\(\int (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5)^3 \, dx\) [63]

Optimal result

Integrand size = 51, antiderivative size = 14 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {(a+b x)^{16}}{16 b} \]

[Out]

1/16*(b*x+a)^16/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2084, 32} \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {(a+b x)^{16}}{16 b} \]

[In]

Int[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^3,x]

[Out]

(a + b*x)^16/(16*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2084

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[u^p, x] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x]
&& IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^{15} \, dx \\ & = \frac {(a+b x)^{16}}{16 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {(a+b x)^{16}}{16 b} \]

[In]

Integrate[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^3,x]

[Out]

(a + b*x)^16/(16*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(163\) vs. \(2(12)=24\).

Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 11.71

method	result	size
default	\(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\)	\(164\)
norman	\(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\)	\(164\)
risch	\(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\)	\(164\)
parallelrisch	\(a^{15} x +\frac {15}{2} a^{14} b \,x^{2}+35 a^{13} b^{2} x^{3}+\frac {455}{4} a^{12} b^{3} x^{4}+273 a^{11} b^{4} x^{5}+\frac {1001}{2} a^{10} b^{5} x^{6}+715 a^{9} b^{6} x^{7}+\frac {6435}{8} a^{8} b^{7} x^{8}+715 a^{7} b^{8} x^{9}+\frac {1001}{2} a^{6} b^{9} x^{10}+273 a^{5} b^{10} x^{11}+\frac {455}{4} a^{4} b^{11} x^{12}+35 a^{3} b^{12} x^{13}+\frac {15}{2} a^{2} b^{13} x^{14}+a \,b^{14} x^{15}+\frac {1}{16} b^{15} x^{16}\)	\(164\)
gosper	\(\frac {x \left (b^{15} x^{15}+16 a \,b^{14} x^{14}+120 a^{2} b^{13} x^{13}+560 a^{3} b^{12} x^{12}+1820 a^{4} b^{11} x^{11}+4368 a^{5} b^{10} x^{10}+8008 a^{6} b^{9} x^{9}+11440 a^{7} b^{8} x^{8}+12870 a^{8} b^{7} x^{7}+11440 a^{9} b^{6} x^{6}+8008 a^{10} b^{5} x^{5}+4368 a^{11} b^{4} x^{4}+1820 a^{12} b^{3} x^{3}+560 a^{13} b^{2} x^{2}+120 a^{14} b x +16 a^{15}\right )}{16}\)	\(165\)

[In]

int((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x,method=_RETURNVERBOSE)

[Out]

a^15*x+15/2*a^14*b*x^2+35*a^13*b^2*x^3+455/4*a^12*b^3*x^4+273*a^11*b^4*x^5+1001/2*a^10*b^5*x^6+715*a^9*b^6*x^7
+6435/8*a^8*b^7*x^8+715*a^7*b^8*x^9+1001/2*a^6*b^9*x^10+273*a^5*b^10*x^11+455/4*a^4*b^11*x^12+35*a^3*b^12*x^13
+15/2*a^2*b^13*x^14+a*b^14*x^15+1/16*b^15*x^16

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 11.64 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {1}{16} \, b^{15} x^{16} + a b^{14} x^{15} + \frac {15}{2} \, a^{2} b^{13} x^{14} + 35 \, a^{3} b^{12} x^{13} + \frac {455}{4} \, a^{4} b^{11} x^{12} + 273 \, a^{5} b^{10} x^{11} + \frac {1001}{2} \, a^{6} b^{9} x^{10} + 715 \, a^{7} b^{8} x^{9} + \frac {6435}{8} \, a^{8} b^{7} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac {1001}{2} \, a^{10} b^{5} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac {455}{4} \, a^{12} b^{3} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac {15}{2} \, a^{14} b x^{2} + a^{15} x \]

[In]

integrate((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x, algorithm="fricas")

[Out]

1/16*b^15*x^16 + a*b^14*x^15 + 15/2*a^2*b^13*x^14 + 35*a^3*b^12*x^13 + 455/4*a^4*b^11*x^12 + 273*a^5*b^10*x^11
+ 1001/2*a^6*b^9*x^10 + 715*a^7*b^8*x^9 + 6435/8*a^8*b^7*x^8 + 715*a^9*b^6*x^7 + 1001/2*a^10*b^5*x^6 + 273*a^
11*b^4*x^5 + 455/4*a^12*b^3*x^4 + 35*a^13*b^2*x^3 + 15/2*a^14*b*x^2 + a^15*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (8) = 16\).

Time = 0.04 (sec) , antiderivative size = 185, normalized size of antiderivative = 13.21 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=a^{15} x + \frac {15 a^{14} b x^{2}}{2} + 35 a^{13} b^{2} x^{3} + \frac {455 a^{12} b^{3} x^{4}}{4} + 273 a^{11} b^{4} x^{5} + \frac {1001 a^{10} b^{5} x^{6}}{2} + 715 a^{9} b^{6} x^{7} + \frac {6435 a^{8} b^{7} x^{8}}{8} + 715 a^{7} b^{8} x^{9} + \frac {1001 a^{6} b^{9} x^{10}}{2} + 273 a^{5} b^{10} x^{11} + \frac {455 a^{4} b^{11} x^{12}}{4} + 35 a^{3} b^{12} x^{13} + \frac {15 a^{2} b^{13} x^{14}}{2} + a b^{14} x^{15} + \frac {b^{15} x^{16}}{16} \]

[In]

integrate((b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*b*x+a**5)**3,x)

[Out]

a**15*x + 15*a**14*b*x**2/2 + 35*a**13*b**2*x**3 + 455*a**12*b**3*x**4/4 + 273*a**11*b**4*x**5 + 1001*a**10*b*
*5*x**6/2 + 715*a**9*b**6*x**7 + 6435*a**8*b**7*x**8/8 + 715*a**7*b**8*x**9 + 1001*a**6*b**9*x**10/2 + 273*a**
5*b**10*x**11 + 455*a**4*b**11*x**12/4 + 35*a**3*b**12*x**13 + 15*a**2*b**13*x**14/2 + a*b**14*x**15 + b**15*x
**16/16

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 592, normalized size of antiderivative = 42.29 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {1}{16} \, b^{15} x^{16} + a b^{14} x^{15} + \frac {75}{14} \, a^{2} b^{13} x^{14} + \frac {125}{13} \, a^{3} b^{12} x^{13} + 100 \, a^{6} b^{9} x^{10} + \frac {1000}{7} \, a^{9} b^{6} x^{7} + \frac {125}{4} \, a^{12} b^{3} x^{4} + a^{15} x + \frac {1}{2} \, {\left (b^{5} x^{6} + 6 \, a b^{4} x^{5} + 15 \, a^{2} b^{3} x^{4} + 20 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x^{2}\right )} a^{10} + \frac {25}{56} \, {\left (21 \, b^{5} x^{8} + 120 \, a b^{4} x^{7} + 280 \, a^{2} b^{3} x^{6} + 336 \, a^{3} b^{2} x^{5}\right )} a^{8} b^{2} + \frac {5}{3} \, {\left (18 \, b^{5} x^{10} + 100 \, a b^{4} x^{9} + 225 \, a^{2} b^{3} x^{8}\right )} a^{6} b^{4} + \frac {25}{11} \, {\left (11 \, b^{5} x^{12} + 60 \, a b^{4} x^{11}\right )} a^{4} b^{6} + \frac {1}{462} \, {\left (126 \, b^{10} x^{11} + 1386 \, a b^{9} x^{10} + 3850 \, a^{2} b^{8} x^{9} + 19800 \, a^{4} b^{6} x^{7} + 27720 \, a^{6} b^{4} x^{5} + 11550 \, a^{8} b^{2} x^{3} + 330 \, {\left (6 \, b^{5} x^{7} + 35 \, a b^{4} x^{6} + 84 \, a^{2} b^{3} x^{5} + 105 \, a^{3} b^{2} x^{4}\right )} a^{4} b + 165 \, {\left (21 \, b^{5} x^{8} + 120 \, a b^{4} x^{7} + 280 \, a^{2} b^{3} x^{6}\right )} a^{3} b^{2} + 385 \, {\left (8 \, b^{5} x^{9} + 45 \, a b^{4} x^{8}\right )} a^{2} b^{3}\right )} a^{5} + \frac {5}{308} \, {\left (77 \, b^{10} x^{12} + 840 \, a b^{9} x^{11} + 4158 \, a^{2} b^{8} x^{10} + 12320 \, a^{3} b^{7} x^{9} + 23100 \, a^{4} b^{6} x^{8} + 26400 \, a^{5} b^{5} x^{7} + 15400 \, a^{6} b^{4} x^{6}\right )} a^{4} b + \frac {5}{429} \, {\left (198 \, b^{10} x^{13} + 2145 \, a b^{9} x^{12} + 10530 \, a^{2} b^{8} x^{11} + 25740 \, a^{3} b^{7} x^{10} + 28600 \, a^{4} b^{6} x^{9}\right )} a^{3} b^{2} + \frac {5}{182} \, {\left (78 \, b^{10} x^{14} + 840 \, a b^{9} x^{13} + 2275 \, a^{2} b^{8} x^{12}\right )} a^{2} b^{3} \]

[In]

integrate((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x, algorithm="maxima")

[Out]

1/16*b^15*x^16 + a*b^14*x^15 + 75/14*a^2*b^13*x^14 + 125/13*a^3*b^12*x^13 + 100*a^6*b^9*x^10 + 1000/7*a^9*b^6*
x^7 + 125/4*a^12*b^3*x^4 + a^15*x + 1/2*(b^5*x^6 + 6*a*b^4*x^5 + 15*a^2*b^3*x^4 + 20*a^3*b^2*x^3 + 15*a^4*b*x^
2)*a^10 + 25/56*(21*b^5*x^8 + 120*a*b^4*x^7 + 280*a^2*b^3*x^6 + 336*a^3*b^2*x^5)*a^8*b^2 + 5/3*(18*b^5*x^10 +
100*a*b^4*x^9 + 225*a^2*b^3*x^8)*a^6*b^4 + 25/11*(11*b^5*x^12 + 60*a*b^4*x^11)*a^4*b^6 + 1/462*(126*b^10*x^11
+ 1386*a*b^9*x^10 + 3850*a^2*b^8*x^9 + 19800*a^4*b^6*x^7 + 27720*a^6*b^4*x^5 + 11550*a^8*b^2*x^3 + 330*(6*b^5*
x^7 + 35*a*b^4*x^6 + 84*a^2*b^3*x^5 + 105*a^3*b^2*x^4)*a^4*b + 165*(21*b^5*x^8 + 120*a*b^4*x^7 + 280*a^2*b^3*x
^6)*a^3*b^2 + 385*(8*b^5*x^9 + 45*a*b^4*x^8)*a^2*b^3)*a^5 + 5/308*(77*b^10*x^12 + 840*a*b^9*x^11 + 4158*a^2*b^
8*x^10 + 12320*a^3*b^7*x^9 + 23100*a^4*b^6*x^8 + 26400*a^5*b^5*x^7 + 15400*a^6*b^4*x^6)*a^4*b + 5/429*(198*b^1
0*x^13 + 2145*a*b^9*x^12 + 10530*a^2*b^8*x^11 + 25740*a^3*b^7*x^10 + 28600*a^4*b^6*x^9)*a^3*b^2 + 5/182*(78*b^
10*x^14 + 840*a*b^9*x^13 + 2275*a^2*b^8*x^12)*a^2*b^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 11.64 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=\frac {1}{16} \, b^{15} x^{16} + a b^{14} x^{15} + \frac {15}{2} \, a^{2} b^{13} x^{14} + 35 \, a^{3} b^{12} x^{13} + \frac {455}{4} \, a^{4} b^{11} x^{12} + 273 \, a^{5} b^{10} x^{11} + \frac {1001}{2} \, a^{6} b^{9} x^{10} + 715 \, a^{7} b^{8} x^{9} + \frac {6435}{8} \, a^{8} b^{7} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac {1001}{2} \, a^{10} b^{5} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac {455}{4} \, a^{12} b^{3} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac {15}{2} \, a^{14} b x^{2} + a^{15} x \]

[In]

integrate((b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5)^3,x, algorithm="giac")

[Out]

1/16*b^15*x^16 + a*b^14*x^15 + 15/2*a^2*b^13*x^14 + 35*a^3*b^12*x^13 + 455/4*a^4*b^11*x^12 + 273*a^5*b^10*x^11
+ 1001/2*a^6*b^9*x^10 + 715*a^7*b^8*x^9 + 6435/8*a^8*b^7*x^8 + 715*a^9*b^6*x^7 + 1001/2*a^10*b^5*x^6 + 273*a^
11*b^4*x^5 + 455/4*a^12*b^3*x^4 + 35*a^13*b^2*x^3 + 15/2*a^14*b*x^2 + a^15*x

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 11.64 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3 \, dx=a^{15}\,x+\frac {15\,a^{14}\,b\,x^2}{2}+35\,a^{13}\,b^2\,x^3+\frac {455\,a^{12}\,b^3\,x^4}{4}+273\,a^{11}\,b^4\,x^5+\frac {1001\,a^{10}\,b^5\,x^6}{2}+715\,a^9\,b^6\,x^7+\frac {6435\,a^8\,b^7\,x^8}{8}+715\,a^7\,b^8\,x^9+\frac {1001\,a^6\,b^9\,x^{10}}{2}+273\,a^5\,b^{10}\,x^{11}+\frac {455\,a^4\,b^{11}\,x^{12}}{4}+35\,a^3\,b^{12}\,x^{13}+\frac {15\,a^2\,b^{13}\,x^{14}}{2}+a\,b^{14}\,x^{15}+\frac {b^{15}\,x^{16}}{16} \]

[In]

int((a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)^3,x)

[Out]

a^15*x + (b^15*x^16)/16 + (15*a^14*b*x^2)/2 + a*b^14*x^15 + 35*a^13*b^2*x^3 + (455*a^12*b^3*x^4)/4 + 273*a^11*
b^4*x^5 + (1001*a^10*b^5*x^6)/2 + 715*a^9*b^6*x^7 + (6435*a^8*b^7*x^8)/8 + 715*a^7*b^8*x^9 + (1001*a^6*b^9*x^1
0)/2 + 273*a^5*b^10*x^11 + (455*a^4*b^11*x^12)/4 + 35*a^3*b^12*x^13 + (15*a^2*b^13*x^14)/2