\(\int (a+b \sqrt {x})^{15} x^2 \, dx\) [2172]

Optimal result

Integrand size = 15, antiderivative size = 122 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=-\frac {a^5 \left (a+b \sqrt {x}\right )^{16}}{8 b^6}+\frac {10 a^4 \left (a+b \sqrt {x}\right )^{17}}{17 b^6}-\frac {10 a^3 \left (a+b \sqrt {x}\right )^{18}}{9 b^6}+\frac {20 a^2 \left (a+b \sqrt {x}\right )^{19}}{19 b^6}-\frac {a \left (a+b \sqrt {x}\right )^{20}}{2 b^6}+\frac {2 \left (a+b \sqrt {x}\right )^{21}}{21 b^6} \]

[Out]

-1/8*a^5*(a+b*x^(1/2))^16/b^6+10/17*a^4*(a+b*x^(1/2))^17/b^6-10/9*a^3*(a+b*x^(1/2))^18/b^6+20/19*a^2*(a+b*x^(1
/2))^19/b^6-1/2*a*(a+b*x^(1/2))^20/b^6+2/21*(a+b*x^(1/2))^21/b^6

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 122, 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} \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=-\frac {a^5 \left (a+b \sqrt {x}\right )^{16}}{8 b^6}+\frac {10 a^4 \left (a+b \sqrt {x}\right )^{17}}{17 b^6}-\frac {10 a^3 \left (a+b \sqrt {x}\right )^{18}}{9 b^6}+\frac {20 a^2 \left (a+b \sqrt {x}\right )^{19}}{19 b^6}+\frac {2 \left (a+b \sqrt {x}\right )^{21}}{21 b^6}-\frac {a \left (a+b \sqrt {x}\right )^{20}}{2 b^6} \]

[In]

Int[(a + b*Sqrt[x])^15*x^2,x]

[Out]

-1/8*(a^5*(a + b*Sqrt[x])^16)/b^6 + (10*a^4*(a + b*Sqrt[x])^17)/(17*b^6) - (10*a^3*(a + b*Sqrt[x])^18)/(9*b^6)
+ (20*a^2*(a + b*Sqrt[x])^19)/(19*b^6) - (a*(a + b*Sqrt[x])^20)/(2*b^6) + (2*(a + b*Sqrt[x])^21)/(21*b^6)

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 x^5 (a+b x)^{15} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {a^5 (a+b x)^{15}}{b^5}+\frac {5 a^4 (a+b x)^{16}}{b^5}-\frac {10 a^3 (a+b x)^{17}}{b^5}+\frac {10 a^2 (a+b x)^{18}}{b^5}-\frac {5 a (a+b x)^{19}}{b^5}+\frac {(a+b x)^{20}}{b^5}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^5 \left (a+b \sqrt {x}\right )^{16}}{8 b^6}+\frac {10 a^4 \left (a+b \sqrt {x}\right )^{17}}{17 b^6}-\frac {10 a^3 \left (a+b \sqrt {x}\right )^{18}}{9 b^6}+\frac {20 a^2 \left (a+b \sqrt {x}\right )^{19}}{19 b^6}-\frac {a \left (a+b \sqrt {x}\right )^{20}}{2 b^6}+\frac {2 \left (a+b \sqrt {x}\right )^{21}}{21 b^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.53 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {54264 a^{15} x^3+697680 a^{14} b x^{7/2}+4273290 a^{13} b^2 x^4+16460080 a^{12} b^3 x^{9/2}[phone] a^{11} b^4 x^5+88884432 a^{10} b^5 x^{11/2}[phone] a^9 b^6 x^6+161164080 a^8 b^7 x^{13/2}[phone] a^7 b^8 x^7+108636528 a^6 b^9 x^{15/2}[phone] a^5 b^{10} x^8+26142480 a^4 b^{11} x^{17/2}+8230040 a^3 b^{12} x^9+1799280 a^2 b^{13} x^{19/2}+244188 a b^{14} x^{10}+15504 b^{15} x^{21/2}}{162792} \]

[In]

Integrate[(a + b*Sqrt[x])^15*x^2,x]

[Out]

(54264*a^15*x^3 + 697680*a^14*b*x^(7/2) + 4273290*a^13*b^2*x^4 + 16460080*a^12*b^3*x^(9/2) + 44442216*a^11*b^4
*x^5 + 88884432*a^10*b^5*x^(11/2) + 135795660*a^9*b^6*x^6 + 161164080*a^8*b^7*x^(13/2) + 149652360*a^7*b^8*x^7
+ 108636528*a^6*b^9*x^(15/2) + 61108047*a^5*b^10*x^8 + 26142480*a^4*b^11*x^(17/2) + 8230040*a^3*b^12*x^9 + 17
99280*a^2*b^13*x^(19/2) + 244188*a*b^14*x^10 + 15504*b^15*x^(21/2))/162792

Maple [A] (verified)

Time = 3.52 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(\frac {2 b^{15} x^{\frac {21}{2}}}{21}+\frac {3 x^{10} a \,b^{14}}{2}+\frac {210 x^{\frac {19}{2}} a^{2} b^{13}}{19}+\frac {455 a^{3} b^{12} x^{9}}{9}+\frac {2730 x^{\frac {17}{2}} a^{4} b^{11}}{17}+\frac {3003 a^{5} b^{10} x^{8}}{8}+\frac {2002 a^{6} b^{9} x^{\frac {15}{2}}}{3}+\frac {6435 a^{7} b^{8} x^{7}}{7}+990 x^{\frac {13}{2}} a^{8} b^{7}+\frac {5005 a^{9} b^{6} x^{6}}{6}+546 x^{\frac {11}{2}} a^{10} b^{5}+273 x^{5} a^{11} b^{4}+\frac {910 x^{\frac {9}{2}} a^{12} b^{3}}{9}+\frac {105 x^{4} a^{13} b^{2}}{4}+\frac {30 x^{\frac {7}{2}} a^{14} b}{7}+\frac {x^{3} a^{15}}{3}\)	\(168\)
default	\(\frac {2 b^{15} x^{\frac {21}{2}}}{21}+\frac {3 x^{10} a \,b^{14}}{2}+\frac {210 x^{\frac {19}{2}} a^{2} b^{13}}{19}+\frac {455 a^{3} b^{12} x^{9}}{9}+\frac {2730 x^{\frac {17}{2}} a^{4} b^{11}}{17}+\frac {3003 a^{5} b^{10} x^{8}}{8}+\frac {2002 a^{6} b^{9} x^{\frac {15}{2}}}{3}+\frac {6435 a^{7} b^{8} x^{7}}{7}+990 x^{\frac {13}{2}} a^{8} b^{7}+\frac {5005 a^{9} b^{6} x^{6}}{6}+546 x^{\frac {11}{2}} a^{10} b^{5}+273 x^{5} a^{11} b^{4}+\frac {910 x^{\frac {9}{2}} a^{12} b^{3}}{9}+\frac {105 x^{4} a^{13} b^{2}}{4}+\frac {30 x^{\frac {7}{2}} a^{14} b}{7}+\frac {x^{3} a^{15}}{3}\)	\(168\)
trager	\(\frac {a \left (756 b^{14} x^{9}+25480 a^{2} b^{12} x^{8}+756 b^{14} x^{8}+189189 a^{4} b^{10} x^{7}+25480 a^{2} b^{12} x^{7}+756 x^{7} b^{14}+463320 a^{6} b^{8} x^{6}+189189 a^{4} b^{10} x^{6}+25480 a^{2} b^{12} x^{6}+756 b^{14} x^{6}+420420 a^{8} b^{6} x^{5}+463320 a^{6} b^{8} x^{5}+189189 a^{4} b^{10} x^{5}+25480 a^{2} b^{12} x^{5}+756 b^{14} x^{5}+137592 x^{4} a^{10} b^{4}+420420 a^{8} b^{6} x^{4}+463320 a^{6} b^{8} x^{4}+189189 x^{4} a^{4} b^{10}+25480 x^{4} a^{2} b^{12}+756 b^{14} x^{4}+13230 a^{12} b^{2} x^{3}+137592 a^{10} b^{4} x^{3}+420420 a^{8} b^{6} x^{3}+463320 a^{6} b^{8} x^{3}+189189 a^{4} b^{10} x^{3}+25480 a^{2} b^{12} x^{3}+756 b^{14} x^{3}+168 a^{14} x^{2}+13230 a^{12} b^{2} x^{2}+137592 a^{10} b^{4} x^{2}+420420 a^{8} b^{6} x^{2}+463320 a^{6} b^{8} x^{2}+189189 a^{4} b^{10} x^{2}+25480 a^{2} b^{12} x^{2}+756 b^{14} x^{2}+168 a^{14} x +13230 a^{12} b^{2} x +137592 a^{10} b^{4} x +420420 a^{8} b^{6} x +463320 a^{6} b^{8} x +189189 a^{4} b^{10} x +25480 a^{2} b^{12} x +756 b^{14} x +168 a^{14}+13230 a^{12} b^{2}+137592 a^{10} b^{4}+420420 a^{8} b^{6}+463320 a^{6} b^{8}+189189 a^{4} b^{10}+25480 a^{2} b^{12}+756 b^{14}\right ) \left (-1+x \right )}{504}+\frac {2 b \,x^{\frac {7}{2}} \left (969 x^{7} b^{14}+112455 a^{2} b^{12} x^{6}+1633905 a^{4} b^{10} x^{5}+6789783 a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}+5555277 a^{10} b^{4} x^{2}+1028755 a^{12} b^{2} x +43605 a^{14}\right )}{20349}\)	\(586\)

[In]

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

[Out]

2/21*b^15*x^(21/2)+3/2*x^10*a*b^14+210/19*x^(19/2)*a^2*b^13+455/9*a^3*b^12*x^9+2730/17*x^(17/2)*a^4*b^11+3003/
8*a^5*b^10*x^8+2002/3*a^6*b^9*x^(15/2)+6435/7*a^7*b^8*x^7+990*x^(13/2)*a^8*b^7+5005/6*a^9*b^6*x^6+546*x^(11/2)
*a^10*b^5+273*x^5*a^11*b^4+910/9*x^(9/2)*a^12*b^3+105/4*x^4*a^13*b^2+30/7*x^(7/2)*a^14*b+1/3*x^3*a^15

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.42 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {3}{2} \, a b^{14} x^{10} + \frac {455}{9} \, a^{3} b^{12} x^{9} + \frac {3003}{8} \, a^{5} b^{10} x^{8} + \frac {6435}{7} \, a^{7} b^{8} x^{7} + \frac {5005}{6} \, a^{9} b^{6} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac {105}{4} \, a^{13} b^{2} x^{4} + \frac {1}{3} \, a^{15} x^{3} + \frac {2}{20349} \, {\left (969 \, b^{15} x^{10} + 112455 \, a^{2} b^{13} x^{9} + 1633905 \, a^{4} b^{11} x^{8} + 6789783 \, a^{6} b^{9} x^{7} + 10072755 \, a^{8} b^{7} x^{6} + 5555277 \, a^{10} b^{5} x^{5} + 1028755 \, a^{12} b^{3} x^{4} + 43605 \, a^{14} b x^{3}\right )} \sqrt {x} \]

[In]

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

[Out]

3/2*a*b^14*x^10 + 455/9*a^3*b^12*x^9 + 3003/8*a^5*b^10*x^8 + 6435/7*a^7*b^8*x^7 + 5005/6*a^9*b^6*x^6 + 273*a^1
1*b^4*x^5 + 105/4*a^13*b^2*x^4 + 1/3*a^15*x^3 + 2/20349*(969*b^15*x^10 + 112455*a^2*b^13*x^9 + 1633905*a^4*b^1
1*x^8 + 6789783*a^6*b^9*x^7 + 10072755*a^8*b^7*x^6 + 5555277*a^10*b^5*x^5 + 1028755*a^12*b^3*x^4 + 43605*a^14*
b*x^3)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.74 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {a^{15} x^{3}}{3} + \frac {30 a^{14} b x^{\frac {7}{2}}}{7} + \frac {105 a^{13} b^{2} x^{4}}{4} + \frac {910 a^{12} b^{3} x^{\frac {9}{2}}}{9} + 273 a^{11} b^{4} x^{5} + 546 a^{10} b^{5} x^{\frac {11}{2}} + \frac {5005 a^{9} b^{6} x^{6}}{6} + 990 a^{8} b^{7} x^{\frac {13}{2}} + \frac {6435 a^{7} b^{8} x^{7}}{7} + \frac {2002 a^{6} b^{9} x^{\frac {15}{2}}}{3} + \frac {3003 a^{5} b^{10} x^{8}}{8} + \frac {2730 a^{4} b^{11} x^{\frac {17}{2}}}{17} + \frac {455 a^{3} b^{12} x^{9}}{9} + \frac {210 a^{2} b^{13} x^{\frac {19}{2}}}{19} + \frac {3 a b^{14} x^{10}}{2} + \frac {2 b^{15} x^{\frac {21}{2}}}{21} \]

[In]

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

[Out]

a**15*x**3/3 + 30*a**14*b*x**(7/2)/7 + 105*a**13*b**2*x**4/4 + 910*a**12*b**3*x**(9/2)/9 + 273*a**11*b**4*x**5
+ 546*a**10*b**5*x**(11/2) + 5005*a**9*b**6*x**6/6 + 990*a**8*b**7*x**(13/2) + 6435*a**7*b**8*x**7/7 + 2002*a
**6*b**9*x**(15/2)/3 + 3003*a**5*b**10*x**8/8 + 2730*a**4*b**11*x**(17/2)/17 + 455*a**3*b**12*x**9/9 + 210*a**
2*b**13*x**(19/2)/19 + 3*a*b**14*x**10/2 + 2*b**15*x**(21/2)/21

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{21}}{21 \, b^{6}} - \frac {{\left (b \sqrt {x} + a\right )}^{20} a}{2 \, b^{6}} + \frac {20 \, {\left (b \sqrt {x} + a\right )}^{19} a^{2}}{19 \, b^{6}} - \frac {10 \, {\left (b \sqrt {x} + a\right )}^{18} a^{3}}{9 \, b^{6}} + \frac {10 \, {\left (b \sqrt {x} + a\right )}^{17} a^{4}}{17 \, b^{6}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{5}}{8 \, b^{6}} \]

[In]

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

[Out]

2/21*(b*sqrt(x) + a)^21/b^6 - 1/2*(b*sqrt(x) + a)^20*a/b^6 + 20/19*(b*sqrt(x) + a)^19*a^2/b^6 - 10/9*(b*sqrt(x
) + a)^18*a^3/b^6 + 10/17*(b*sqrt(x) + a)^17*a^4/b^6 - 1/8*(b*sqrt(x) + a)^16*a^5/b^6

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {2}{21} \, b^{15} x^{\frac {21}{2}} + \frac {3}{2} \, a b^{14} x^{10} + \frac {210}{19} \, a^{2} b^{13} x^{\frac {19}{2}} + \frac {455}{9} \, a^{3} b^{12} x^{9} + \frac {2730}{17} \, a^{4} b^{11} x^{\frac {17}{2}} + \frac {3003}{8} \, a^{5} b^{10} x^{8} + \frac {2002}{3} \, a^{6} b^{9} x^{\frac {15}{2}} + \frac {6435}{7} \, a^{7} b^{8} x^{7} + 990 \, a^{8} b^{7} x^{\frac {13}{2}} + \frac {5005}{6} \, a^{9} b^{6} x^{6} + 546 \, a^{10} b^{5} x^{\frac {11}{2}} + 273 \, a^{11} b^{4} x^{5} + \frac {910}{9} \, a^{12} b^{3} x^{\frac {9}{2}} + \frac {105}{4} \, a^{13} b^{2} x^{4} + \frac {30}{7} \, a^{14} b x^{\frac {7}{2}} + \frac {1}{3} \, a^{15} x^{3} \]

[In]

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

[Out]

2/21*b^15*x^(21/2) + 3/2*a*b^14*x^10 + 210/19*a^2*b^13*x^(19/2) + 455/9*a^3*b^12*x^9 + 2730/17*a^4*b^11*x^(17/
2) + 3003/8*a^5*b^10*x^8 + 2002/3*a^6*b^9*x^(15/2) + 6435/7*a^7*b^8*x^7 + 990*a^8*b^7*x^(13/2) + 5005/6*a^9*b^
6*x^6 + 546*a^10*b^5*x^(11/2) + 273*a^11*b^4*x^5 + 910/9*a^12*b^3*x^(9/2) + 105/4*a^13*b^2*x^4 + 30/7*a^14*b*x
^(7/2) + 1/3*a^15*x^3

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {a^{15}\,x^3}{3}+\frac {2\,b^{15}\,x^{21/2}}{21}+\frac {30\,a^{14}\,b\,x^{7/2}}{7}+\frac {3\,a\,b^{14}\,x^{10}}{2}+\frac {105\,a^{13}\,b^2\,x^4}{4}+273\,a^{11}\,b^4\,x^5+\frac {5005\,a^9\,b^6\,x^6}{6}+\frac {6435\,a^7\,b^8\,x^7}{7}+\frac {3003\,a^5\,b^{10}\,x^8}{8}+\frac {455\,a^3\,b^{12}\,x^9}{9}+\frac {910\,a^{12}\,b^3\,x^{9/2}}{9}+546\,a^{10}\,b^5\,x^{11/2}+990\,a^8\,b^7\,x^{13/2}+\frac {2002\,a^6\,b^9\,x^{15/2}}{3}+\frac {2730\,a^4\,b^{11}\,x^{17/2}}{17}+\frac {210\,a^2\,b^{13}\,x^{19/2}}{19} \]

[In]

int(x^2*(a + b*x^(1/2))^15,x)

[Out]

(a^15*x^3)/3 + (2*b^15*x^(21/2))/21 + (30*a^14*b*x^(7/2))/7 + (3*a*b^14*x^10)/2 + (105*a^13*b^2*x^4)/4 + 273*a
^11*b^4*x^5 + (5005*a^9*b^6*x^6)/6 + (6435*a^7*b^8*x^7)/7 + (3003*a^5*b^10*x^8)/8 + (455*a^3*b^12*x^9)/9 + (91
0*a^12*b^3*x^(9/2))/9 + 546*a^10*b^5*x^(11/2) + 990*a^8*b^7*x^(13/2) + (2002*a^6*b^9*x^(15/2))/3 + (2730*a^4*b
^11*x^(17/2))/17 + (210*a^2*b^13*x^(19/2))/19