\(\int x^{12} (a+b x^{13})^{12} \, dx\) [349]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {\left (a+b x^{13}\right )^{13}}{169 b} \]

[Out]

1/169*(b*x^13+a)^13/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {\left (a+b x^{13}\right )^{13}}{169 b} \]

[In]

Int[x^12*(a + b*x^13)^12,x]

[Out]

(a + b*x^13)^13/(169*b)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x^{13}\right )^{13}}{169 b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(16)=32\).

Time = 0.00 (sec) , antiderivative size = 160, normalized size of antiderivative = 10.00 \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {a^{12} x^{13}}{13}+\frac {6}{13} a^{11} b x^{26}+\frac {22}{13} a^{10} b^2 x^{39}+\frac {55}{13} a^9 b^3 x^{52}+\frac {99}{13} a^8 b^4 x^{65}+\frac {132}{13} a^7 b^5 x^{78}+\frac {132}{13} a^6 b^6 x^{91}+\frac {99}{13} a^5 b^7 x^{104}+\frac {55}{13} a^4 b^8 x^{117}+\frac {22}{13} a^3 b^9 x^{130}+\frac {6}{13} a^2 b^{10} x^{143}+\frac {1}{13} a b^{11} x^{156}+\frac {b^{12} x^{169}}{169} \]

[In]

Integrate[x^12*(a + b*x^13)^12,x]

[Out]

(a^12*x^13)/13 + (6*a^11*b*x^26)/13 + (22*a^10*b^2*x^39)/13 + (55*a^9*b^3*x^52)/13 + (99*a^8*b^4*x^65)/13 + (1
32*a^7*b^5*x^78)/13 + (132*a^6*b^6*x^91)/13 + (99*a^5*b^7*x^104)/13 + (55*a^4*b^8*x^117)/13 + (22*a^3*b^9*x^13
0)/13 + (6*a^2*b^10*x^143)/13 + (a*b^11*x^156)/13 + (b^12*x^169)/169

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (b \,x^{13}+a \right )^{13}}{169 b}\)	\(15\)
gosper	\(\frac {1}{169} b^{12} x^{169}+\frac {1}{13} a \,b^{11} x^{156}+\frac {6}{13} a^{2} b^{10} x^{143}+\frac {22}{13} a^{3} b^{9} x^{130}+\frac {55}{13} a^{4} b^{8} x^{117}+\frac {99}{13} a^{5} b^{7} x^{104}+\frac {132}{13} a^{6} b^{6} x^{91}+\frac {132}{13} a^{7} b^{5} x^{78}+\frac {99}{13} a^{8} b^{4} x^{65}+\frac {55}{13} a^{9} b^{3} x^{52}+\frac {22}{13} a^{10} b^{2} x^{39}+\frac {6}{13} b \,a^{11} x^{26}+\frac {1}{13} a^{12} x^{13}\)	\(135\)
parallelrisch	\(\frac {1}{169} b^{12} x^{169}+\frac {1}{13} a \,b^{11} x^{156}+\frac {6}{13} a^{2} b^{10} x^{143}+\frac {22}{13} a^{3} b^{9} x^{130}+\frac {55}{13} a^{4} b^{8} x^{117}+\frac {99}{13} a^{5} b^{7} x^{104}+\frac {132}{13} a^{6} b^{6} x^{91}+\frac {132}{13} a^{7} b^{5} x^{78}+\frac {99}{13} a^{8} b^{4} x^{65}+\frac {55}{13} a^{9} b^{3} x^{52}+\frac {22}{13} a^{10} b^{2} x^{39}+\frac {6}{13} b \,a^{11} x^{26}+\frac {1}{13} a^{12} x^{13}\)	\(135\)
risch	\(\frac {b^{12} x^{169}}{169}+\frac {a \,b^{11} x^{156}}{13}+\frac {6 a^{2} b^{10} x^{143}}{13}+\frac {22 a^{3} b^{9} x^{130}}{13}+\frac {55 a^{4} b^{8} x^{117}}{13}+\frac {99 a^{5} b^{7} x^{104}}{13}+\frac {132 a^{6} b^{6} x^{91}}{13}+\frac {132 a^{7} b^{5} x^{78}}{13}+\frac {99 a^{8} b^{4} x^{65}}{13}+\frac {55 a^{9} b^{3} x^{52}}{13}+\frac {22 a^{10} b^{2} x^{39}}{13}+\frac {6 b \,a^{11} x^{26}}{13}+\frac {a^{12} x^{13}}{13}+\frac {a^{13}}{169 b}\)	\(143\)

[In]

int(x^12*(b*x^13+a)^12,x,method=_RETURNVERBOSE)

[Out]

1/169*(b*x^13+a)^13/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (14) = 28\).

Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 8.38 \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {1}{169} \, b^{12} x^{169} + \frac {1}{13} \, a b^{11} x^{156} + \frac {6}{13} \, a^{2} b^{10} x^{143} + \frac {22}{13} \, a^{3} b^{9} x^{130} + \frac {55}{13} \, a^{4} b^{8} x^{117} + \frac {99}{13} \, a^{5} b^{7} x^{104} + \frac {132}{13} \, a^{6} b^{6} x^{91} + \frac {132}{13} \, a^{7} b^{5} x^{78} + \frac {99}{13} \, a^{8} b^{4} x^{65} + \frac {55}{13} \, a^{9} b^{3} x^{52} + \frac {22}{13} \, a^{10} b^{2} x^{39} + \frac {6}{13} \, a^{11} b x^{26} + \frac {1}{13} \, a^{12} x^{13} \]

[In]

integrate(x^12*(b*x^13+a)^12,x, algorithm="fricas")

[Out]

1/169*b^12*x^169 + 1/13*a*b^11*x^156 + 6/13*a^2*b^10*x^143 + 22/13*a^3*b^9*x^130 + 55/13*a^4*b^8*x^117 + 99/13
*a^5*b^7*x^104 + 132/13*a^6*b^6*x^91 + 132/13*a^7*b^5*x^78 + 99/13*a^8*b^4*x^65 + 55/13*a^9*b^3*x^52 + 22/13*a
^10*b^2*x^39 + 6/13*a^11*b*x^26 + 1/13*a^12*x^13

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (10) = 20\).

Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 10.00 \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {a^{12} x^{13}}{13} + \frac {6 a^{11} b x^{26}}{13} + \frac {22 a^{10} b^{2} x^{39}}{13} + \frac {55 a^{9} b^{3} x^{52}}{13} + \frac {99 a^{8} b^{4} x^{65}}{13} + \frac {132 a^{7} b^{5} x^{78}}{13} + \frac {132 a^{6} b^{6} x^{91}}{13} + \frac {99 a^{5} b^{7} x^{104}}{13} + \frac {55 a^{4} b^{8} x^{117}}{13} + \frac {22 a^{3} b^{9} x^{130}}{13} + \frac {6 a^{2} b^{10} x^{143}}{13} + \frac {a b^{11} x^{156}}{13} + \frac {b^{12} x^{169}}{169} \]

[In]

integrate(x**12*(b*x**13+a)**12,x)

[Out]

a**12*x**13/13 + 6*a**11*b*x**26/13 + 22*a**10*b**2*x**39/13 + 55*a**9*b**3*x**52/13 + 99*a**8*b**4*x**65/13 +
132*a**7*b**5*x**78/13 + 132*a**6*b**6*x**91/13 + 99*a**5*b**7*x**104/13 + 55*a**4*b**8*x**117/13 + 22*a**3*b
**9*x**130/13 + 6*a**2*b**10*x**143/13 + a*b**11*x**156/13 + b**12*x**169/169

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {{\left (b x^{13} + a\right )}^{13}}{169 \, b} \]

[In]

integrate(x^12*(b*x^13+a)^12,x, algorithm="maxima")

[Out]

1/169*(b*x^13 + a)^13/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {{\left (b x^{13} + a\right )}^{13}}{169 \, b} \]

[In]

integrate(x^12*(b*x^13+a)^12,x, algorithm="giac")

[Out]

1/169*(b*x^13 + a)^13/b

Mupad [B] (verification not implemented)

Time = 8.89 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^{12} \left (a+b x^{13}\right )^{12} \, dx=\frac {{\left (b\,x^{13}+a\right )}^{13}}{169\,b} \]

[In]

int(x^12*(a + b*x^13)^12,x)

[Out]

(a + b*x^13)^13/(169*b)