3.22.75 \(\int \frac {(a+b \sqrt {x})^{15}}{x} \, dx\) [2175]

3.22.75.1 Optimal result

Integrand size = 15, antiderivative size = 205 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=30 a^{14} b \sqrt {x}+105 a^{13} b^2 x+\frac {910}{3} a^{12} b^3 x^{3/2}+\frac {1365}{2} a^{11} b^4 x^2+\frac {6006}{5} a^{10} b^5 x^{5/2}+\frac {5005}{3} a^9 b^6 x^3+\frac {12870}{7} a^8 b^7 x^{7/2}+\frac {6435}{4} a^7 b^8 x^4+\frac {10010}{9} a^6 b^9 x^{9/2}+\frac {3003}{5} a^5 b^{10} x^5+\frac {2730}{11} a^4 b^{11} x^{11/2}+\frac {455}{6} a^3 b^{12} x^6+\frac {210}{13} a^2 b^{13} x^{13/2}+\frac {15}{7} a b^{14} x^7+\frac {2}{15} b^{15} x^{15/2}+a^{15} \log (x) \]

output

105*a^13*b^2*x+910/3*a^12*b^3*x^(3/2)+1365/2*a^11*b^4*x^2+6006/5*a^10*b^5* 
x^(5/2)+5005/3*a^9*b^6*x^3+12870/7*a^8*b^7*x^(7/2)+6435/4*a^7*b^8*x^4+1001 
0/9*a^6*b^9*x^(9/2)+3003/5*a^5*b^10*x^5+2730/11*a^4*b^11*x^(11/2)+455/6*a^ 
3*b^12*x^6+210/13*a^2*b^13*x^(13/2)+15/7*a*b^14*x^7+2/15*b^15*x^(15/2)+a^1 
5*ln(x)+30*a^14*b*x^(1/2)
 

3.22.75.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {5405400 a^{14} b \sqrt {x}[phone] a^{13} b^2 x+54654600 a^{12} b^3 x^{3/2}[phone] a^{11} b^4 x^2+216432216 a^{10} b^5 x^{5/2}[phone] a^9 b^6 x^3+331273800 a^8 b^7 x^{7/2}[phone] a^7 b^8 x^4+200400200 a^6 b^9 x^{9/2}[phone] a^5 b^{10} x^5+44717400 a^4 b^{11} x^{11/2}[phone] a^3 b^{12} x^6+2910600 a^2 b^{13} x^{13/2}+386100 a b^{14} x^7+24024 b^{15} x^{15/2}}{180180}+2 a^{15} \log \left (\sqrt {x}\right ) \]

input

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

output

(5405400*a^14*b*Sqrt[x] + 18918900*a^13*b^2*x + 54654600*a^12*b^3*x^(3/2) 
+ 122972850*a^11*b^4*x^2 + 216432216*a^10*b^5*x^(5/2) + 300600300*a^9*b^6* 
x^3 + 331273800*a^8*b^7*x^(7/2) + 289864575*a^7*b^8*x^4 + 200400200*a^6*b^ 
9*x^(9/2) + 108216108*a^5*b^10*x^5 + 44717400*a^4*b^11*x^(11/2) + 13663650 
*a^3*b^12*x^6 + 2910600*a^2*b^13*x^(13/2) + 386100*a*b^14*x^7 + 24024*b^15 
*x^(15/2))/180180 + 2*a^15*Log[Sqrt[x]]
 

3.22.75.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 49, 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.

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

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (a^{15} \log \left (\sqrt {x}\right )+15 a^{14} b \sqrt {x}+\frac {105}{2} a^{13} b^2 x+\frac {455}{3} a^{12} b^3 x^{3/2}+\frac {1365}{4} a^{11} b^4 x^2+\frac {3003}{5} a^{10} b^5 x^{5/2}+\frac {5005}{6} a^9 b^6 x^3+\frac {6435}{7} a^8 b^7 x^{7/2}+\frac {6435}{8} a^7 b^8 x^4+\frac {5005}{9} a^6 b^9 x^{9/2}+\frac {3003}{10} a^5 b^{10} x^5+\frac {1365}{11} a^4 b^{11} x^{11/2}+\frac {455}{12} a^3 b^{12} x^6+\frac {105}{13} a^2 b^{13} x^{13/2}+\frac {15}{14} a b^{14} x^7+\frac {1}{15} b^{15} x^{15/2}\right )\)

input

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

output

2*(15*a^14*b*Sqrt[x] + (105*a^13*b^2*x)/2 + (455*a^12*b^3*x^(3/2))/3 + (13 
65*a^11*b^4*x^2)/4 + (3003*a^10*b^5*x^(5/2))/5 + (5005*a^9*b^6*x^3)/6 + (6 
435*a^8*b^7*x^(7/2))/7 + (6435*a^7*b^8*x^4)/8 + (5005*a^6*b^9*x^(9/2))/9 + 
 (3003*a^5*b^10*x^5)/10 + (1365*a^4*b^11*x^(11/2))/11 + (455*a^3*b^12*x^6) 
/12 + (105*a^2*b^13*x^(13/2))/13 + (15*a*b^14*x^7)/14 + (b^15*x^(15/2))/15 
 + a^15*Log[Sqrt[x]])
 

3.22.75.3.1 Defintions of rubi rules used

rule 49

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}, x] 
&& IGtQ[m, 0] && IGtQ[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]
 

3.22.75.4 Maple [A] (verified)

Time = 3.54 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(105 a^{13} b^{2} x +\frac {910 a^{12} b^{3} x^{\frac {3}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{2}}{2}+\frac {6006 a^{10} b^{5} x^{\frac {5}{2}}}{5}+\frac {5005 a^{9} b^{6} x^{3}}{3}+\frac {12870 a^{8} b^{7} x^{\frac {7}{2}}}{7}+\frac {6435 a^{7} b^{8} x^{4}}{4}+\frac {10010 a^{6} b^{9} x^{\frac {9}{2}}}{9}+\frac {3003 a^{5} b^{10} x^{5}}{5}+\frac {2730 a^{4} b^{11} x^{\frac {11}{2}}}{11}+\frac {455 a^{3} b^{12} x^{6}}{6}+\frac {210 a^{2} b^{13} x^{\frac {13}{2}}}{13}+\frac {15 a \,b^{14} x^{7}}{7}+\frac {2 b^{15} x^{\frac {15}{2}}}{15}+a^{15} \ln \left (x \right )+30 a^{14} b \sqrt {x}\)	\(164\)
default	\(105 a^{13} b^{2} x +\frac {910 a^{12} b^{3} x^{\frac {3}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{2}}{2}+\frac {6006 a^{10} b^{5} x^{\frac {5}{2}}}{5}+\frac {5005 a^{9} b^{6} x^{3}}{3}+\frac {12870 a^{8} b^{7} x^{\frac {7}{2}}}{7}+\frac {6435 a^{7} b^{8} x^{4}}{4}+\frac {10010 a^{6} b^{9} x^{\frac {9}{2}}}{9}+\frac {3003 a^{5} b^{10} x^{5}}{5}+\frac {2730 a^{4} b^{11} x^{\frac {11}{2}}}{11}+\frac {455 a^{3} b^{12} x^{6}}{6}+\frac {210 a^{2} b^{13} x^{\frac {13}{2}}}{13}+\frac {15 a \,b^{14} x^{7}}{7}+\frac {2 b^{15} x^{\frac {15}{2}}}{15}+a^{15} \ln \left (x \right )+30 a^{14} b \sqrt {x}\)	\(164\)
trager	\(\frac {a \,b^{2} \left (900 b^{12} x^{6}+31850 a^{2} b^{10} x^{5}+900 b^{12} x^{5}+252252 a^{4} b^{8} x^{4}+31850 a^{2} b^{10} x^{4}+900 b^{12} x^{4}+675675 a^{6} b^{6} x^{3}+252252 a^{4} b^{8} x^{3}+31850 a^{2} b^{10} x^{3}+900 b^{12} x^{3}+700700 a^{8} b^{4} x^{2}+675675 a^{6} b^{6} x^{2}+252252 a^{4} b^{8} x^{2}+31850 a^{2} b^{10} x^{2}+900 b^{12} x^{2}+286650 a^{10} b^{2} x +700700 a^{8} b^{4} x +675675 a^{6} b^{6} x +252252 a^{4} b^{8} x +31850 a^{2} b^{10} x +900 b^{12} x +44100 a^{12}+286650 a^{10} b^{2}+700700 a^{8} b^{4}+675675 a^{6} b^{6}+252252 a^{4} b^{8}+31850 a^{2} b^{10}+900 b^{12}\right ) \left (-1+x \right )}{420}+\frac {2 b \left (3003 x^{7} b^{14}+363825 a^{2} b^{12} x^{6}+5589675 a^{4} b^{10} x^{5}[phone] a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}[phone] a^{10} b^{4} x^{2}+6831825 a^{12} b^{2} x +675675 a^{14}\right ) \sqrt {x}}{45045}+a^{15} \ln \left (x \right )\)	\(353\)

input

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

output

105*a^13*b^2*x+910/3*a^12*b^3*x^(3/2)+1365/2*a^11*b^4*x^2+6006/5*a^10*b^5* 
x^(5/2)+5005/3*a^9*b^6*x^3+12870/7*a^8*b^7*x^(7/2)+6435/4*a^7*b^8*x^4+1001 
0/9*a^6*b^9*x^(9/2)+3003/5*a^5*b^10*x^5+2730/11*a^4*b^11*x^(11/2)+455/6*a^ 
3*b^12*x^6+210/13*a^2*b^13*x^(13/2)+15/7*a*b^14*x^7+2/15*b^15*x^(15/2)+a^1 
5*ln(x)+30*a^14*b*x^(1/2)
 

3.22.75.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {15}{7} \, a b^{14} x^{7} + \frac {455}{6} \, a^{3} b^{12} x^{6} + \frac {3003}{5} \, a^{5} b^{10} x^{5} + \frac {6435}{4} \, a^{7} b^{8} x^{4} + \frac {5005}{3} \, a^{9} b^{6} x^{3} + \frac {1365}{2} \, a^{11} b^{4} x^{2} + 105 \, a^{13} b^{2} x + 2 \, a^{15} \log \left (\sqrt {x}\right ) + \frac {2}{45045} \, {\left (3003 \, b^{15} x^{7} + 363825 \, a^{2} b^{13} x^{6} + 5589675 \, a^{4} b^{11} x^{5} + 25050025 \, a^{6} b^{9} x^{4} + 41409225 \, a^{8} b^{7} x^{3} + 27054027 \, a^{10} b^{5} x^{2} + 6831825 \, a^{12} b^{3} x + 675675 \, a^{14} b\right )} \sqrt {x} \]

input

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

output

15/7*a*b^14*x^7 + 455/6*a^3*b^12*x^6 + 3003/5*a^5*b^10*x^5 + 6435/4*a^7*b^ 
8*x^4 + 5005/3*a^9*b^6*x^3 + 1365/2*a^11*b^4*x^2 + 105*a^13*b^2*x + 2*a^15 
*log(sqrt(x)) + 2/45045*(3003*b^15*x^7 + 363825*a^2*b^13*x^6 + 5589675*a^4 
*b^11*x^5 + 25050025*a^6*b^9*x^4 + 41409225*a^8*b^7*x^3 + 27054027*a^10*b^ 
5*x^2 + 6831825*a^12*b^3*x + 675675*a^14*b)*sqrt(x)
 

3.22.75.6 Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=a^{15} \log {\left (x \right )} + 30 a^{14} b \sqrt {x} + 105 a^{13} b^{2} x + \frac {910 a^{12} b^{3} x^{\frac {3}{2}}}{3} + \frac {1365 a^{11} b^{4} x^{2}}{2} + \frac {6006 a^{10} b^{5} x^{\frac {5}{2}}}{5} + \frac {5005 a^{9} b^{6} x^{3}}{3} + \frac {12870 a^{8} b^{7} x^{\frac {7}{2}}}{7} + \frac {6435 a^{7} b^{8} x^{4}}{4} + \frac {10010 a^{6} b^{9} x^{\frac {9}{2}}}{9} + \frac {3003 a^{5} b^{10} x^{5}}{5} + \frac {2730 a^{4} b^{11} x^{\frac {11}{2}}}{11} + \frac {455 a^{3} b^{12} x^{6}}{6} + \frac {210 a^{2} b^{13} x^{\frac {13}{2}}}{13} + \frac {15 a b^{14} x^{7}}{7} + \frac {2 b^{15} x^{\frac {15}{2}}}{15} \]

input

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

output

a**15*log(x) + 30*a**14*b*sqrt(x) + 105*a**13*b**2*x + 910*a**12*b**3*x**( 
3/2)/3 + 1365*a**11*b**4*x**2/2 + 6006*a**10*b**5*x**(5/2)/5 + 5005*a**9*b 
**6*x**3/3 + 12870*a**8*b**7*x**(7/2)/7 + 6435*a**7*b**8*x**4/4 + 10010*a* 
*6*b**9*x**(9/2)/9 + 3003*a**5*b**10*x**5/5 + 2730*a**4*b**11*x**(11/2)/11 
 + 455*a**3*b**12*x**6/6 + 210*a**2*b**13*x**(13/2)/13 + 15*a*b**14*x**7/7 
 + 2*b**15*x**(15/2)/15
 

3.22.75.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {2}{15} \, b^{15} x^{\frac {15}{2}} + \frac {15}{7} \, a b^{14} x^{7} + \frac {210}{13} \, a^{2} b^{13} x^{\frac {13}{2}} + \frac {455}{6} \, a^{3} b^{12} x^{6} + \frac {2730}{11} \, a^{4} b^{11} x^{\frac {11}{2}} + \frac {3003}{5} \, a^{5} b^{10} x^{5} + \frac {10010}{9} \, a^{6} b^{9} x^{\frac {9}{2}} + \frac {6435}{4} \, a^{7} b^{8} x^{4} + \frac {12870}{7} \, a^{8} b^{7} x^{\frac {7}{2}} + \frac {5005}{3} \, a^{9} b^{6} x^{3} + \frac {6006}{5} \, a^{10} b^{5} x^{\frac {5}{2}} + \frac {1365}{2} \, a^{11} b^{4} x^{2} + \frac {910}{3} \, a^{12} b^{3} x^{\frac {3}{2}} + 105 \, a^{13} b^{2} x + a^{15} \log \left (x\right ) + 30 \, a^{14} b \sqrt {x} \]

input

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

output

2/15*b^15*x^(15/2) + 15/7*a*b^14*x^7 + 210/13*a^2*b^13*x^(13/2) + 455/6*a^ 
3*b^12*x^6 + 2730/11*a^4*b^11*x^(11/2) + 3003/5*a^5*b^10*x^5 + 10010/9*a^6 
*b^9*x^(9/2) + 6435/4*a^7*b^8*x^4 + 12870/7*a^8*b^7*x^(7/2) + 5005/3*a^9*b 
^6*x^3 + 6006/5*a^10*b^5*x^(5/2) + 1365/2*a^11*b^4*x^2 + 910/3*a^12*b^3*x^ 
(3/2) + 105*a^13*b^2*x + a^15*log(x) + 30*a^14*b*sqrt(x)
 

3.22.75.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=\frac {2}{15} \, b^{15} x^{\frac {15}{2}} + \frac {15}{7} \, a b^{14} x^{7} + \frac {210}{13} \, a^{2} b^{13} x^{\frac {13}{2}} + \frac {455}{6} \, a^{3} b^{12} x^{6} + \frac {2730}{11} \, a^{4} b^{11} x^{\frac {11}{2}} + \frac {3003}{5} \, a^{5} b^{10} x^{5} + \frac {10010}{9} \, a^{6} b^{9} x^{\frac {9}{2}} + \frac {6435}{4} \, a^{7} b^{8} x^{4} + \frac {12870}{7} \, a^{8} b^{7} x^{\frac {7}{2}} + \frac {5005}{3} \, a^{9} b^{6} x^{3} + \frac {6006}{5} \, a^{10} b^{5} x^{\frac {5}{2}} + \frac {1365}{2} \, a^{11} b^{4} x^{2} + \frac {910}{3} \, a^{12} b^{3} x^{\frac {3}{2}} + 105 \, a^{13} b^{2} x + a^{15} \log \left ({\left | x \right |}\right ) + 30 \, a^{14} b \sqrt {x} \]

input

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

output

2/15*b^15*x^(15/2) + 15/7*a*b^14*x^7 + 210/13*a^2*b^13*x^(13/2) + 455/6*a^ 
3*b^12*x^6 + 2730/11*a^4*b^11*x^(11/2) + 3003/5*a^5*b^10*x^5 + 10010/9*a^6 
*b^9*x^(9/2) + 6435/4*a^7*b^8*x^4 + 12870/7*a^8*b^7*x^(7/2) + 5005/3*a^9*b 
^6*x^3 + 6006/5*a^10*b^5*x^(5/2) + 1365/2*a^11*b^4*x^2 + 910/3*a^12*b^3*x^ 
(3/2) + 105*a^13*b^2*x + a^15*log(abs(x)) + 30*a^14*b*sqrt(x)
 

3.22.75.9 Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=2\,a^{15}\,\ln \left (\sqrt {x}\right )+\frac {2\,b^{15}\,x^{15/2}}{15}+105\,a^{13}\,b^2\,x+30\,a^{14}\,b\,\sqrt {x}+\frac {15\,a\,b^{14}\,x^7}{7}+\frac {1365\,a^{11}\,b^4\,x^2}{2}+\frac {5005\,a^9\,b^6\,x^3}{3}+\frac {6435\,a^7\,b^8\,x^4}{4}+\frac {3003\,a^5\,b^{10}\,x^5}{5}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{3}+\frac {455\,a^3\,b^{12}\,x^6}{6}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{5}+\frac {12870\,a^8\,b^7\,x^{7/2}}{7}+\frac {10010\,a^6\,b^9\,x^{9/2}}{9}+\frac {2730\,a^4\,b^{11}\,x^{11/2}}{11}+\frac {210\,a^2\,b^{13}\,x^{13/2}}{13} \]

input

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

output

2*a^15*log(x^(1/2)) + (2*b^15*x^(15/2))/15 + 105*a^13*b^2*x + 30*a^14*b*x^ 
(1/2) + (15*a*b^14*x^7)/7 + (1365*a^11*b^4*x^2)/2 + (5005*a^9*b^6*x^3)/3 + 
 (6435*a^7*b^8*x^4)/4 + (3003*a^5*b^10*x^5)/5 + (910*a^12*b^3*x^(3/2))/3 + 
 (455*a^3*b^12*x^6)/6 + (6006*a^10*b^5*x^(5/2))/5 + (12870*a^8*b^7*x^(7/2) 
)/7 + (10010*a^6*b^9*x^(9/2))/9 + (2730*a^4*b^11*x^(11/2))/11 + (210*a^2*b 
^13*x^(13/2))/13