\(\int (a+b \sqrt {x})^{15} x^2 \, dx\) [62]

Optimal result

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

-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
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.53 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {54264 a^{15} x^3+697680 a^{14} b x^{7/2}+4273290 a^{13} b^2 x^4+16460080 a^{12} b^3 x^{9/2}[phone] a^{11} b^4 x^5+88884432 a^{10} b^5 x^{11/2}[phone] a^9 b^6 x^6+161164080 a^8 b^7 x^{13/2}[phone] a^7 b^8 x^7+108636528 a^6 b^9 x^{15/2}[phone] a^5 b^{10} x^8+26142480 a^4 b^{11} x^{17/2}+8230040 a^3 b^{12} x^9+1799280 a^2 b^{13} x^{19/2}+244188 a b^{14} x^{10}+15504 b^{15} x^{21/2}}{162792} \] Input:

Integrate[(a + b*Sqrt[x])^15*x^2,x]
 

Output:

(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) + 13 
5795660*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 + 1799280*a^2*b^13*x^(19/2) + 244188*a*b^14* 
x^10 + 15504*b^15*x^(21/2))/162792
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 124, 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, 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.

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

\(\Big \downarrow \) 798

\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{15} x^{5/2}d\sqrt {x}\)

\(\Big \downarrow \) 49

\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{20}}{b^5}-\frac {5 a \left (a+b \sqrt {x}\right )^{19}}{b^5}+\frac {10 a^2 \left (a+b \sqrt {x}\right )^{18}}{b^5}-\frac {10 a^3 \left (a+b \sqrt {x}\right )^{17}}{b^5}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{16}}{b^5}-\frac {a^5 \left (a+b \sqrt {x}\right )^{15}}{b^5}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {a^5 \left (a+b \sqrt {x}\right )^{16}}{16 b^6}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{17}}{17 b^6}-\frac {5 a^3 \left (a+b \sqrt {x}\right )^{18}}{9 b^6}+\frac {10 a^2 \left (a+b \sqrt {x}\right )^{19}}{19 b^6}+\frac {\left (a+b \sqrt {x}\right )^{21}}{21 b^6}-\frac {a \left (a+b \sqrt {x}\right )^{20}}{4 b^6}\right )\)

Input:

Int[(a + b*Sqrt[x])^15*x^2,x]
 

Output:

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

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]
 

Maple [A] (verified)

Time = 22.96 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(\frac {2 b^{15} x^{\frac {21}{2}}}{21}+\frac {3 a \,b^{14} x^{10}}{2}+\frac {210 a^{2} b^{13} x^{\frac {19}{2}}}{19}+\frac {455 a^{3} b^{12} x^{9}}{9}+\frac {2730 a^{4} b^{11} x^{\frac {17}{2}}}{17}+\frac {3003 a^{5} b^{10} x^{8}}{8}+\frac {2002 a^{6} b^{9} x^{\frac {15}{2}}}{3}+\frac {6435 x^{7} b^{8} a^{7}}{7}+990 a^{8} b^{7} x^{\frac {13}{2}}+\frac {5005 a^{9} b^{6} x^{6}}{6}+546 a^{10} b^{5} x^{\frac {11}{2}}+273 a^{11} b^{4} x^{5}+\frac {910 a^{12} b^{3} x^{\frac {9}{2}}}{9}+\frac {105 a^{13} b^{2} x^{4}}{4}+\frac {30 a^{14} b \,x^{\frac {7}{2}}}{7}+\frac {a^{15} x^{3}}{3}\)	\(168\)
default	\(\frac {2 b^{15} x^{\frac {21}{2}}}{21}+\frac {3 a \,b^{14} x^{10}}{2}+\frac {210 a^{2} b^{13} x^{\frac {19}{2}}}{19}+\frac {455 a^{3} b^{12} x^{9}}{9}+\frac {2730 a^{4} b^{11} x^{\frac {17}{2}}}{17}+\frac {3003 a^{5} b^{10} x^{8}}{8}+\frac {2002 a^{6} b^{9} x^{\frac {15}{2}}}{3}+\frac {6435 x^{7} b^{8} a^{7}}{7}+990 a^{8} b^{7} x^{\frac {13}{2}}+\frac {5005 a^{9} b^{6} x^{6}}{6}+546 a^{10} b^{5} x^{\frac {11}{2}}+273 a^{11} b^{4} x^{5}+\frac {910 a^{12} b^{3} x^{\frac {9}{2}}}{9}+\frac {105 a^{13} b^{2} x^{4}}{4}+\frac {30 a^{14} b \,x^{\frac {7}{2}}}{7}+\frac {a^{15} x^{3}}{3}\)	\(168\)
orering	\(-\frac {\left (-453492 b^{34} x^{17}+6428380 a^{2} b^{32} x^{16}-42376425 a^{4} b^{30} x^{15}[phone] a^{6} b^{28} x^{14}-482496245 a^{8} b^{26} x^{13}[phone] a^{10} b^{24} x^{12}-1516767525 a^{12} b^{22} x^{11}[phone] a^{14} b^{20} x^{10}-1622085465 a^{16} b^{18} x^{9}[phone] a^{18} b^{16} x^{8}[phone] a^{22} b^{12} x^{6}[phone] a^{24} b^{10} x^{5}[phone] a^{26} b^{8} x^{4}[phone] a^{28} x^{3} b^{6}[phone] a^{30} x^{2} b^{4}[phone] a^{32} x \,b^{2}[phone] a^{34}\right ) \left (a +b \sqrt {x}\right )^{15}}{2441880 b^{6} \left (-b^{2} x +a^{2}\right )^{14}}+\frac {\left (-11628 b^{34} x^{17}+173740 a^{2} b^{32} x^{16}-1210755 a^{4} b^{30} x^{15}+5219565 a^{6} b^{28} x^{14}-15564395 a^{8} b^{26} x^{13}[phone] a^{10} b^{24} x^{12}-56176575 a^{12} b^{22} x^{11}[phone] a^{14} b^{20} x^{10}-70525455 a^{16} b^{18} x^{9}[phone] a^{18} b^{16} x^{8}[phone] a^{22} b^{12} x^{6}[phone] a^{24} b^{10} x^{5}[phone] a^{26} b^{8} x^{4}[phone] a^{28} x^{3} b^{6}[phone] a^{30} x^{2} b^{4}[phone] a^{32} x \,b^{2}[phone] a^{34}\right ) \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} x^{\frac {3}{2}} b}{2}+2 \left (a +b \sqrt {x}\right )^{15} x \right )}{1220940 b^{6} \left (-b^{2} x +a^{2}\right )^{14} x}\)	\(430\)
trager	\(\frac {a \left (756 b^{14} x^{9}+25480 b^{12} x^{8} a^{2}+756 b^{14} x^{8}+189189 b^{10} x^{7} a^{4}+25480 a^{2} b^{12} x^{7}+756 x^{7} b^{14}+463320 b^{8} x^{6} a^{6}+189189 a^{4} b^{10} x^{6}+25480 a^{2} b^{12} x^{6}+756 b^{14} x^{6}+420420 b^{6} x^{5} a^{8}+463320 a^{6} b^{8} x^{5}+189189 a^{4} b^{10} x^{5}+25480 a^{2} b^{12} x^{5}+756 b^{14} x^{5}+137592 b^{4} x^{4} a^{10}+420420 x^{4} b^{6} a^{8}+463320 a^{6} b^{8} x^{4}+189189 a^{4} b^{10} x^{4}+25480 b^{12} x^{4} a^{2}+756 b^{14} x^{4}+13230 a^{12} b^{2} x^{3}+137592 a^{10} b^{4} x^{3}+420420 a^{8} b^{6} x^{3}+463320 a^{6} b^{8} x^{3}+189189 b^{10} x^{3} a^{4}+25480 b^{12} x^{3} a^{2}+756 b^{14} x^{3}+168 a^{14} x^{2}+13230 a^{12} b^{2} x^{2}+137592 a^{10} b^{4} x^{2}+420420 a^{8} b^{6} x^{2}+463320 b^{8} x^{2} a^{6}+189189 b^{10} x^{2} a^{4}+25480 b^{12} x^{2} a^{2}+756 b^{14} x^{2}+168 a^{14} x +13230 a^{12} b^{2} x +137592 a^{10} b^{4} x +420420 a^{8} b^{6} x +463320 a^{6} b^{8} x +189189 b^{10} x \,a^{4}+25480 b^{12} x \,a^{2}+756 b^{14} x +168 a^{14}+13230 a^{12} b^{2}+137592 a^{10} b^{4}+420420 a^{8} b^{6}+463320 a^{6} b^{8}+189189 a^{4} b^{10}+25480 a^{2} b^{12}+756 b^{14}\right ) \left (-1+x \right )}{504}+\frac {2 b \,x^{\frac {7}{2}} \left (969 x^{7} b^{14}+112455 a^{2} b^{12} x^{6}+1633905 a^{4} b^{10} x^{5}+6789783 a^{6} b^{8} x^{4}[phone] a^{8} b^{6} x^{3}+5555277 a^{10} b^{4} x^{2}+1028755 a^{12} b^{2} x +43605 a^{14}\right )}{20349}\)	\(586\)

Input:

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

Output:

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*x^7*b^8*a^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
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.42 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {3}{2} \, a b^{14} x^{10} + \frac {455}{9} \, a^{3} b^{12} x^{9} + \frac {3003}{8} \, a^{5} b^{10} x^{8} + \frac {6435}{7} \, a^{7} b^{8} x^{7} + \frac {5005}{6} \, a^{9} b^{6} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac {105}{4} \, a^{13} b^{2} x^{4} + \frac {1}{3} \, a^{15} x^{3} + \frac {2}{20349} \, {\left (969 \, b^{15} x^{10} + 112455 \, a^{2} b^{13} x^{9} + 1633905 \, a^{4} b^{11} x^{8} + 6789783 \, a^{6} b^{9} x^{7} + 10072755 \, a^{8} b^{7} x^{6} + 5555277 \, a^{10} b^{5} x^{5} + 1028755 \, a^{12} b^{3} x^{4} + 43605 \, a^{14} b x^{3}\right )} \sqrt {x} \] Input:

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

Output:

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^11*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^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)*sqrt(x)
 

Sympy [A] (verification not implemented)

Time = 1.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.74 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {a^{15} x^{3}}{3} + \frac {30 a^{14} b x^{\frac {7}{2}}}{7} + \frac {105 a^{13} b^{2} x^{4}}{4} + \frac {910 a^{12} b^{3} x^{\frac {9}{2}}}{9} + 273 a^{11} b^{4} x^{5} + 546 a^{10} b^{5} x^{\frac {11}{2}} + \frac {5005 a^{9} b^{6} x^{6}}{6} + 990 a^{8} b^{7} x^{\frac {13}{2}} + \frac {6435 a^{7} b^{8} x^{7}}{7} + \frac {2002 a^{6} b^{9} x^{\frac {15}{2}}}{3} + \frac {3003 a^{5} b^{10} x^{8}}{8} + \frac {2730 a^{4} b^{11} x^{\frac {17}{2}}}{17} + \frac {455 a^{3} b^{12} x^{9}}{9} + \frac {210 a^{2} b^{13} x^{\frac {19}{2}}}{19} + \frac {3 a b^{14} x^{10}}{2} + \frac {2 b^{15} x^{\frac {21}{2}}}{21} \] Input:

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

Output:

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)

Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{21}}{21 \, b^{6}} - \frac {{\left (b \sqrt {x} + a\right )}^{20} a}{2 \, b^{6}} + \frac {20 \, {\left (b \sqrt {x} + a\right )}^{19} a^{2}}{19 \, b^{6}} - \frac {10 \, {\left (b \sqrt {x} + a\right )}^{18} a^{3}}{9 \, b^{6}} + \frac {10 \, {\left (b \sqrt {x} + a\right )}^{17} a^{4}}{17 \, b^{6}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{5}}{8 \, b^{6}} \] Input:

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

Output:

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)

Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {2}{21} \, b^{15} x^{\frac {21}{2}} + \frac {3}{2} \, a b^{14} x^{10} + \frac {210}{19} \, a^{2} b^{13} x^{\frac {19}{2}} + \frac {455}{9} \, a^{3} b^{12} x^{9} + \frac {2730}{17} \, a^{4} b^{11} x^{\frac {17}{2}} + \frac {3003}{8} \, a^{5} b^{10} x^{8} + \frac {2002}{3} \, a^{6} b^{9} x^{\frac {15}{2}} + \frac {6435}{7} \, a^{7} b^{8} x^{7} + 990 \, a^{8} b^{7} x^{\frac {13}{2}} + \frac {5005}{6} \, a^{9} b^{6} x^{6} + 546 \, a^{10} b^{5} x^{\frac {11}{2}} + 273 \, a^{11} b^{4} x^{5} + \frac {910}{9} \, a^{12} b^{3} x^{\frac {9}{2}} + \frac {105}{4} \, a^{13} b^{2} x^{4} + \frac {30}{7} \, a^{14} b x^{\frac {7}{2}} + \frac {1}{3} \, a^{15} x^{3} \] Input:

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

Output:

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

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

Output:

(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 + (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) + (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
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.46 \[ \int \left (a+b \sqrt {x}\right )^{15} x^2 \, dx=\frac {x^{3} \left (697680 \sqrt {x}\, a^{14} b [phone] \sqrt {x}\, a^{12} b^{3} x [phone] \sqrt {x}\, a^{10} b^{5} x^{2}[phone] \sqrt {x}\, a^{8} b^{7} x^{3}[phone] \sqrt {x}\, a^{6} b^{9} x^{4}[phone] \sqrt {x}\, a^{4} b^{11} x^{5}+1799280 \sqrt {x}\, a^{2} b^{13} x^{6}+15504 \sqrt {x}\, b^{15} x^{7}+54264 a^{15}+4273290 a^{13} b^{2} x [phone] a^{11} b^{4} x^{2}[phone] a^{9} b^{6} x^{3}[phone] a^{7} b^{8} x^{4}[phone] a^{5} b^{10} x^{5}+8230040 a^{3} b^{12} x^{6}+244188 a \,b^{14} x^{7}\right )}{162792} \] Input:

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

Output:

(x**3*(697680*sqrt(x)*a**14*b + 16460080*sqrt(x)*a**12*b**3*x + 88884432*s 
qrt(x)*a**10*b**5*x**2 + 161164080*sqrt(x)*a**8*b**7*x**3 + 108636528*sqrt 
(x)*a**6*b**9*x**4 + 26142480*sqrt(x)*a**4*b**11*x**5 + 1799280*sqrt(x)*a* 
*2*b**13*x**6 + 15504*sqrt(x)*b**15*x**7 + 54264*a**15 + 4273290*a**13*b** 
2*x + 44442216*a**11*b**4*x**2 + 135795660*a**9*b**6*x**3 + 149652360*a**7 
*b**8*x**4 + 61108047*a**5*b**10*x**5 + 8230040*a**3*b**12*x**6 + 244188*a 
*b**14*x**7))/162792