\(\int \frac {(a+b \sqrt {x})^{15}}{x^{14}} \, dx\) [77]

Optimal result

Integrand size = 15, antiderivative size = 270 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {\left (a+b \sqrt {x}\right )^{16}}{13 a x^{13}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{65 a^2 x^{25/2}}-\frac {3 b^2 \left (a+b \sqrt {x}\right )^{16}}{260 a^3 x^{12}}+\frac {6 b^3 \left (a+b \sqrt {x}\right )^{16}}{1495 a^4 x^{23/2}}-\frac {21 b^4 \left (a+b \sqrt {x}\right )^{16}}{16445 a^5 x^{11}}+\frac {6 b^5 \left (a+b \sqrt {x}\right )^{16}}{16445 a^6 x^{21/2}}-\frac {3 b^6 \left (a+b \sqrt {x}\right )^{16}}{32890 a^7 x^{10}}+\frac {6 b^7 \left (a+b \sqrt {x}\right )^{16}}{312455 a^8 x^{19/2}}-\frac {b^8 \left (a+b \sqrt {x}\right )^{16}}{312455 a^9 x^9}+\frac {2 b^9 \left (a+b \sqrt {x}\right )^{16}}{5311735 a^{10} x^{17/2}}-\frac {b^{10} \left (a+b \sqrt {x}\right )^{16}}{42493880 a^{11} x^8} \] Output:

-1/13*(a+b*x^(1/2))^16/a/x^13+2/65*b*(a+b*x^(1/2))^16/a^2/x^(25/2)-3/260*b 
^2*(a+b*x^(1/2))^16/a^3/x^12+6/1495*b^3*(a+b*x^(1/2))^16/a^4/x^(23/2)-21/1 
6445*b^4*(a+b*x^(1/2))^16/a^5/x^11+6/16445*b^5*(a+b*x^(1/2))^16/a^6/x^(21/ 
2)-3/32890*b^6*(a+b*x^(1/2))^16/a^7/x^10+6/312455*b^7*(a+b*x^(1/2))^16/a^8 
/x^(19/2)-1/312455*b^8*(a+b*x^(1/2))^16/a^9/x^9+2/5311735*b^9*(a+b*x^(1/2) 
)^16/a^10/x^(17/2)-1/42493880*b^10*(a+b*x^(1/2))^16/a^11/x^8
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=\frac {-3268760 a^{15}-50992656 a^{14} b \sqrt {x}-371821450 a^{13} b^2 x-1681279600 a^{12} b^3 x^{3/2}-5273104200 a^{11} b^4 x^2-12153249680 a^{10} b^5 x^{5/2}-21268186940 a^9 b^6 x^3-28784012400 a^8 b^7 x^{7/2}-30383124200 a^7 b^8 x^4-25021396400 a^6 b^9 x^{9/2}-15951140205 a^5 b^{10} x^5-7733886160 a^4 b^{11} x^{11/2}-2762102200 a^3 b^{12} x^6-686439600 a^2 b^{13} x^{13/2}-106234700 a b^{14} x^7-7726160 b^{15} x^{15/2}}{42493880 x^{13}} \] Input:

Integrate[(a + b*Sqrt[x])^15/x^14,x]
 

Output:

(-3268760*a^15 - 50992656*a^14*b*Sqrt[x] - 371821450*a^13*b^2*x - 16812796 
00*a^12*b^3*x^(3/2) - 5273104200*a^11*b^4*x^2 - 12153249680*a^10*b^5*x^(5/ 
2) - 21268186940*a^9*b^6*x^3 - 28784012400*a^8*b^7*x^(7/2) - 30383124200*a 
^7*b^8*x^4 - 25021396400*a^6*b^9*x^(9/2) - 15951140205*a^5*b^10*x^5 - 7733 
886160*a^4*b^11*x^(11/2) - 2762102200*a^3*b^12*x^6 - 686439600*a^2*b^13*x^ 
(13/2) - 106234700*a*b^14*x^7 - 7726160*b^15*x^(15/2))/(42493880*x^13)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.21, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {798, 55, 55, 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.

\(\displaystyle \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{27/2}}d\sqrt {x}\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{13}}d\sqrt {x}}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{25/2}}d\sqrt {x}}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{12}}d\sqrt {x}}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{23/2}}d\sqrt {x}}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}}d\sqrt {x}}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{21/2}}d\sqrt {x}}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{10}}d\sqrt {x}}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{19/2}}d\sqrt {x}}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^9}d\sqrt {x}}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 55

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (-\frac {b \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{17/2}}d\sqrt {x}}{17 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

\(\Big \downarrow \) 48

\(\displaystyle 2 \left (-\frac {5 b \left (-\frac {9 b \left (-\frac {b \left (-\frac {7 b \left (-\frac {3 b \left (-\frac {5 b \left (-\frac {b \left (-\frac {3 b \left (-\frac {b \left (\frac {b \left (a+b \sqrt {x}\right )^{16}}{272 a^2 x^8}-\frac {\left (a+b \sqrt {x}\right )^{16}}{17 a x^{17/2}}\right )}{9 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{18 a x^9}\right )}{19 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{19 a x^{19/2}}\right )}{5 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{20 a x^{10}}\right )}{21 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{21 a x^{21/2}}\right )}{11 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{22 a x^{11}}\right )}{23 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{23 a x^{23/2}}\right )}{3 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{24 a x^{12}}\right )}{25 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{25 a x^{25/2}}\right )}{13 a}-\frac {\left (a+b \sqrt {x}\right )^{16}}{26 a x^{13}}\right )\)

Input:

Int[(a + b*Sqrt[x])^15/x^14,x]
 

Output:

2*((-5*b*((-9*b*(-1/3*(b*((-7*b*((-3*b*((-5*b*(-1/5*(b*((-3*b*(-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)))/(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])^1 
6/(21*a*x^(21/2))))/(11*a) - (a + b*Sqrt[x])^16/(22*a*x^11)))/(23*a) - (a 
+ b*Sqrt[x])^16/(23*a*x^(23/2))))/a - (a + b*Sqrt[x])^16/(24*a*x^12)))/(25 
*a) - (a + b*Sqrt[x])^16/(25*a*x^(25/2))))/(13*a) - (a + b*Sqrt[x])^16/(26 
*a*x^13))
 

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 = 23.63 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.62

method	result	size
derivativedivides	\(-\frac {12870 a^{8} b^{7}}{19 x^{\frac {19}{2}}}-\frac {3003 a^{5} b^{10}}{8 x^{8}}-\frac {35 a^{13} b^{2}}{4 x^{12}}-\frac {2 b^{15}}{11 x^{\frac {11}{2}}}-\frac {1001 a^{9} b^{6}}{2 x^{10}}-\frac {715 a^{7} b^{8}}{x^{9}}-\frac {1365 a^{11} b^{4}}{11 x^{11}}-\frac {10010 a^{6} b^{9}}{17 x^{\frac {17}{2}}}-\frac {65 a^{3} b^{12}}{x^{7}}-\frac {6 a^{14} b}{5 x^{\frac {25}{2}}}-\frac {5 a \,b^{14}}{2 x^{6}}-\frac {910 a^{12} b^{3}}{23 x^{\frac {23}{2}}}-\frac {182 a^{4} b^{11}}{x^{\frac {15}{2}}}-\frac {a^{15}}{13 x^{13}}-\frac {286 a^{10} b^{5}}{x^{\frac {21}{2}}}-\frac {210 a^{2} b^{13}}{13 x^{\frac {13}{2}}}\)	\(168\)
default	\(-\frac {12870 a^{8} b^{7}}{19 x^{\frac {19}{2}}}-\frac {3003 a^{5} b^{10}}{8 x^{8}}-\frac {35 a^{13} b^{2}}{4 x^{12}}-\frac {2 b^{15}}{11 x^{\frac {11}{2}}}-\frac {1001 a^{9} b^{6}}{2 x^{10}}-\frac {715 a^{7} b^{8}}{x^{9}}-\frac {1365 a^{11} b^{4}}{11 x^{11}}-\frac {10010 a^{6} b^{9}}{17 x^{\frac {17}{2}}}-\frac {65 a^{3} b^{12}}{x^{7}}-\frac {6 a^{14} b}{5 x^{\frac {25}{2}}}-\frac {5 a \,b^{14}}{2 x^{6}}-\frac {910 a^{12} b^{3}}{23 x^{\frac {23}{2}}}-\frac {182 a^{4} b^{11}}{x^{\frac {15}{2}}}-\frac {a^{15}}{13 x^{13}}-\frac {286 a^{10} b^{5}}{x^{\frac {21}{2}}}-\frac {210 a^{2} b^{13}}{13 x^{\frac {13}{2}}}\)	\(168\)
orering	\(-\frac {\left (241442500 b^{28} x^{14}-2828220300 a^{2} b^{26} x^{13}[phone] a^{4} b^{24} x^{12}-58649365775 a^{6} b^{22} x^{11}[phone] a^{8} b^{20} x^{10}-279749071875 a^{10} b^{18} x^{9}[phone] a^{12} b^{16} x^{8}-434167788015 a^{14} b^{14} x^{7}[phone] a^{16} b^{12} x^{6}-235230726731 a^{18} b^{10} x^{5}[phone] a^{20} b^{8} x^{4}-40390576590 a^{22} b^{6} x^{3}[phone] a^{24} b^{4} x^{2}-1484052570 a^{26} b^{2} x [phone] a^{28}\right ) \left (a +b \sqrt {x}\right )^{15}}{637408200 x^{13} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (9657700 b^{28} x^{14}-104748900 a^{2} b^{26} x^{13}[phone] a^{4} b^{24} x^{12}-1891915025 a^{6} b^{22} x^{11}[phone] a^{8} b^{20} x^{10}-7992830625 a^{10} b^{18} x^{9}[phone] a^{12} b^{16} x^{8}-11132507385 a^{14} b^{14} x^{7}[phone] a^{16} b^{12} x^{6}-5470482017 a^{18} b^{10} x^{5}[phone] a^{20} b^{8} x^{4}-859373970 a^{22} b^{6} x^{3}[phone] a^{24} b^{4} x^{2}-29099070 a^{26} b^{2} x +1961256 a^{28}\right ) x^{2} \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} b}{2 x^{\frac {29}{2}}}-\frac {14 \left (a +b \sqrt {x}\right )^{15}}{x^{15}}\right )}{318704100 \left (-b^{2} x +a^{2}\right )^{14}}\)	\(385\)
trager	\(\frac {\left (-1+x \right ) \left (88 a^{14} x^{12}+10010 a^{12} b^{2} x^{12}+141960 a^{10} b^{4} x^{12}+572572 a^{8} b^{6} x^{12}+817960 a^{6} b^{8} x^{12}+429429 a^{4} b^{10} x^{12}+74360 a^{2} b^{12} x^{12}+2860 b^{14} x^{12}+88 a^{14} x^{11}+10010 a^{12} b^{2} x^{11}+141960 a^{10} b^{4} x^{11}+572572 a^{8} b^{6} x^{11}+817960 a^{6} b^{8} x^{11}+429429 a^{4} b^{10} x^{11}+74360 a^{2} b^{12} x^{11}+2860 b^{14} x^{11}+88 a^{14} x^{10}+10010 a^{12} b^{2} x^{10}+141960 a^{10} b^{4} x^{10}+572572 a^{8} b^{6} x^{10}+817960 a^{6} b^{8} x^{10}+429429 a^{4} b^{10} x^{10}+74360 a^{2} b^{12} x^{10}+2860 b^{14} x^{10}+88 a^{14} x^{9}+10010 a^{12} b^{2} x^{9}+141960 a^{10} b^{4} x^{9}+572572 a^{8} b^{6} x^{9}+817960 a^{6} b^{8} x^{9}+429429 a^{4} b^{10} x^{9}+74360 a^{2} b^{12} x^{9}+2860 b^{14} x^{9}+88 a^{14} x^{8}+10010 a^{12} b^{2} x^{8}+141960 a^{10} b^{4} x^{8}+572572 a^{8} b^{6} x^{8}+817960 a^{6} b^{8} x^{8}+429429 a^{4} b^{10} x^{8}+74360 a^{2} b^{12} x^{8}+2860 b^{14} x^{8}+88 a^{14} x^{7}+10010 a^{12} b^{2} x^{7}+141960 a^{10} b^{4} x^{7}+572572 a^{8} b^{6} x^{7}+817960 a^{6} b^{8} x^{7}+429429 b^{10} x^{7} a^{4}+74360 a^{2} b^{12} x^{7}+2860 x^{7} b^{14}+88 a^{14} x^{6}+10010 a^{12} b^{2} x^{6}+141960 a^{10} b^{4} x^{6}+572572 a^{8} b^{6} x^{6}+817960 b^{8} x^{6} a^{6}+429429 a^{4} b^{10} x^{6}+74360 a^{2} b^{12} x^{6}+88 a^{14} x^{5}+10010 a^{12} b^{2} x^{5}+141960 a^{10} b^{4} x^{5}+572572 a^{8} b^{6} x^{5}+817960 a^{6} b^{8} x^{5}+429429 a^{4} b^{10} x^{5}+88 a^{14} x^{4}+10010 a^{12} b^{2} x^{4}+141960 b^{4} x^{4} a^{10}+572572 x^{4} b^{6} a^{8}+817960 a^{6} b^{8} x^{4}+88 a^{14} x^{3}+10010 a^{12} b^{2} x^{3}+141960 a^{10} b^{4} x^{3}+572572 a^{8} b^{6} x^{3}+88 a^{14} x^{2}+10010 a^{12} b^{2} x^{2}+141960 a^{10} b^{4} x^{2}+88 a^{14} x +10010 a^{12} b^{2} x +88 a^{14}\right ) a}{1144 x^{13}}-\frac {2 \left (482885 x^{7} b^{14}[phone] a^{2} b^{12} x^{6}[phone] 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}[phone] a^{12} b^{2} x +3187041 a^{14}\right ) b}{5311735 x^{\frac {25}{2}}}\)	\(868\)

Input:

int((a+b*x^(1/2))^15/x^14,x,method=_RETURNVERBOSE)
 

Output:

-12870/19*a^8*b^7/x^(19/2)-3003/8*a^5*b^10/x^8-35/4*a^13*b^2/x^12-2/11*b^1 
5/x^(11/2)-1001/2*a^9*b^6/x^10-715*a^7*b^8/x^9-1365/11*a^11*b^4/x^11-10010 
/17*a^6*b^9/x^(17/2)-65*a^3*b^12/x^7-6/5*a^14*b/x^(25/2)-5/2*a*b^14/x^6-91 
0/23*a^12*b^3/x^(23/2)-182*a^4*b^11/x^(15/2)-1/13*a^15/x^13-286*a^10*b^5/x 
^(21/2)-210/13*a^2*b^13/x^(13/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {106234700 \, a b^{14} x^{7} + 2762102200 \, a^{3} b^{12} x^{6} + 15951140205 \, a^{5} b^{10} x^{5} + 30383124200 \, a^{7} b^{8} x^{4} + 21268186940 \, a^{9} b^{6} x^{3} + 5273104200 \, a^{11} b^{4} x^{2} + 371821450 \, a^{13} b^{2} x + 3268760 \, a^{15} + 16 \, {\left (482885 \, b^{15} x^{7} + 42902475 \, a^{2} b^{13} x^{6} + 483367885 \, a^{4} b^{11} x^{5} + 1563837275 \, a^{6} b^{9} x^{4} + 1799000775 \, a^{8} b^{7} x^{3} + 759578105 \, a^{10} b^{5} x^{2} + 105079975 \, a^{12} b^{3} x + 3187041 \, a^{14} b\right )} \sqrt {x}}{42493880 \, x^{13}} \] Input:

integrate((a+b*x^(1/2))^15/x^14,x, algorithm="fricas")
 

Output:

-1/42493880*(106234700*a*b^14*x^7 + 2762102200*a^3*b^12*x^6 + 15951140205* 
a^5*b^10*x^5 + 30383124200*a^7*b^8*x^4 + 21268186940*a^9*b^6*x^3 + 5273104 
200*a^11*b^4*x^2 + 371821450*a^13*b^2*x + 3268760*a^15 + 16*(482885*b^15*x 
^7 + 42902475*a^2*b^13*x^6 + 483367885*a^4*b^11*x^5 + 1563837275*a^6*b^9*x 
^4 + 1799000775*a^8*b^7*x^3 + 759578105*a^10*b^5*x^2 + 105079975*a^12*b^3* 
x + 3187041*a^14*b)*sqrt(x))/x^13
 

Sympy [A] (verification not implemented)

Time = 2.36 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=- \frac {a^{15}}{13 x^{13}} - \frac {6 a^{14} b}{5 x^{\frac {25}{2}}} - \frac {35 a^{13} b^{2}}{4 x^{12}} - \frac {910 a^{12} b^{3}}{23 x^{\frac {23}{2}}} - \frac {1365 a^{11} b^{4}}{11 x^{11}} - \frac {286 a^{10} b^{5}}{x^{\frac {21}{2}}} - \frac {1001 a^{9} b^{6}}{2 x^{10}} - \frac {12870 a^{8} b^{7}}{19 x^{\frac {19}{2}}} - \frac {715 a^{7} b^{8}}{x^{9}} - \frac {10010 a^{6} b^{9}}{17 x^{\frac {17}{2}}} - \frac {3003 a^{5} b^{10}}{8 x^{8}} - \frac {182 a^{4} b^{11}}{x^{\frac {15}{2}}} - \frac {65 a^{3} b^{12}}{x^{7}} - \frac {210 a^{2} b^{13}}{13 x^{\frac {13}{2}}} - \frac {5 a b^{14}}{2 x^{6}} - \frac {2 b^{15}}{11 x^{\frac {11}{2}}} \] Input:

integrate((a+b*x**(1/2))**15/x**14,x)
 

Output:

-a**15/(13*x**13) - 6*a**14*b/(5*x**(25/2)) - 35*a**13*b**2/(4*x**12) - 91 
0*a**12*b**3/(23*x**(23/2)) - 1365*a**11*b**4/(11*x**11) - 286*a**10*b**5/ 
x**(21/2) - 1001*a**9*b**6/(2*x**10) - 12870*a**8*b**7/(19*x**(19/2)) - 71 
5*a**7*b**8/x**9 - 10010*a**6*b**9/(17*x**(17/2)) - 3003*a**5*b**10/(8*x** 
8) - 182*a**4*b**11/x**(15/2) - 65*a**3*b**12/x**7 - 210*a**2*b**13/(13*x* 
*(13/2)) - 5*a*b**14/(2*x**6) - 2*b**15/(11*x**(11/2))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {7726160 \, b^{15} x^{\frac {15}{2}} + 106234700 \, a b^{14} x^{7} + 686439600 \, a^{2} b^{13} x^{\frac {13}{2}} + 2762102200 \, a^{3} b^{12} x^{6} + 7733886160 \, a^{4} b^{11} x^{\frac {11}{2}} + 15951140205 \, a^{5} b^{10} x^{5} + 25021396400 \, a^{6} b^{9} x^{\frac {9}{2}} + 30383124200 \, a^{7} b^{8} x^{4} + 28784012400 \, a^{8} b^{7} x^{\frac {7}{2}} + 21268186940 \, a^{9} b^{6} x^{3} + 12153249680 \, a^{10} b^{5} x^{\frac {5}{2}} + 5273104200 \, a^{11} b^{4} x^{2} + 1681279600 \, a^{12} b^{3} x^{\frac {3}{2}} + 371821450 \, a^{13} b^{2} x + 50992656 \, a^{14} b \sqrt {x} + 3268760 \, a^{15}}{42493880 \, x^{13}} \] Input:

integrate((a+b*x^(1/2))^15/x^14,x, algorithm="maxima")
 

Output:

-1/42493880*(7726160*b^15*x^(15/2) + 106234700*a*b^14*x^7 + 686439600*a^2* 
b^13*x^(13/2) + 2762102200*a^3*b^12*x^6 + 7733886160*a^4*b^11*x^(11/2) + 1 
5951140205*a^5*b^10*x^5 + 25021396400*a^6*b^9*x^(9/2) + 30383124200*a^7*b^ 
8*x^4 + 28784012400*a^8*b^7*x^(7/2) + 21268186940*a^9*b^6*x^3 + 1215324968 
0*a^10*b^5*x^(5/2) + 5273104200*a^11*b^4*x^2 + 1681279600*a^12*b^3*x^(3/2) 
 + 371821450*a^13*b^2*x + 50992656*a^14*b*sqrt(x) + 3268760*a^15)/x^13
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {7726160 \, b^{15} x^{\frac {15}{2}} + 106234700 \, a b^{14} x^{7} + 686439600 \, a^{2} b^{13} x^{\frac {13}{2}} + 2762102200 \, a^{3} b^{12} x^{6} + 7733886160 \, a^{4} b^{11} x^{\frac {11}{2}} + 15951140205 \, a^{5} b^{10} x^{5} + 25021396400 \, a^{6} b^{9} x^{\frac {9}{2}} + 30383124200 \, a^{7} b^{8} x^{4} + 28784012400 \, a^{8} b^{7} x^{\frac {7}{2}} + 21268186940 \, a^{9} b^{6} x^{3} + 12153249680 \, a^{10} b^{5} x^{\frac {5}{2}} + 5273104200 \, a^{11} b^{4} x^{2} + 1681279600 \, a^{12} b^{3} x^{\frac {3}{2}} + 371821450 \, a^{13} b^{2} x + 50992656 \, a^{14} b \sqrt {x} + 3268760 \, a^{15}}{42493880 \, x^{13}} \] Input:

integrate((a+b*x^(1/2))^15/x^14,x, algorithm="giac")
 

Output:

-1/42493880*(7726160*b^15*x^(15/2) + 106234700*a*b^14*x^7 + 686439600*a^2* 
b^13*x^(13/2) + 2762102200*a^3*b^12*x^6 + 7733886160*a^4*b^11*x^(11/2) + 1 
5951140205*a^5*b^10*x^5 + 25021396400*a^6*b^9*x^(9/2) + 30383124200*a^7*b^ 
8*x^4 + 28784012400*a^8*b^7*x^(7/2) + 21268186940*a^9*b^6*x^3 + 1215324968 
0*a^10*b^5*x^(5/2) + 5273104200*a^11*b^4*x^2 + 1681279600*a^12*b^3*x^(3/2) 
 + 371821450*a^13*b^2*x + 50992656*a^14*b*sqrt(x) + 3268760*a^15)/x^13
 

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=-\frac {\frac {a^{15}}{13}+\frac {2\,b^{15}\,x^{15/2}}{11}+\frac {35\,a^{13}\,b^2\,x}{4}+\frac {6\,a^{14}\,b\,\sqrt {x}}{5}+\frac {5\,a\,b^{14}\,x^7}{2}+\frac {1365\,a^{11}\,b^4\,x^2}{11}+\frac {1001\,a^9\,b^6\,x^3}{2}+715\,a^7\,b^8\,x^4+\frac {3003\,a^5\,b^{10}\,x^5}{8}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{23}+65\,a^3\,b^{12}\,x^6+286\,a^{10}\,b^5\,x^{5/2}+\frac {12870\,a^8\,b^7\,x^{7/2}}{19}+\frac {10010\,a^6\,b^9\,x^{9/2}}{17}+182\,a^4\,b^{11}\,x^{11/2}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{13}}{x^{13}} \] Input:

int((a + b*x^(1/2))^15/x^14,x)
 

Output:

-(a^15/13 + (2*b^15*x^(15/2))/11 + (35*a^13*b^2*x)/4 + (6*a^14*b*x^(1/2))/ 
5 + (5*a*b^14*x^7)/2 + (1365*a^11*b^4*x^2)/11 + (1001*a^9*b^6*x^3)/2 + 715 
*a^7*b^8*x^4 + (3003*a^5*b^10*x^5)/8 + (910*a^12*b^3*x^(3/2))/23 + 65*a^3* 
b^12*x^6 + 286*a^10*b^5*x^(5/2) + (12870*a^8*b^7*x^(7/2))/19 + (10010*a^6* 
b^9*x^(9/2))/17 + 182*a^4*b^11*x^(11/2) + (210*a^2*b^13*x^(13/2))/13)/x^13
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{14}} \, dx=\frac {-3268760 \sqrt {x}\, a^{15}-371821450 \sqrt {x}\, a^{13} b^{2} x -5273104200 \sqrt {x}\, a^{11} b^{4} x^{2}-21268186940 \sqrt {x}\, a^{9} b^{6} x^{3}-30383124200 \sqrt {x}\, a^{7} b^{8} x^{4}-15951140205 \sqrt {x}\, a^{5} b^{10} x^{5}-2762102200 \sqrt {x}\, a^{3} b^{12} x^{6}-106234700 \sqrt {x}\, a \,b^{14} x^{7}-50992656 a^{14} b x -1681279600 a^{12} b^{3} x^{2}-12153249680 a^{10} b^{5} x^{3}-28784012400 a^{8} b^{7} x^{4}-25021396400 a^{6} b^{9} x^{5}-7733886160 a^{4} b^{11} x^{6}-686439600 a^{2} b^{13} x^{7}-7726160 b^{15} x^{8}}{42493880 \sqrt {x}\, x^{13}} \] Input:

int((a+b*x^(1/2))^15/x^14,x)
 

Output:

( - 3268760*sqrt(x)*a**15 - 371821450*sqrt(x)*a**13*b**2*x - 5273104200*sq 
rt(x)*a**11*b**4*x**2 - 21268186940*sqrt(x)*a**9*b**6*x**3 - 30383124200*s 
qrt(x)*a**7*b**8*x**4 - 15951140205*sqrt(x)*a**5*b**10*x**5 - 2762102200*s 
qrt(x)*a**3*b**12*x**6 - 106234700*sqrt(x)*a*b**14*x**7 - 50992656*a**14*b 
*x - 1681279600*a**12*b**3*x**2 - 12153249680*a**10*b**5*x**3 - 2878401240 
0*a**8*b**7*x**4 - 25021396400*a**6*b**9*x**5 - 7733886160*a**4*b**11*x**6 
 - 686439600*a**2*b**13*x**7 - 7726160*b**15*x**8)/(42493880*sqrt(x)*x**13 
)