\(\int \frac {(a+b \sqrt {x})^{15}}{x^{12}} \, dx\) [75]

Optimal result

Integrand size = 15, antiderivative size = 170 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{11 a x^{11}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{77 a^2 x^{21/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{154 a^3 x^{10}}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{1463 a^4 x^{19/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{4389 a^5 x^9}+\frac {2 b^5 \left (a+b \sqrt {x}\right )^{16}}{74613 a^6 x^{17/2}}-\frac {b^6 \left (a+b \sqrt {x}\right )^{16}}{596904 a^7 x^8} \] Output:

-1/11*(a+b*x^(1/2))^16/a/x^11+2/77*b*(a+b*x^(1/2))^16/a^2/x^(21/2)-1/154*b 
^2*(a+b*x^(1/2))^16/a^3/x^10+2/1463*b^3*(a+b*x^(1/2))^16/a^4/x^(19/2)-1/43 
89*b^4*(a+b*x^(1/2))^16/a^5/x^9+2/74613*b^5*(a+b*x^(1/2))^16/a^6/x^(17/2)- 
1/596904*b^6*(a+b*x^(1/2))^16/a^7/x^8
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=\frac {-54264 a^{15}-852720 a^{14} b \sqrt {x}-6267492 a^{13} b^2 x-28588560 a^{12} b^3 x^{3/2}-90530440 a^{11} b^4 x^2-210882672 a^{10} b^5 x^{5/2}-373438065 a^9 b^6 x^3-512143632 a^8 b^7 x^{7/2}-548725320 a^7 b^8 x^4-459616080 a^6 b^9 x^{9/2}-298750452 a^5 b^{10} x^5-148140720 a^4 b^{11} x^{11/2}-54318264 a^3 b^{12} x^6-13927760 a^2 b^{13} x^{13/2}-2238390 a b^{14} x^7-170544 b^{15} x^{15/2}}{596904 x^{11}} \] Input:

Integrate[(a + b*Sqrt[x])^15/x^12,x]
 

Output:

(-54264*a^15 - 852720*a^14*b*Sqrt[x] - 6267492*a^13*b^2*x - 28588560*a^12* 
b^3*x^(3/2) - 90530440*a^11*b^4*x^2 - 210882672*a^10*b^5*x^(5/2) - 3734380 
65*a^9*b^6*x^3 - 512143632*a^8*b^7*x^(7/2) - 548725320*a^7*b^8*x^4 - 45961 
6080*a^6*b^9*x^(9/2) - 298750452*a^5*b^10*x^5 - 148140720*a^4*b^11*x^(11/2 
) - 54318264*a^3*b^12*x^6 - 13927760*a^2*b^13*x^(13/2) - 2238390*a*b^14*x^ 
7 - 170544*b^15*x^(15/2))/(596904*x^11)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {798, 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.

\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{23/2}}d\sqrt {x}\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}}d\sqrt {x}}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {5 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{21/2}}d\sqrt {x}}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}}d\sqrt {x}}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{19/2}}d\sqrt {x}}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9}d\sqrt {x}}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17/2}}d\sqrt {x}}{17 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )\)

\(\Big \downarrow \) 48

\(\displaystyle 2 \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (\frac {b \left (a+b \sqrt {x}\right )^{16}}{272 a^2 x^8}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )\)

Input:

Int[(a + b*Sqrt[x])^15/x^12,x]
 

Output:

2*((-3*b*((-5*b*(-1/5*(b*((-3*b*(-1/9*(b*(-1/17*(a + b*Sqrt[x])^16/(a*x^(1 
7/2)) + (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))
 

Defintions of rubi rules used

rule 48

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]
 

rule 55

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])
 

rule 798

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]]
 

Maple [A] (verified)

Time = 23.22 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(-\frac {858 a^{8} b^{7}}{x^{\frac {15}{2}}}-\frac {910 a^{12} b^{3}}{19 x^{\frac {19}{2}}}-\frac {770 a^{6} b^{9}}{x^{\frac {13}{2}}}-\frac {455 a^{11} b^{4}}{3 x^{9}}-\frac {2 b^{15}}{7 x^{\frac {7}{2}}}-\frac {6006 a^{10} b^{5}}{17 x^{\frac {17}{2}}}-\frac {1001 a^{5} b^{10}}{2 x^{6}}-\frac {5005 a^{9} b^{6}}{8 x^{8}}-\frac {70 a^{2} b^{13}}{3 x^{\frac {9}{2}}}-\frac {a^{15}}{11 x^{11}}-\frac {21 a^{13} b^{2}}{2 x^{10}}-\frac {10 a^{14} b}{7 x^{\frac {21}{2}}}-\frac {2730 a^{4} b^{11}}{11 x^{\frac {11}{2}}}-\frac {6435 a^{7} b^{8}}{7 x^{7}}-\frac {15 a \,b^{14}}{4 x^{4}}-\frac {91 a^{3} b^{12}}{x^{5}}\)	\(168\)
default	\(-\frac {858 a^{8} b^{7}}{x^{\frac {15}{2}}}-\frac {910 a^{12} b^{3}}{19 x^{\frac {19}{2}}}-\frac {770 a^{6} b^{9}}{x^{\frac {13}{2}}}-\frac {455 a^{11} b^{4}}{3 x^{9}}-\frac {2 b^{15}}{7 x^{\frac {7}{2}}}-\frac {6006 a^{10} b^{5}}{17 x^{\frac {17}{2}}}-\frac {1001 a^{5} b^{10}}{2 x^{6}}-\frac {5005 a^{9} b^{6}}{8 x^{8}}-\frac {70 a^{2} b^{13}}{3 x^{\frac {9}{2}}}-\frac {a^{15}}{11 x^{11}}-\frac {21 a^{13} b^{2}}{2 x^{10}}-\frac {10 a^{14} b}{7 x^{\frac {21}{2}}}-\frac {2730 a^{4} b^{11}}{11 x^{\frac {11}{2}}}-\frac {6435 a^{7} b^{8}}{7 x^{7}}-\frac {15 a \,b^{14}}{4 x^{4}}-\frac {91 a^{3} b^{12}}{x^{5}}\)	\(168\)
orering	\(-\frac {\left (1812030 b^{28} x^{14}-17956862 a^{2} b^{26} x^{13}[phone] a^{4} b^{24} x^{12}-339728170 a^{6} b^{22} x^{11}[phone] a^{8} b^{20} x^{10}-1575629055 a^{10} b^{18} x^{9}[phone] a^{12} b^{16} x^{8}-2422797715 a^{14} b^{14} x^{7}[phone] a^{16} b^{12} x^{6}-1309861875 a^{18} b^{10} x^{5}[phone] a^{20} b^{8} x^{4}-225111887 a^{22} b^{6} x^{3}[phone] a^{24} b^{4} x^{2}-8289540 a^{26} b^{2} x +581400 a^{28}\right ) \left (a +b \sqrt {x}\right )^{15}}{2984520 x^{11} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (319770 b^{28} x^{14}-2835294 a^{2} b^{26} x^{13}[phone] a^{4} b^{24} x^{12}-44312370 a^{6} b^{22} x^{11}[phone] a^{8} b^{20} x^{10}-175069895 a^{10} b^{18} x^{9}[phone] a^{12} b^{16} x^{8}-234464295 a^{14} b^{14} x^{7}[phone] a^{16} b^{12} x^{6}-112273875 a^{18} b^{10} x^{5}[phone] a^{20} b^{8} x^{4}-17316299 a^{22} b^{6} x^{3}+4024020 a^{24} b^{4} x^{2}-578340 a^{26} b^{2} x +38760 a^{28}\right ) x^{2} \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} b}{2 x^{\frac {25}{2}}}-\frac {12 \left (a +b \sqrt {x}\right )^{15}}{x^{13}}\right )}{4476780 \left (-b^{2} x +a^{2}\right )^{14}}\)	\(385\)
trager	\(\frac {\left (-1+x \right ) \left (168 a^{14} x^{10}+19404 a^{12} b^{2} x^{10}+280280 a^{10} b^{4} x^{10}+1156155 a^{8} b^{6} x^{10}+1698840 a^{6} b^{8} x^{10}+924924 a^{4} b^{10} x^{10}+168168 a^{2} b^{12} x^{10}+6930 b^{14} x^{10}+168 a^{14} x^{9}+19404 a^{12} b^{2} x^{9}+280280 a^{10} b^{4} x^{9}+1156155 a^{8} b^{6} x^{9}+1698840 a^{6} b^{8} x^{9}+924924 a^{4} b^{10} x^{9}+168168 a^{2} b^{12} x^{9}+6930 b^{14} x^{9}+168 a^{14} x^{8}+19404 a^{12} b^{2} x^{8}+280280 a^{10} b^{4} x^{8}+1156155 a^{8} b^{6} x^{8}+1698840 a^{6} b^{8} x^{8}+924924 a^{4} b^{10} x^{8}+168168 a^{2} b^{12} x^{8}+6930 b^{14} x^{8}+168 a^{14} x^{7}+19404 a^{12} b^{2} x^{7}+280280 a^{10} b^{4} x^{7}+1156155 a^{8} b^{6} x^{7}+1698840 a^{6} b^{8} x^{7}+924924 b^{10} x^{7} a^{4}+168168 a^{2} b^{12} x^{7}+6930 x^{7} b^{14}+168 a^{14} x^{6}+19404 a^{12} b^{2} x^{6}+280280 a^{10} b^{4} x^{6}+1156155 a^{8} b^{6} x^{6}+1698840 b^{8} x^{6} a^{6}+924924 a^{4} b^{10} x^{6}+168168 a^{2} b^{12} x^{6}+168 a^{14} x^{5}+19404 a^{12} b^{2} x^{5}+280280 a^{10} b^{4} x^{5}+1156155 a^{8} b^{6} x^{5}+1698840 a^{6} b^{8} x^{5}+924924 a^{4} b^{10} x^{5}+168 a^{14} x^{4}+19404 a^{12} b^{2} x^{4}+280280 b^{4} x^{4} a^{10}+1156155 x^{4} b^{6} a^{8}+1698840 a^{6} b^{8} x^{4}+168 a^{14} x^{3}+19404 a^{12} b^{2} x^{3}+280280 a^{10} b^{4} x^{3}+1156155 a^{8} b^{6} x^{3}+168 a^{14} x^{2}+19404 a^{12} b^{2} x^{2}+280280 a^{10} b^{4} x^{2}+168 a^{14} x +19404 a^{12} b^{2} x +168 a^{14}\right ) a}{1848 x^{11}}-\frac {2 \left (10659 x^{7} b^{14}+870485 a^{2} b^{12} x^{6}+9258795 a^{4} b^{10} x^{5}[phone] a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}[phone] a^{10} b^{4} x^{2}+1786785 a^{12} b^{2} x +53295 a^{14}\right ) b}{74613 x^{\frac {21}{2}}}\)	\(704\)

Input:

int((a+b*x^(1/2))^15/x^12,x,method=_RETURNVERBOSE)
 

Output:

-858*a^8*b^7/x^(15/2)-910/19*a^12*b^3/x^(19/2)-770*a^6*b^9/x^(13/2)-455/3* 
a^11*b^4/x^9-2/7*b^15/x^(7/2)-6006/17*a^10*b^5/x^(17/2)-1001/2*a^5*b^10/x^ 
6-5005/8*a^9*b^6/x^8-70/3*a^2*b^13/x^(9/2)-1/11*a^15/x^11-21/2*a^13*b^2/x^ 
10-10/7*a^14*b/x^(21/2)-2730/11*a^4*b^11/x^(11/2)-6435/7*a^7*b^8/x^7-15/4* 
a*b^14/x^4-91*a^3*b^12/x^5
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=-\frac {2238390 \, a b^{14} x^{7} + 54318264 \, a^{3} b^{12} x^{6} + 298750452 \, a^{5} b^{10} x^{5} + 548725320 \, a^{7} b^{8} x^{4} + 373438065 \, a^{9} b^{6} x^{3} + 90530440 \, a^{11} b^{4} x^{2} + 6267492 \, a^{13} b^{2} x + 54264 \, a^{15} + 16 \, {\left (10659 \, b^{15} x^{7} + 870485 \, a^{2} b^{13} x^{6} + 9258795 \, a^{4} b^{11} x^{5} + 28726005 \, a^{6} b^{9} x^{4} + 32008977 \, a^{8} b^{7} x^{3} + 13180167 \, a^{10} b^{5} x^{2} + 1786785 \, a^{12} b^{3} x + 53295 \, a^{14} b\right )} \sqrt {x}}{596904 \, x^{11}} \] Input:

integrate((a+b*x^(1/2))^15/x^12,x, algorithm="fricas")
 

Output:

-1/596904*(2238390*a*b^14*x^7 + 54318264*a^3*b^12*x^6 + 298750452*a^5*b^10 
*x^5 + 548725320*a^7*b^8*x^4 + 373438065*a^9*b^6*x^3 + 90530440*a^11*b^4*x 
^2 + 6267492*a^13*b^2*x + 54264*a^15 + 16*(10659*b^15*x^7 + 870485*a^2*b^1 
3*x^6 + 9258795*a^4*b^11*x^5 + 28726005*a^6*b^9*x^4 + 32008977*a^8*b^7*x^3 
 + 13180167*a^10*b^5*x^2 + 1786785*a^12*b^3*x + 53295*a^14*b)*sqrt(x))/x^1 
1
 

Sympy [A] (verification not implemented)

Time = 1.58 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=- \frac {a^{15}}{11 x^{11}} - \frac {10 a^{14} b}{7 x^{\frac {21}{2}}} - \frac {21 a^{13} b^{2}}{2 x^{10}} - \frac {910 a^{12} b^{3}}{19 x^{\frac {19}{2}}} - \frac {455 a^{11} b^{4}}{3 x^{9}} - \frac {6006 a^{10} b^{5}}{17 x^{\frac {17}{2}}} - \frac {5005 a^{9} b^{6}}{8 x^{8}} - \frac {858 a^{8} b^{7}}{x^{\frac {15}{2}}} - \frac {6435 a^{7} b^{8}}{7 x^{7}} - \frac {770 a^{6} b^{9}}{x^{\frac {13}{2}}} - \frac {1001 a^{5} b^{10}}{2 x^{6}} - \frac {2730 a^{4} b^{11}}{11 x^{\frac {11}{2}}} - \frac {91 a^{3} b^{12}}{x^{5}} - \frac {70 a^{2} b^{13}}{3 x^{\frac {9}{2}}} - \frac {15 a b^{14}}{4 x^{4}} - \frac {2 b^{15}}{7 x^{\frac {7}{2}}} \] Input:

integrate((a+b*x**(1/2))**15/x**12,x)
 

Output:

-a**15/(11*x**11) - 10*a**14*b/(7*x**(21/2)) - 21*a**13*b**2/(2*x**10) - 9 
10*a**12*b**3/(19*x**(19/2)) - 455*a**11*b**4/(3*x**9) - 6006*a**10*b**5/( 
17*x**(17/2)) - 5005*a**9*b**6/(8*x**8) - 858*a**8*b**7/x**(15/2) - 6435*a 
**7*b**8/(7*x**7) - 770*a**6*b**9/x**(13/2) - 1001*a**5*b**10/(2*x**6) - 2 
730*a**4*b**11/(11*x**(11/2)) - 91*a**3*b**12/x**5 - 70*a**2*b**13/(3*x**( 
9/2)) - 15*a*b**14/(4*x**4) - 2*b**15/(7*x**(7/2))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=-\frac {170544 \, b^{15} x^{\frac {15}{2}} + 2238390 \, a b^{14} x^{7} + 13927760 \, a^{2} b^{13} x^{\frac {13}{2}} + 54318264 \, a^{3} b^{12} x^{6} + 148140720 \, a^{4} b^{11} x^{\frac {11}{2}} + 298750452 \, a^{5} b^{10} x^{5} + 459616080 \, a^{6} b^{9} x^{\frac {9}{2}} + 548725320 \, a^{7} b^{8} x^{4} + 512143632 \, a^{8} b^{7} x^{\frac {7}{2}} + 373438065 \, a^{9} b^{6} x^{3} + 210882672 \, a^{10} b^{5} x^{\frac {5}{2}} + 90530440 \, a^{11} b^{4} x^{2} + 28588560 \, a^{12} b^{3} x^{\frac {3}{2}} + 6267492 \, a^{13} b^{2} x + 852720 \, a^{14} b \sqrt {x} + 54264 \, a^{15}}{596904 \, x^{11}} \] Input:

integrate((a+b*x^(1/2))^15/x^12,x, algorithm="maxima")
 

Output:

-1/596904*(170544*b^15*x^(15/2) + 2238390*a*b^14*x^7 + 13927760*a^2*b^13*x 
^(13/2) + 54318264*a^3*b^12*x^6 + 148140720*a^4*b^11*x^(11/2) + 298750452* 
a^5*b^10*x^5 + 459616080*a^6*b^9*x^(9/2) + 548725320*a^7*b^8*x^4 + 5121436 
32*a^8*b^7*x^(7/2) + 373438065*a^9*b^6*x^3 + 210882672*a^10*b^5*x^(5/2) + 
90530440*a^11*b^4*x^2 + 28588560*a^12*b^3*x^(3/2) + 6267492*a^13*b^2*x + 8 
52720*a^14*b*sqrt(x) + 54264*a^15)/x^11
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=-\frac {170544 \, b^{15} x^{\frac {15}{2}} + 2238390 \, a b^{14} x^{7} + 13927760 \, a^{2} b^{13} x^{\frac {13}{2}} + 54318264 \, a^{3} b^{12} x^{6} + 148140720 \, a^{4} b^{11} x^{\frac {11}{2}} + 298750452 \, a^{5} b^{10} x^{5} + 459616080 \, a^{6} b^{9} x^{\frac {9}{2}} + 548725320 \, a^{7} b^{8} x^{4} + 512143632 \, a^{8} b^{7} x^{\frac {7}{2}} + 373438065 \, a^{9} b^{6} x^{3} + 210882672 \, a^{10} b^{5} x^{\frac {5}{2}} + 90530440 \, a^{11} b^{4} x^{2} + 28588560 \, a^{12} b^{3} x^{\frac {3}{2}} + 6267492 \, a^{13} b^{2} x + 852720 \, a^{14} b \sqrt {x} + 54264 \, a^{15}}{596904 \, x^{11}} \] Input:

integrate((a+b*x^(1/2))^15/x^12,x, algorithm="giac")
 

Output:

-1/596904*(170544*b^15*x^(15/2) + 2238390*a*b^14*x^7 + 13927760*a^2*b^13*x 
^(13/2) + 54318264*a^3*b^12*x^6 + 148140720*a^4*b^11*x^(11/2) + 298750452* 
a^5*b^10*x^5 + 459616080*a^6*b^9*x^(9/2) + 548725320*a^7*b^8*x^4 + 5121436 
32*a^8*b^7*x^(7/2) + 373438065*a^9*b^6*x^3 + 210882672*a^10*b^5*x^(5/2) + 
90530440*a^11*b^4*x^2 + 28588560*a^12*b^3*x^(3/2) + 6267492*a^13*b^2*x + 8 
52720*a^14*b*sqrt(x) + 54264*a^15)/x^11
 

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=-\frac {\frac {a^{15}}{11}+\frac {2\,b^{15}\,x^{15/2}}{7}+\frac {21\,a^{13}\,b^2\,x}{2}+\frac {10\,a^{14}\,b\,\sqrt {x}}{7}+\frac {15\,a\,b^{14}\,x^7}{4}+\frac {455\,a^{11}\,b^4\,x^2}{3}+\frac {5005\,a^9\,b^6\,x^3}{8}+\frac {6435\,a^7\,b^8\,x^4}{7}+\frac {1001\,a^5\,b^{10}\,x^5}{2}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{19}+91\,a^3\,b^{12}\,x^6+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{17}+858\,a^8\,b^7\,x^{7/2}+770\,a^6\,b^9\,x^{9/2}+\frac {2730\,a^4\,b^{11}\,x^{11/2}}{11}+\frac {70\,a^2\,b^{13}\,x^{13/2}}{3}}{x^{11}} \] Input:

int((a + b*x^(1/2))^15/x^12,x)
 

Output:

-(a^15/11 + (2*b^15*x^(15/2))/7 + (21*a^13*b^2*x)/2 + (10*a^14*b*x^(1/2))/ 
7 + (15*a*b^14*x^7)/4 + (455*a^11*b^4*x^2)/3 + (5005*a^9*b^6*x^3)/8 + (643 
5*a^7*b^8*x^4)/7 + (1001*a^5*b^10*x^5)/2 + (910*a^12*b^3*x^(3/2))/19 + 91* 
a^3*b^12*x^6 + (6006*a^10*b^5*x^(5/2))/17 + 858*a^8*b^7*x^(7/2) + 770*a^6* 
b^9*x^(9/2) + (2730*a^4*b^11*x^(11/2))/11 + (70*a^2*b^13*x^(13/2))/3)/x^11
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}} \, dx=\frac {-54264 \sqrt {x}\, a^{15}-6267492 \sqrt {x}\, a^{13} b^{2} x -90530440 \sqrt {x}\, a^{11} b^{4} x^{2}-373438065 \sqrt {x}\, a^{9} b^{6} x^{3}-548725320 \sqrt {x}\, a^{7} b^{8} x^{4}-298750452 \sqrt {x}\, a^{5} b^{10} x^{5}-54318264 \sqrt {x}\, a^{3} b^{12} x^{6}-2238390 \sqrt {x}\, a \,b^{14} x^{7}-852720 a^{14} b x -28588560 a^{12} b^{3} x^{2}-210882672 a^{10} b^{5} x^{3}-512143632 a^{8} b^{7} x^{4}-459616080 a^{6} b^{9} x^{5}-148140720 a^{4} b^{11} x^{6}-13927760 a^{2} b^{13} x^{7}-170544 b^{15} x^{8}}{596904 \sqrt {x}\, x^{11}} \] Input:

int((a+b*x^(1/2))^15/x^12,x)
 

Output:

( - 54264*sqrt(x)*a**15 - 6267492*sqrt(x)*a**13*b**2*x - 90530440*sqrt(x)* 
a**11*b**4*x**2 - 373438065*sqrt(x)*a**9*b**6*x**3 - 548725320*sqrt(x)*a** 
7*b**8*x**4 - 298750452*sqrt(x)*a**5*b**10*x**5 - 54318264*sqrt(x)*a**3*b* 
*12*x**6 - 2238390*sqrt(x)*a*b**14*x**7 - 852720*a**14*b*x - 28588560*a**1 
2*b**3*x**2 - 210882672*a**10*b**5*x**3 - 512143632*a**8*b**7*x**4 - 45961 
6080*a**6*b**9*x**5 - 148140720*a**4*b**11*x**6 - 13927760*a**2*b**13*x**7 
 - 170544*b**15*x**8)/(596904*sqrt(x)*x**11)