/2}+42\,a^2\,b^{13}\,x^{13/2}}{x^9} \]
-(a^15/9 + (2*b^15*x^(15/2))/3 + (105*a^13*b^2*x)/8 + (30*a^14*b*x^(1/2))/ 17 + (15*a*b^14*x^7)/2 + 195*a^11*b^4*x^2 + (5005*a^9*b^6*x^3)/6 + 1287*a^ 7*b^8*x^4 + (3003*a^5*b^10*x^5)/4 + (182*a^12*b^3*x^(3/2))/3 + (455*a^3*b^ 12*x^6)/3 + 462*a^10*b^5*x^(5/2) + 1170*a^8*b^7*x^(7/2) + (10010*a^6*b^9*x ^(9/2))/9 + 390*a^4*b^11*x^(11/2) + 42*a^2*b^13*x^(13/2))/x^9