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

Optimal result

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

[Out]

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+10010/9*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^15*ln(x)+30*a^14*b*x^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x} \, dx=a^{15} \log (x)+30 a^{14} b \sqrt {x}+105 a^{13} b^2 x+\frac {910}{3} a^{12} b^3 x^{3/2}+\frac {1365}{2} a^{11} b^4 x^2+\frac {6006}{5} a^{10} b^5 x^{5/2}+\frac {5005}{3} a^9 b^6 x^3+\frac {12870}{7} a^8 b^7 x^{7/2}+\frac {6435}{4} a^7 b^8 x^4+\frac {10010}{9} a^6 b^9 x^{9/2}+\frac {3003}{5} a^5 b^{10} x^5+\frac {2730}{11} a^4 b^{11} x^{11/2}+\frac {455}{6} a^3 b^{12} x^6+\frac {210}{13} a^2 b^{13} x^{13/2}+\frac {15}{7} a b^{14} x^7+\frac {2}{15} b^{15} x^{15/2} \]

[In]

Int[(a + b*Sqrt[x])^15/x,x]

[Out]

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 + (3
003*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 + a^15*Log[x]

Rule 45

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] && NeQ[b*c - a*d, 0] && 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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (15 a^{14} b+\frac {a^{15}}{x}+105 a^{13} b^2 x+455 a^{12} b^3 x^2+1365 a^{11} b^4 x^3+3003 a^{10} b^5 x^4+5005 a^9 b^6 x^5+6435 a^8 b^7 x^6+6435 a^7 b^8 x^7+5005 a^6 b^9 x^8+3003 a^5 b^{10} x^9+1365 a^4 b^{11} x^{10}+455 a^3 b^{12} x^{11}+105 a^2 b^{13} x^{12}+15 a b^{14} x^{13}+b^{15} x^{14}\right ) \, dx,x,\sqrt {x}\right ) \\ & = 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) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[(a + b*Sqrt[x])^15/x,x]

[Out]

(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]]

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	\(\frac {a \,b^{2} \left (900 b^{12} x^{6}+31850 a^{2} b^{10} x^{5}+900 b^{12} x^{5}+252252 a^{4} b^{8} x^{4}+31850 a^{2} b^{10} x^{4}+900 b^{12} x^{4}+675675 a^{6} b^{6} x^{3}+252252 a^{4} b^{8} x^{3}+31850 a^{2} b^{10} x^{3}+900 b^{12} x^{3}+700700 a^{8} b^{4} x^{2}+675675 a^{6} b^{6} x^{2}+252252 a^{4} b^{8} x^{2}+31850 a^{2} b^{10} x^{2}+900 b^{12} x^{2}+286650 a^{10} b^{2} x +700700 a^{8} b^{4} x +675675 a^{6} b^{6} x +252252 a^{4} b^{8} x +31850 a^{2} b^{10} x +900 b^{12} x +44100 a^{12}+286650 a^{10} b^{2}+700700 a^{8} b^{4}+675675 a^{6} b^{6}+252252 a^{4} b^{8}+31850 a^{2} b^{10}+900 b^{12}\right ) \left (-1+x \right )}{420}+\frac {2 b \left (3003 x^{7} b^{14}+363825 a^{2} b^{12} x^{6}+5589675 a^{4} b^{10} x^{5}[phone] a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}[phone] a^{10} b^{4} x^{2}+6831825 a^{12} b^{2} x +675675 a^{14}\right ) \sqrt {x}}{45045}+a^{15} \ln \left (x \right )\)	\(353\)

[In]

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

[Out]

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+10010/9*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^15*ln(x)+30*a^14*b*x^(1/2)

Fricas [A] (verification not implemented)

none

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

[In]

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

[Out]

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 + 67567
5*a^14*b)*sqrt(x)

Sympy [A] (verification not implemented)

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

[In]

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

[Out]

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 + 60
06*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

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

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)

Giac [A] (verification not implemented)

none

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

[In]

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

[Out]

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)

Mupad [B] (verification not implemented)

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

[In]

int((a + b*x^(1/2))^15/x,x)

[Out]

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