\(\int \frac {(a+b \sqrt {x})^{15}}{x^{15}} \, dx\) [78]

Optimal result

Integrand size = 15, antiderivative size = 211 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=-\frac {a^{15}}{14 x^{14}}-\frac {10 a^{14} b}{9 x^{27/2}}-\frac {105 a^{13} b^2}{13 x^{13}}-\frac {182 a^{12} b^3}{5 x^{25/2}}-\frac {455 a^{11} b^4}{4 x^{12}}-\frac {6006 a^{10} b^5}{23 x^{23/2}}-\frac {455 a^9 b^6}{x^{11}}-\frac {4290 a^8 b^7}{7 x^{21/2}}-\frac {1287 a^7 b^8}{2 x^{10}}-\frac {10010 a^6 b^9}{19 x^{19/2}}-\frac {1001 a^5 b^{10}}{3 x^9}-\frac {2730 a^4 b^{11}}{17 x^{17/2}}-\frac {455 a^3 b^{12}}{8 x^8}-\frac {14 a^2 b^{13}}{x^{15/2}}-\frac {15 a b^{14}}{7 x^7}-\frac {2 b^{15}}{13 x^{13/2}} \] Output:

-1/14*a^15/x^14-10/9*a^14*b/x^(27/2)-105/13*a^13*b^2/x^13-182/5*a^12*b^3/x 
^(25/2)-455/4*a^11*b^4/x^12-6006/23*a^10*b^5/x^(23/2)-455*a^9*b^6/x^11-429 
0/7*a^8*b^7/x^(21/2)-1287/2*a^7*b^8/x^10-10010/19*a^6*b^9/x^(19/2)-1001/3* 
a^5*b^10/x^9-2730/17*a^4*b^11/x^(17/2)-455/8*a^3*b^12/x^8-14*a^2*b^13/x^(1 
5/2)-15/7*a*b^14/x^7-2/13*b^15/x^(13/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=\frac {-17383860 a^{15}-270415600 a^{14} b \sqrt {x}-1965713400 a^{13} b^2 x-8858815056 a^{12} b^3 x^{3/2}-27683797050 a^{11} b^4 x^2-63552368880 a^{10} b^5 x^{5/2}-110735188200 a^9 b^6 x^3-149153518800 a^8 b^7 x^{7/2}-156611194740 a^7 b^8 x^4-128219691600 a^6 b^9 x^{9/2}-81205804680 a^5 b^{10} x^5-39083007600 a^4 b^{11} x^{11/2}-13841898525 a^3 b^{12} x^6-3407236560 a^2 b^{13} x^{13/2}-521515800 a b^{14} x^7-37442160 b^{15} x^{15/2}}{243374040 x^{14}} \] Input:

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

Output:

(-17383860*a^15 - 270415600*a^14*b*Sqrt[x] - 1965713400*a^13*b^2*x - 88588 
15056*a^12*b^3*x^(3/2) - 27683797050*a^11*b^4*x^2 - 63552368880*a^10*b^5*x 
^(5/2) - 110735188200*a^9*b^6*x^3 - 149153518800*a^8*b^7*x^(7/2) - 1566111 
94740*a^7*b^8*x^4 - 128219691600*a^6*b^9*x^(9/2) - 81205804680*a^5*b^10*x^ 
5 - 39083007600*a^4*b^11*x^(11/2) - 13841898525*a^3*b^12*x^6 - 3407236560* 
a^2*b^13*x^(13/2) - 521515800*a*b^14*x^7 - 37442160*b^15*x^(15/2))/(243374 
040*x^14)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 2009}

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.

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

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{29/2}}d\sqrt {x}\)

\(\Big \downarrow \) 53

\(\displaystyle 2 \int \left (\frac {a^{15}}{x^{29/2}}+\frac {15 b a^{14}}{x^{14}}+\frac {105 b^2 a^{13}}{x^{27/2}}+\frac {455 b^3 a^{12}}{x^{13}}+\frac {1365 b^4 a^{11}}{x^{25/2}}+\frac {3003 b^5 a^{10}}{x^{12}}+\frac {5005 b^6 a^9}{x^{23/2}}+\frac {6435 b^7 a^8}{x^{11}}+\frac {6435 b^8 a^7}{x^{21/2}}+\frac {5005 b^9 a^6}{x^{10}}+\frac {3003 b^{10} a^5}{x^{19/2}}+\frac {1365 b^{11} a^4}{x^9}+\frac {455 b^{12} a^3}{x^{17/2}}+\frac {105 b^{13} a^2}{x^8}+\frac {15 b^{14} a}{x^{15/2}}+\frac {b^{15}}{x^7}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {a^{15}}{28 x^{14}}-\frac {5 a^{14} b}{9 x^{27/2}}-\frac {105 a^{13} b^2}{26 x^{13}}-\frac {91 a^{12} b^3}{5 x^{25/2}}-\frac {455 a^{11} b^4}{8 x^{12}}-\frac {3003 a^{10} b^5}{23 x^{23/2}}-\frac {455 a^9 b^6}{2 x^{11}}-\frac {2145 a^8 b^7}{7 x^{21/2}}-\frac {1287 a^7 b^8}{4 x^{10}}-\frac {5005 a^6 b^9}{19 x^{19/2}}-\frac {1001 a^5 b^{10}}{6 x^9}-\frac {1365 a^4 b^{11}}{17 x^{17/2}}-\frac {455 a^3 b^{12}}{16 x^8}-\frac {7 a^2 b^{13}}{x^{15/2}}-\frac {15 a b^{14}}{14 x^7}-\frac {b^{15}}{13 x^{13/2}}\right )\)

Input:

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

Output:

2*(-1/28*a^15/x^14 - (5*a^14*b)/(9*x^(27/2)) - (105*a^13*b^2)/(26*x^13) - 
(91*a^12*b^3)/(5*x^(25/2)) - (455*a^11*b^4)/(8*x^12) - (3003*a^10*b^5)/(23 
*x^(23/2)) - (455*a^9*b^6)/(2*x^11) - (2145*a^8*b^7)/(7*x^(21/2)) - (1287* 
a^7*b^8)/(4*x^10) - (5005*a^6*b^9)/(19*x^(19/2)) - (1001*a^5*b^10)/(6*x^9) 
 - (1365*a^4*b^11)/(17*x^(17/2)) - (455*a^3*b^12)/(16*x^8) - (7*a^2*b^13)/ 
x^(15/2) - (15*a*b^14)/(14*x^7) - b^15/(13*x^(13/2)))
 

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

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]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 23.62 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {a^{15}}{14 x^{14}}-\frac {10 a^{14} b}{9 x^{\frac {27}{2}}}-\frac {105 a^{13} b^{2}}{13 x^{13}}-\frac {182 a^{12} b^{3}}{5 x^{\frac {25}{2}}}-\frac {455 a^{11} b^{4}}{4 x^{12}}-\frac {6006 a^{10} b^{5}}{23 x^{\frac {23}{2}}}-\frac {455 a^{9} b^{6}}{x^{11}}-\frac {4290 a^{8} b^{7}}{7 x^{\frac {21}{2}}}-\frac {1287 a^{7} b^{8}}{2 x^{10}}-\frac {10010 a^{6} b^{9}}{19 x^{\frac {19}{2}}}-\frac {1001 a^{5} b^{10}}{3 x^{9}}-\frac {2730 a^{4} b^{11}}{17 x^{\frac {17}{2}}}-\frac {455 a^{3} b^{12}}{8 x^{8}}-\frac {14 a^{2} b^{13}}{x^{\frac {15}{2}}}-\frac {15 a \,b^{14}}{7 x^{7}}-\frac {2 b^{15}}{13 x^{\frac {13}{2}}}\)	\(168\)
default	\(-\frac {a^{15}}{14 x^{14}}-\frac {10 a^{14} b}{9 x^{\frac {27}{2}}}-\frac {105 a^{13} b^{2}}{13 x^{13}}-\frac {182 a^{12} b^{3}}{5 x^{\frac {25}{2}}}-\frac {455 a^{11} b^{4}}{4 x^{12}}-\frac {6006 a^{10} b^{5}}{23 x^{\frac {23}{2}}}-\frac {455 a^{9} b^{6}}{x^{11}}-\frac {4290 a^{8} b^{7}}{7 x^{\frac {21}{2}}}-\frac {1287 a^{7} b^{8}}{2 x^{10}}-\frac {10010 a^{6} b^{9}}{19 x^{\frac {19}{2}}}-\frac {1001 a^{5} b^{10}}{3 x^{9}}-\frac {2730 a^{4} b^{11}}{17 x^{\frac {17}{2}}}-\frac {455 a^{3} b^{12}}{8 x^{8}}-\frac {14 a^{2} b^{13}}{x^{\frac {15}{2}}}-\frac {15 a \,b^{14}}{7 x^{7}}-\frac {2 b^{15}}{13 x^{\frac {13}{2}}}\)	\(168\)
orering	\(-\frac {\left (77558760 b^{28} x^{14}-943074405 a^{2} b^{26} x^{13}[phone] a^{4} b^{24} x^{12}-20214204465 a^{6} b^{22} x^{11}[phone] a^{8} b^{20} x^{10}-97731031125 a^{10} b^{18} x^{9}[phone] a^{12} b^{16} x^{8}-152621203889 a^{14} b^{14} x^{7}[phone] a^{16} b^{12} x^{6}-82928147302 a^{18} b^{10} x^{5}[phone] a^{20} b^{8} x^{4}-14256469994 a^{22} b^{6} x^{3}[phone] a^{24} b^{4} x^{2}-523982228 a^{26} b^{2} x [phone] a^{28}\right ) \left (a +b \sqrt {x}\right )^{15}}{243374040 x^{14} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (40116600 b^{28} x^{14}-456326325 a^{2} b^{26} x^{13}[phone] a^{4} b^{24} x^{12}-8663230485 a^{6} b^{22} x^{11}[phone] a^{8} b^{20} x^{10}-37588858125 a^{10} b^{18} x^{9}[phone] a^{12} b^{16} x^{8}-53239954845 a^{14} b^{14} x^{7}[phone] a^{16} b^{12} x^{6}-26466429990 a^{18} b^{10} x^{5}[phone] a^{20} b^{8} x^{4}-4193079410 a^{22} b^{6} x^{3}[phone] a^{24} b^{4} x^{2}-142904244 a^{26} b^{2} x +9657700 a^{28}\right ) x^{2} \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} b}{2 x^{\frac {31}{2}}}-\frac {15 \left (a +b \sqrt {x}\right )^{15}}{x^{16}}\right )}{1825305300 \left (-b^{2} x +a^{2}\right )^{14}}\)	\(385\)
trager	\(\frac {\left (-1+x \right ) \left (156 a^{14} x^{13}+17640 a^{12} b^{2} x^{13}+248430 a^{10} b^{4} x^{13}+993720 a^{8} b^{6} x^{13}+1405404 a^{6} b^{8} x^{13}+728728 a^{4} b^{10} x^{13}+124215 a^{2} b^{12} x^{13}+4680 b^{14} x^{13}+156 a^{14} x^{12}+17640 a^{12} b^{2} x^{12}+248430 a^{10} b^{4} x^{12}+993720 a^{8} b^{6} x^{12}+1405404 a^{6} b^{8} x^{12}+728728 a^{4} b^{10} x^{12}+124215 a^{2} b^{12} x^{12}+4680 b^{14} x^{12}+156 a^{14} x^{11}+17640 a^{12} b^{2} x^{11}+248430 a^{10} b^{4} x^{11}+993720 a^{8} b^{6} x^{11}+1405404 a^{6} b^{8} x^{11}+728728 a^{4} b^{10} x^{11}+124215 a^{2} b^{12} x^{11}+4680 b^{14} x^{11}+156 a^{14} x^{10}+17640 a^{12} b^{2} x^{10}+248430 a^{10} b^{4} x^{10}+993720 a^{8} b^{6} x^{10}+1405404 a^{6} b^{8} x^{10}+728728 a^{4} b^{10} x^{10}+124215 a^{2} b^{12} x^{10}+4680 b^{14} x^{10}+156 a^{14} x^{9}+17640 a^{12} b^{2} x^{9}+248430 a^{10} b^{4} x^{9}+993720 a^{8} b^{6} x^{9}+1405404 a^{6} b^{8} x^{9}+728728 a^{4} b^{10} x^{9}+124215 a^{2} b^{12} x^{9}+4680 b^{14} x^{9}+156 a^{14} x^{8}+17640 a^{12} b^{2} x^{8}+248430 a^{10} b^{4} x^{8}+993720 a^{8} b^{6} x^{8}+1405404 a^{6} b^{8} x^{8}+728728 a^{4} b^{10} x^{8}+124215 a^{2} b^{12} x^{8}+4680 b^{14} x^{8}+156 a^{14} x^{7}+17640 a^{12} b^{2} x^{7}+248430 a^{10} b^{4} x^{7}+993720 a^{8} b^{6} x^{7}+1405404 a^{6} b^{8} x^{7}+728728 b^{10} x^{7} a^{4}+124215 a^{2} b^{12} x^{7}+4680 x^{7} b^{14}+156 a^{14} x^{6}+17640 a^{12} b^{2} x^{6}+248430 a^{10} b^{4} x^{6}+993720 a^{8} b^{6} x^{6}+1405404 b^{8} x^{6} a^{6}+728728 a^{4} b^{10} x^{6}+124215 a^{2} b^{12} x^{6}+156 a^{14} x^{5}+17640 a^{12} b^{2} x^{5}+248430 a^{10} b^{4} x^{5}+993720 a^{8} b^{6} x^{5}+1405404 a^{6} b^{8} x^{5}+728728 a^{4} b^{10} x^{5}+156 a^{14} x^{4}+17640 a^{12} b^{2} x^{4}+248430 b^{4} x^{4} a^{10}+993720 x^{4} b^{6} a^{8}+1405404 a^{6} b^{8} x^{4}+156 a^{14} x^{3}+17640 a^{12} b^{2} x^{3}+248430 a^{10} b^{4} x^{3}+993720 a^{8} b^{6} x^{3}+156 a^{14} x^{2}+17640 a^{12} b^{2} x^{2}+248430 a^{10} b^{4} x^{2}+156 a^{14} x +17640 a^{12} b^{2} x +156 a^{14}\right ) a}{2184 x^{14}}-\frac {2 \left (2340135 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 [phone] a^{14}\right ) b}{30421755 x^{\frac {27}{2}}}\)	\(950\)

Input:

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

Output:

-1/14*a^15/x^14-10/9*a^14*b/x^(27/2)-105/13*a^13*b^2/x^13-182/5*a^12*b^3/x 
^(25/2)-455/4*a^11*b^4/x^12-6006/23*a^10*b^5/x^(23/2)-455*a^9*b^6/x^11-429 
0/7*a^8*b^7/x^(21/2)-1287/2*a^7*b^8/x^10-10010/19*a^6*b^9/x^(19/2)-1001/3* 
a^5*b^10/x^9-2730/17*a^4*b^11/x^(17/2)-455/8*a^3*b^12/x^8-14*a^2*b^13/x^(1 
5/2)-15/7*a*b^14/x^7-2/13*b^15/x^(13/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=-\frac {521515800 \, a b^{14} x^{7} + 13841898525 \, a^{3} b^{12} x^{6} + 81205804680 \, a^{5} b^{10} x^{5} + 156611194740 \, a^{7} b^{8} x^{4} + 110735188200 \, a^{9} b^{6} x^{3} + 27683797050 \, a^{11} b^{4} x^{2} + 1965713400 \, a^{13} b^{2} x + 17383860 \, a^{15} + 16 \, {\left (2340135 \, b^{15} x^{7} + 212952285 \, a^{2} b^{13} x^{6} + 2442687975 \, a^{4} b^{11} x^{5} + 8013730725 \, a^{6} b^{9} x^{4} + 9322094925 \, a^{8} b^{7} x^{3} + 3972023055 \, a^{10} b^{5} x^{2} + 553675941 \, a^{12} b^{3} x + 16900975 \, a^{14} b\right )} \sqrt {x}}{243374040 \, x^{14}} \] Input:

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

Output:

-1/243374040*(521515800*a*b^14*x^7 + 13841898525*a^3*b^12*x^6 + 8120580468 
0*a^5*b^10*x^5 + 156611194740*a^7*b^8*x^4 + 110735188200*a^9*b^6*x^3 + 276 
83797050*a^11*b^4*x^2 + 1965713400*a^13*b^2*x + 17383860*a^15 + 16*(234013 
5*b^15*x^7 + 212952285*a^2*b^13*x^6 + 2442687975*a^4*b^11*x^5 + 8013730725 
*a^6*b^9*x^4 + 9322094925*a^8*b^7*x^3 + 3972023055*a^10*b^5*x^2 + 55367594 
1*a^12*b^3*x + 16900975*a^14*b)*sqrt(x))/x^14
 

Sympy [A] (verification not implemented)

Time = 2.61 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=- \frac {a^{15}}{14 x^{14}} - \frac {10 a^{14} b}{9 x^{\frac {27}{2}}} - \frac {105 a^{13} b^{2}}{13 x^{13}} - \frac {182 a^{12} b^{3}}{5 x^{\frac {25}{2}}} - \frac {455 a^{11} b^{4}}{4 x^{12}} - \frac {6006 a^{10} b^{5}}{23 x^{\frac {23}{2}}} - \frac {455 a^{9} b^{6}}{x^{11}} - \frac {4290 a^{8} b^{7}}{7 x^{\frac {21}{2}}} - \frac {1287 a^{7} b^{8}}{2 x^{10}} - \frac {10010 a^{6} b^{9}}{19 x^{\frac {19}{2}}} - \frac {1001 a^{5} b^{10}}{3 x^{9}} - \frac {2730 a^{4} b^{11}}{17 x^{\frac {17}{2}}} - \frac {455 a^{3} b^{12}}{8 x^{8}} - \frac {14 a^{2} b^{13}}{x^{\frac {15}{2}}} - \frac {15 a b^{14}}{7 x^{7}} - \frac {2 b^{15}}{13 x^{\frac {13}{2}}} \] Input:

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

Output:

-a**15/(14*x**14) - 10*a**14*b/(9*x**(27/2)) - 105*a**13*b**2/(13*x**13) - 
 182*a**12*b**3/(5*x**(25/2)) - 455*a**11*b**4/(4*x**12) - 6006*a**10*b**5 
/(23*x**(23/2)) - 455*a**9*b**6/x**11 - 4290*a**8*b**7/(7*x**(21/2)) - 128 
7*a**7*b**8/(2*x**10) - 10010*a**6*b**9/(19*x**(19/2)) - 1001*a**5*b**10/( 
3*x**9) - 2730*a**4*b**11/(17*x**(17/2)) - 455*a**3*b**12/(8*x**8) - 14*a* 
*2*b**13/x**(15/2) - 15*a*b**14/(7*x**7) - 2*b**15/(13*x**(13/2))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=-\frac {37442160 \, b^{15} x^{\frac {15}{2}} + 521515800 \, a b^{14} x^{7} + 3407236560 \, a^{2} b^{13} x^{\frac {13}{2}} + 13841898525 \, a^{3} b^{12} x^{6} + 39083007600 \, a^{4} b^{11} x^{\frac {11}{2}} + 81205804680 \, a^{5} b^{10} x^{5} + 128219691600 \, a^{6} b^{9} x^{\frac {9}{2}} + 156611194740 \, a^{7} b^{8} x^{4} + 149153518800 \, a^{8} b^{7} x^{\frac {7}{2}} + 110735188200 \, a^{9} b^{6} x^{3} + 63552368880 \, a^{10} b^{5} x^{\frac {5}{2}} + 27683797050 \, a^{11} b^{4} x^{2} + 8858815056 \, a^{12} b^{3} x^{\frac {3}{2}} + 1965713400 \, a^{13} b^{2} x + 270415600 \, a^{14} b \sqrt {x} + 17383860 \, a^{15}}{243374040 \, x^{14}} \] Input:

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

Output:

-1/243374040*(37442160*b^15*x^(15/2) + 521515800*a*b^14*x^7 + 3407236560*a 
^2*b^13*x^(13/2) + 13841898525*a^3*b^12*x^6 + 39083007600*a^4*b^11*x^(11/2 
) + 81205804680*a^5*b^10*x^5 + 128219691600*a^6*b^9*x^(9/2) + 156611194740 
*a^7*b^8*x^4 + 149153518800*a^8*b^7*x^(7/2) + 110735188200*a^9*b^6*x^3 + 6 
3552368880*a^10*b^5*x^(5/2) + 27683797050*a^11*b^4*x^2 + 8858815056*a^12*b 
^3*x^(3/2) + 1965713400*a^13*b^2*x + 270415600*a^14*b*sqrt(x) + 17383860*a 
^15)/x^14
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=-\frac {37442160 \, b^{15} x^{\frac {15}{2}} + 521515800 \, a b^{14} x^{7} + 3407236560 \, a^{2} b^{13} x^{\frac {13}{2}} + 13841898525 \, a^{3} b^{12} x^{6} + 39083007600 \, a^{4} b^{11} x^{\frac {11}{2}} + 81205804680 \, a^{5} b^{10} x^{5} + 128219691600 \, a^{6} b^{9} x^{\frac {9}{2}} + 156611194740 \, a^{7} b^{8} x^{4} + 149153518800 \, a^{8} b^{7} x^{\frac {7}{2}} + 110735188200 \, a^{9} b^{6} x^{3} + 63552368880 \, a^{10} b^{5} x^{\frac {5}{2}} + 27683797050 \, a^{11} b^{4} x^{2} + 8858815056 \, a^{12} b^{3} x^{\frac {3}{2}} + 1965713400 \, a^{13} b^{2} x + 270415600 \, a^{14} b \sqrt {x} + 17383860 \, a^{15}}{243374040 \, x^{14}} \] Input:

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

Output:

-1/243374040*(37442160*b^15*x^(15/2) + 521515800*a*b^14*x^7 + 3407236560*a 
^2*b^13*x^(13/2) + 13841898525*a^3*b^12*x^6 + 39083007600*a^4*b^11*x^(11/2 
) + 81205804680*a^5*b^10*x^5 + 128219691600*a^6*b^9*x^(9/2) + 156611194740 
*a^7*b^8*x^4 + 149153518800*a^8*b^7*x^(7/2) + 110735188200*a^9*b^6*x^3 + 6 
3552368880*a^10*b^5*x^(5/2) + 27683797050*a^11*b^4*x^2 + 8858815056*a^12*b 
^3*x^(3/2) + 1965713400*a^13*b^2*x + 270415600*a^14*b*sqrt(x) + 17383860*a 
^15)/x^14
 

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=-\frac {\frac {a^{15}}{14}+\frac {2\,b^{15}\,x^{15/2}}{13}+\frac {105\,a^{13}\,b^2\,x}{13}+\frac {10\,a^{14}\,b\,\sqrt {x}}{9}+\frac {15\,a\,b^{14}\,x^7}{7}+\frac {455\,a^{11}\,b^4\,x^2}{4}+455\,a^9\,b^6\,x^3+\frac {1287\,a^7\,b^8\,x^4}{2}+\frac {1001\,a^5\,b^{10}\,x^5}{3}+\frac {182\,a^{12}\,b^3\,x^{3/2}}{5}+\frac {455\,a^3\,b^{12}\,x^6}{8}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{23}+\frac {4290\,a^8\,b^7\,x^{7/2}}{7}+\frac {10010\,a^6\,b^9\,x^{9/2}}{19}+\frac {2730\,a^4\,b^{11}\,x^{11/2}}{17}+14\,a^2\,b^{13}\,x^{13/2}}{x^{14}} \] Input:

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

Output:

-(a^15/14 + (2*b^15*x^(15/2))/13 + (105*a^13*b^2*x)/13 + (10*a^14*b*x^(1/2 
))/9 + (15*a*b^14*x^7)/7 + (455*a^11*b^4*x^2)/4 + 455*a^9*b^6*x^3 + (1287* 
a^7*b^8*x^4)/2 + (1001*a^5*b^10*x^5)/3 + (182*a^12*b^3*x^(3/2))/5 + (455*a 
^3*b^12*x^6)/8 + (6006*a^10*b^5*x^(5/2))/23 + (4290*a^8*b^7*x^(7/2))/7 + ( 
10010*a^6*b^9*x^(9/2))/19 + (2730*a^4*b^11*x^(11/2))/17 + 14*a^2*b^13*x^(1 
3/2))/x^14
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{15}} \, dx=\frac {-17383860 \sqrt {x}\, a^{15}-1965713400 \sqrt {x}\, a^{13} b^{2} x -27683797050 \sqrt {x}\, a^{11} b^{4} x^{2}-110735188200 \sqrt {x}\, a^{9} b^{6} x^{3}-156611194740 \sqrt {x}\, a^{7} b^{8} x^{4}-81205804680 \sqrt {x}\, a^{5} b^{10} x^{5}-13841898525 \sqrt {x}\, a^{3} b^{12} x^{6}-521515800 \sqrt {x}\, a \,b^{14} x^{7}-270415600 a^{14} b x -8858815056 a^{12} b^{3} x^{2}-63552368880 a^{10} b^{5} x^{3}-149153518800 a^{8} b^{7} x^{4}-128219691600 a^{6} b^{9} x^{5}-39083007600 a^{4} b^{11} x^{6}-3407236560 a^{2} b^{13} x^{7}-37442160 b^{15} x^{8}}{243374040 \sqrt {x}\, x^{14}} \] Input:

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

Output:

( - 17383860*sqrt(x)*a**15 - 1965713400*sqrt(x)*a**13*b**2*x - 27683797050 
*sqrt(x)*a**11*b**4*x**2 - 110735188200*sqrt(x)*a**9*b**6*x**3 - 156611194 
740*sqrt(x)*a**7*b**8*x**4 - 81205804680*sqrt(x)*a**5*b**10*x**5 - 1384189 
8525*sqrt(x)*a**3*b**12*x**6 - 521515800*sqrt(x)*a*b**14*x**7 - 270415600* 
a**14*b*x - 8858815056*a**12*b**3*x**2 - 63552368880*a**10*b**5*x**3 - 149 
153518800*a**8*b**7*x**4 - 128219691600*a**6*b**9*x**5 - 39083007600*a**4* 
b**11*x**6 - 3407236560*a**2*b**13*x**7 - 37442160*b**15*x**8)/(243374040* 
sqrt(x)*x**14)