\(\int (a+b \sqrt {x})^{15} \, dx\) [2174]

Optimal result

Integrand size = 11, antiderivative size = 38 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=-\frac {a \left (a+b \sqrt {x}\right )^{16}}{8 b^2}+\frac {2 \left (a+b \sqrt {x}\right )^{17}}{17 b^2} \]

[Out]

-1/8*a*(a+b*x^(1/2))^16/b^2+2/17*(a+b*x^(1/2))^17/b^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 45} \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {2 \left (a+b \sqrt {x}\right )^{17}}{17 b^2}-\frac {a \left (a+b \sqrt {x}\right )^{16}}{8 b^2} \]

[In]

Int[(a + b*Sqrt[x])^15,x]

[Out]

-1/8*(a*(a + b*Sqrt[x])^16)/b^2 + (2*(a + b*Sqrt[x])^17)/(17*b^2)

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 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x (a+b x)^{15} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {a (a+b x)^{15}}{b}+\frac {(a+b x)^{16}}{b}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a \left (a+b \sqrt {x}\right )^{16}}{8 b^2}+\frac {2 \left (a+b \sqrt {x}\right )^{17}}{17 b^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(185\) vs. \(2(38)=76\).

Time = 0.04 (sec) , antiderivative size = 185, normalized size of antiderivative = 4.87 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {1}{136} \left (136 a^{15} x+1360 a^{14} b x^{3/2}+7140 a^{13} b^2 x^2+24752 a^{12} b^3 x^{5/2}+61880 a^{11} b^4 x^3+116688 a^{10} b^5 x^{7/2}+170170 a^9 b^6 x^4+194480 a^8 b^7 x^{9/2}+175032 a^7 b^8 x^5+123760 a^6 b^9 x^{11/2}+68068 a^5 b^{10} x^6+28560 a^4 b^{11} x^{13/2}+8840 a^3 b^{12} x^7+1904 a^2 b^{13} x^{15/2}+255 a b^{14} x^8+16 b^{15} x^{17/2}\right ) \]

[In]

Integrate[(a + b*Sqrt[x])^15,x]

[Out]

(136*a^15*x + 1360*a^14*b*x^(3/2) + 7140*a^13*b^2*x^2 + 24752*a^12*b^3*x^(5/2) + 61880*a^11*b^4*x^3 + 116688*a
^10*b^5*x^(7/2) + 170170*a^9*b^6*x^4 + 194480*a^8*b^7*x^(9/2) + 175032*a^7*b^8*x^5 + 123760*a^6*b^9*x^(11/2) +
68068*a^5*b^10*x^6 + 28560*a^4*b^11*x^(13/2) + 8840*a^3*b^12*x^7 + 1904*a^2*b^13*x^(15/2) + 255*a*b^14*x^8 +
16*b^15*x^(17/2))/136

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(30)=60\).

Time = 3.41 (sec) , antiderivative size = 165, normalized size of antiderivative = 4.34

method	result	size
derivativedivides	\(\frac {2 x^{\frac {17}{2}} b^{15}}{17}+\frac {15 a \,b^{14} x^{8}}{8}+14 x^{\frac {15}{2}} a^{2} b^{13}+65 a^{3} b^{12} x^{7}+210 x^{\frac {13}{2}} a^{4} b^{11}+\frac {1001 x^{6} a^{5} b^{10}}{2}+910 a^{6} b^{9} x^{\frac {11}{2}}+1287 a^{7} b^{8} x^{5}+1430 x^{\frac {9}{2}} a^{8} b^{7}+\frac {5005 a^{9} b^{6} x^{4}}{4}+858 x^{\frac {7}{2}} a^{10} b^{5}+455 a^{11} b^{4} x^{3}+182 a^{12} b^{3} x^{\frac {5}{2}}+\frac {105 a^{13} b^{2} x^{2}}{2}+10 x^{\frac {3}{2}} a^{14} b +x \,a^{15}\)	\(165\)
default	\(\frac {2 x^{\frac {17}{2}} b^{15}}{17}+\frac {15 a \,b^{14} x^{8}}{8}+14 x^{\frac {15}{2}} a^{2} b^{13}+65 a^{3} b^{12} x^{7}+210 x^{\frac {13}{2}} a^{4} b^{11}+\frac {1001 x^{6} a^{5} b^{10}}{2}+910 a^{6} b^{9} x^{\frac {11}{2}}+1287 a^{7} b^{8} x^{5}+1430 x^{\frac {9}{2}} a^{8} b^{7}+\frac {5005 a^{9} b^{6} x^{4}}{4}+858 x^{\frac {7}{2}} a^{10} b^{5}+455 a^{11} b^{4} x^{3}+182 a^{12} b^{3} x^{\frac {5}{2}}+\frac {105 a^{13} b^{2} x^{2}}{2}+10 x^{\frac {3}{2}} a^{14} b +x \,a^{15}\)	\(165\)
trager	\(\frac {a \left (15 x^{7} b^{14}+520 a^{2} b^{12} x^{6}+15 b^{14} x^{6}+4004 a^{4} b^{10} x^{5}+520 a^{2} b^{12} x^{5}+15 b^{14} x^{5}+10296 a^{6} b^{8} x^{4}+4004 x^{4} a^{4} b^{10}+520 x^{4} a^{2} b^{12}+15 b^{14} x^{4}+10010 a^{8} b^{6} x^{3}+10296 a^{6} b^{8} x^{3}+4004 a^{4} b^{10} x^{3}+520 a^{2} b^{12} x^{3}+15 b^{14} x^{3}+3640 a^{10} b^{4} x^{2}+10010 a^{8} b^{6} x^{2}+10296 a^{6} b^{8} x^{2}+4004 a^{4} b^{10} x^{2}+520 a^{2} b^{12} x^{2}+15 b^{14} x^{2}+420 a^{12} b^{2} x +3640 a^{10} b^{4} x +10010 a^{8} b^{6} x +10296 a^{6} b^{8} x +4004 a^{4} b^{10} x +520 a^{2} b^{12} x +15 b^{14} x +8 a^{14}+420 a^{12} b^{2}+3640 a^{10} b^{4}+10010 a^{8} b^{6}+10296 a^{6} b^{8}+4004 a^{4} b^{10}+520 a^{2} b^{12}+15 b^{14}\right ) \left (-1+x \right )}{8}+\frac {2 b \,x^{\frac {3}{2}} \left (x^{7} b^{14}+119 a^{2} b^{12} x^{6}+1785 a^{4} b^{10} x^{5}+7735 a^{6} b^{8} x^{4}+12155 a^{8} b^{6} x^{3}+7293 a^{10} b^{4} x^{2}+1547 a^{12} b^{2} x +85 a^{14}\right )}{17}\)	\(423\)

[In]

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

[Out]

2/17*x^(17/2)*b^15+15/8*a*b^14*x^8+14*x^(15/2)*a^2*b^13+65*a^3*b^12*x^7+210*x^(13/2)*a^4*b^11+1001/2*x^6*a^5*b
^10+910*a^6*b^9*x^(11/2)+1287*a^7*b^8*x^5+1430*x^(9/2)*a^8*b^7+5005/4*a^9*b^6*x^4+858*x^(7/2)*a^10*b^5+455*a^1
1*b^4*x^3+182*a^12*b^3*x^(5/2)+105/2*a^13*b^2*x^2+10*x^(3/2)*a^14*b+x*a^15

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (30) = 60\).

Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.39 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {15}{8} \, a b^{14} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac {1001}{2} \, a^{5} b^{10} x^{6} + 1287 \, a^{7} b^{8} x^{5} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + 455 \, a^{11} b^{4} x^{3} + \frac {105}{2} \, a^{13} b^{2} x^{2} + a^{15} x + \frac {2}{17} \, {\left (b^{15} x^{8} + 119 \, a^{2} b^{13} x^{7} + 1785 \, a^{4} b^{11} x^{6} + 7735 \, a^{6} b^{9} x^{5} + 12155 \, a^{8} b^{7} x^{4} + 7293 \, a^{10} b^{5} x^{3} + 1547 \, a^{12} b^{3} x^{2} + 85 \, a^{14} b x\right )} \sqrt {x} \]

[In]

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

[Out]

15/8*a*b^14*x^8 + 65*a^3*b^12*x^7 + 1001/2*a^5*b^10*x^6 + 1287*a^7*b^8*x^5 + 5005/4*a^9*b^6*x^4 + 455*a^11*b^4
*x^3 + 105/2*a^13*b^2*x^2 + a^15*x + 2/17*(b^15*x^8 + 119*a^2*b^13*x^7 + 1785*a^4*b^11*x^6 + 7735*a^6*b^9*x^5
+ 12155*a^8*b^7*x^4 + 7293*a^10*b^5*x^3 + 1547*a^12*b^3*x^2 + 85*a^14*b*x)*sqrt(x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (32) = 64\).

Time = 0.65 (sec) , antiderivative size = 197, normalized size of antiderivative = 5.18 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=a^{15} x + 10 a^{14} b x^{\frac {3}{2}} + \frac {105 a^{13} b^{2} x^{2}}{2} + 182 a^{12} b^{3} x^{\frac {5}{2}} + 455 a^{11} b^{4} x^{3} + 858 a^{10} b^{5} x^{\frac {7}{2}} + \frac {5005 a^{9} b^{6} x^{4}}{4} + 1430 a^{8} b^{7} x^{\frac {9}{2}} + 1287 a^{7} b^{8} x^{5} + 910 a^{6} b^{9} x^{\frac {11}{2}} + \frac {1001 a^{5} b^{10} x^{6}}{2} + 210 a^{4} b^{11} x^{\frac {13}{2}} + 65 a^{3} b^{12} x^{7} + 14 a^{2} b^{13} x^{\frac {15}{2}} + \frac {15 a b^{14} x^{8}}{8} + \frac {2 b^{15} x^{\frac {17}{2}}}{17} \]

[In]

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

[Out]

a**15*x + 10*a**14*b*x**(3/2) + 105*a**13*b**2*x**2/2 + 182*a**12*b**3*x**(5/2) + 455*a**11*b**4*x**3 + 858*a*
*10*b**5*x**(7/2) + 5005*a**9*b**6*x**4/4 + 1430*a**8*b**7*x**(9/2) + 1287*a**7*b**8*x**5 + 910*a**6*b**9*x**(
11/2) + 1001*a**5*b**10*x**6/2 + 210*a**4*b**11*x**(13/2) + 65*a**3*b**12*x**7 + 14*a**2*b**13*x**(15/2) + 15*
a*b**14*x**8/8 + 2*b**15*x**(17/2)/17

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{17}}{17 \, b^{2}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a}{8 \, b^{2}} \]

[In]

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

[Out]

2/17*(b*sqrt(x) + a)^17/b^2 - 1/8*(b*sqrt(x) + a)^16*a/b^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (30) = 60\).

Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.32 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {2}{17} \, b^{15} x^{\frac {17}{2}} + \frac {15}{8} \, a b^{14} x^{8} + 14 \, a^{2} b^{13} x^{\frac {15}{2}} + 65 \, a^{3} b^{12} x^{7} + 210 \, a^{4} b^{11} x^{\frac {13}{2}} + \frac {1001}{2} \, a^{5} b^{10} x^{6} + 910 \, a^{6} b^{9} x^{\frac {11}{2}} + 1287 \, a^{7} b^{8} x^{5} + 1430 \, a^{8} b^{7} x^{\frac {9}{2}} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + 858 \, a^{10} b^{5} x^{\frac {7}{2}} + 455 \, a^{11} b^{4} x^{3} + 182 \, a^{12} b^{3} x^{\frac {5}{2}} + \frac {105}{2} \, a^{13} b^{2} x^{2} + 10 \, a^{14} b x^{\frac {3}{2}} + a^{15} x \]

[In]

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

[Out]

2/17*b^15*x^(17/2) + 15/8*a*b^14*x^8 + 14*a^2*b^13*x^(15/2) + 65*a^3*b^12*x^7 + 210*a^4*b^11*x^(13/2) + 1001/2
*a^5*b^10*x^6 + 910*a^6*b^9*x^(11/2) + 1287*a^7*b^8*x^5 + 1430*a^8*b^7*x^(9/2) + 5005/4*a^9*b^6*x^4 + 858*a^10
*b^5*x^(7/2) + 455*a^11*b^4*x^3 + 182*a^12*b^3*x^(5/2) + 105/2*a^13*b^2*x^2 + 10*a^14*b*x^(3/2) + a^15*x

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.32 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=a^{15}\,x+\frac {2\,b^{15}\,x^{17/2}}{17}+10\,a^{14}\,b\,x^{3/2}+\frac {15\,a\,b^{14}\,x^8}{8}+\frac {105\,a^{13}\,b^2\,x^2}{2}+455\,a^{11}\,b^4\,x^3+\frac {5005\,a^9\,b^6\,x^4}{4}+1287\,a^7\,b^8\,x^5+\frac {1001\,a^5\,b^{10}\,x^6}{2}+65\,a^3\,b^{12}\,x^7+182\,a^{12}\,b^3\,x^{5/2}+858\,a^{10}\,b^5\,x^{7/2}+1430\,a^8\,b^7\,x^{9/2}+910\,a^6\,b^9\,x^{11/2}+210\,a^4\,b^{11}\,x^{13/2}+14\,a^2\,b^{13}\,x^{15/2} \]

[In]

int((a + b*x^(1/2))^15,x)

[Out]

a^15*x + (2*b^15*x^(17/2))/17 + 10*a^14*b*x^(3/2) + (15*a*b^14*x^8)/8 + (105*a^13*b^2*x^2)/2 + 455*a^11*b^4*x^
3 + (5005*a^9*b^6*x^4)/4 + 1287*a^7*b^8*x^5 + (1001*a^5*b^10*x^6)/2 + 65*a^3*b^12*x^7 + 182*a^12*b^3*x^(5/2) +
858*a^10*b^5*x^(7/2) + 1430*a^8*b^7*x^(9/2) + 910*a^6*b^9*x^(11/2) + 210*a^4*b^11*x^(13/2) + 14*a^2*b^13*x^(1
5/2)