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

Optimal. Leaf size=14 \[ \frac{(a+b x)^{16}}{16 b} \]

[Out]

(a + b*x)^16/(16*b)

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Rubi [A]  time = 0.0169574, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039, Rules used = {2059, 32} \[ \frac{(a+b x)^{16}}{16 b} \]

Antiderivative was successfully verified.

[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 2059

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0013318, size = 14, normalized size = 1. \[ \frac{(a+b x)^{16}}{16 b} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.001, size = 164, normalized size = 11.7 \begin{align*}{\frac{{b}^{15}{x}^{16}}{16}}+a{b}^{14}{x}^{15}+{\frac{15\,{a}^{2}{b}^{13}{x}^{14}}{2}}+35\,{a}^{3}{b}^{12}{x}^{13}+{\frac{455\,{a}^{4}{b}^{11}{x}^{12}}{4}}+273\,{a}^{5}{b}^{10}{x}^{11}+{\frac{1001\,{a}^{6}{b}^{9}{x}^{10}}{2}}+715\,{a}^{7}{b}^{8}{x}^{9}+{\frac{6435\,{a}^{8}{b}^{7}{x}^{8}}{8}}+715\,{a}^{9}{b}^{6}{x}^{7}+{\frac{1001\,{a}^{10}{b}^{5}{x}^{6}}{2}}+273\,{a}^{11}{b}^{4}{x}^{5}+{\frac{455\,{a}^{12}{b}^{3}{x}^{4}}{4}}+35\,{a}^{13}{b}^{2}{x}^{3}+{\frac{15\,{a}^{14}b{x}^{2}}{2}}+{a}^{15}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[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

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Maxima [B]  time = 1.19535, size = 799, normalized size = 57.07 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.59064, size = 402, normalized size = 28.71 \begin{align*} \frac{1}{16} x^{16} b^{15} + x^{15} b^{14} a + \frac{15}{2} x^{14} b^{13} a^{2} + 35 x^{13} b^{12} a^{3} + \frac{455}{4} x^{12} b^{11} a^{4} + 273 x^{11} b^{10} a^{5} + \frac{1001}{2} x^{10} b^{9} a^{6} + 715 x^{9} b^{8} a^{7} + \frac{6435}{8} x^{8} b^{7} a^{8} + 715 x^{7} b^{6} a^{9} + \frac{1001}{2} x^{6} b^{5} a^{10} + 273 x^{5} b^{4} a^{11} + \frac{455}{4} x^{4} b^{3} a^{12} + 35 x^{3} b^{2} a^{13} + \frac{15}{2} x^{2} b a^{14} + x a^{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [B]  time = 0.105733, size = 185, normalized size = 13.21 \begin{align*} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.14015, size = 220, normalized size = 15.71 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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