\(\int x^{12 m} (a+b x^{1+12 m})^{12} \, dx\) [2595]

Optimal result

Integrand size = 19, antiderivative size = 27 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b (1+12 m)} \]

[Out]

1/13*(a+b*x^(1+12*m))^13/b/(1+12*m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {267} \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \]

[In]

Int[x^(12*m)*(a + b*x^(1 + 12*m))^12,x]

[Out]

(a + b*x^(1 + 12*m))^13/(13*b*(1 + 12*m))

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^{1+12 m}\right )^{13}}{13 b (1+12 m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b+156 b m} \]

[In]

Integrate[x^(12*m)*(a + b*x^(1 + 12*m))^12,x]

[Out]

(a + b*x^(1 + 12*m))^13/(13*b + 156*b*m)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(25)=50\).

Time = 45.19 (sec) , antiderivative size = 288, normalized size of antiderivative = 10.67

method	result	size
parallelrisch	\(\frac {b^{12} x^{12+144 m} x^{12 m} x +13 a \,b^{11} x^{11+132 m} x^{12 m} x +78 a^{2} b^{10} x^{10+120 m} x^{12 m} x +286 a^{3} b^{9} x^{9+108 m} x^{12 m} x +715 a^{4} b^{8} x^{8+96 m} x^{12 m} x +1287 a^{5} b^{7} x^{7+84 m} x^{12 m} x +1716 a^{6} b^{6} x^{6+72 m} x^{12 m} x +1716 a^{7} b^{5} x^{5+60 m} x^{12 m} x +1287 a^{8} b^{4} x^{4+48 m} x^{12 m} x +715 a^{9} b^{3} x^{3+36 m} x^{12 m} x +286 a^{10} b^{2} x^{2+24 m} x^{12 m} x +78 b \,a^{11} x^{1+12 m} x^{12 m} x +13 a^{12} x^{12 m} x}{13+156 m}\)	\(288\)
risch	\(\frac {b^{12} x^{13} x^{156 m}}{13+156 m}+\frac {a \,b^{11} x^{12} x^{144 m}}{1+12 m}+\frac {6 a^{2} b^{10} x^{11} x^{132 m}}{1+12 m}+\frac {22 a^{3} b^{9} x^{10} x^{120 m}}{1+12 m}+\frac {55 a^{4} b^{8} x^{9} x^{108 m}}{1+12 m}+\frac {99 a^{5} b^{7} x^{8} x^{96 m}}{1+12 m}+\frac {132 a^{6} b^{6} x^{7} x^{84 m}}{1+12 m}+\frac {132 a^{7} b^{5} x^{6} x^{72 m}}{1+12 m}+\frac {99 a^{8} b^{4} x^{5} x^{60 m}}{1+12 m}+\frac {55 a^{9} b^{3} x^{4} x^{48 m}}{1+12 m}+\frac {22 a^{10} b^{2} x^{3} x^{36 m}}{1+12 m}+\frac {6 b \,a^{11} x^{2} x^{24 m}}{1+12 m}+\frac {a^{12} x \,x^{12 m}}{1+12 m}\)	\(311\)

[In]

int(x^(12*m)*(a+b*x^(1+12*m))^12,x,method=_RETURNVERBOSE)

[Out]

1/13*(b^12*(x^(1+12*m))^12*x^(12*m)*x+13*a*b^11*(x^(1+12*m))^11*x^(12*m)*x+78*a^2*b^10*(x^(1+12*m))^10*x^(12*m
)*x+286*a^3*b^9*(x^(1+12*m))^9*x^(12*m)*x+715*a^4*b^8*(x^(1+12*m))^8*x^(12*m)*x+1287*a^5*b^7*(x^(1+12*m))^7*x^
(12*m)*x+1716*a^6*b^6*(x^(1+12*m))^6*x^(12*m)*x+1716*a^7*b^5*(x^(1+12*m))^5*x^(12*m)*x+1287*a^8*b^4*(x^(1+12*m
))^4*x^(12*m)*x+715*a^9*b^3*(x^(1+12*m))^3*x^(12*m)*x+286*a^10*b^2*(x^(1+12*m))^2*x^(12*m)*x+78*b*a^11*x^(1+12
*m)*x^(12*m)*x+13*a^12*x^(12*m)*x)/(1+12*m)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 7.19 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {b^{12} x^{156 \, m + 13} + 13 \, a b^{11} x^{144 \, m + 12} + 78 \, a^{2} b^{10} x^{132 \, m + 11} + 286 \, a^{3} b^{9} x^{120 \, m + 10} + 715 \, a^{4} b^{8} x^{108 \, m + 9} + 1287 \, a^{5} b^{7} x^{96 \, m + 8} + 1716 \, a^{6} b^{6} x^{84 \, m + 7} + 1716 \, a^{7} b^{5} x^{72 \, m + 6} + 1287 \, a^{8} b^{4} x^{60 \, m + 5} + 715 \, a^{9} b^{3} x^{48 \, m + 4} + 286 \, a^{10} b^{2} x^{36 \, m + 3} + 78 \, a^{11} b x^{24 \, m + 2} + 13 \, a^{12} x^{12 \, m + 1}}{13 \, {\left (12 \, m + 1\right )}} \]

[In]

integrate(x^(12*m)*(a+b*x^(1+12*m))^12,x, algorithm="fricas")

[Out]

1/13*(b^12*x^(156*m + 13) + 13*a*b^11*x^(144*m + 12) + 78*a^2*b^10*x^(132*m + 11) + 286*a^3*b^9*x^(120*m + 10)
+ 715*a^4*b^8*x^(108*m + 9) + 1287*a^5*b^7*x^(96*m + 8) + 1716*a^6*b^6*x^(84*m + 7) + 1716*a^7*b^5*x^(72*m +
6) + 1287*a^8*b^4*x^(60*m + 5) + 715*a^9*b^3*x^(48*m + 4) + 286*a^10*b^2*x^(36*m + 3) + 78*a^11*b*x^(24*m + 2)
+ 13*a^12*x^(12*m + 1))/(12*m + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (19) = 38\).

Time = 6.22 (sec) , antiderivative size = 345, normalized size of antiderivative = 12.78 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\begin {cases} \frac {13 a^{12} x x^{12 m}}{156 m + 13} + \frac {78 a^{11} b x x^{12 m} x^{12 m + 1}}{156 m + 13} + \frac {286 a^{10} b^{2} x x^{12 m} x^{24 m + 2}}{156 m + 13} + \frac {715 a^{9} b^{3} x x^{12 m} x^{36 m + 3}}{156 m + 13} + \frac {1287 a^{8} b^{4} x x^{12 m} x^{48 m + 4}}{156 m + 13} + \frac {1716 a^{7} b^{5} x x^{12 m} x^{60 m + 5}}{156 m + 13} + \frac {1716 a^{6} b^{6} x x^{12 m} x^{72 m + 6}}{156 m + 13} + \frac {1287 a^{5} b^{7} x x^{12 m} x^{84 m + 7}}{156 m + 13} + \frac {715 a^{4} b^{8} x x^{12 m} x^{96 m + 8}}{156 m + 13} + \frac {286 a^{3} b^{9} x x^{12 m} x^{108 m + 9}}{156 m + 13} + \frac {78 a^{2} b^{10} x x^{12 m} x^{120 m + 10}}{156 m + 13} + \frac {13 a b^{11} x x^{12 m} x^{132 m + 11}}{156 m + 13} + \frac {b^{12} x x^{12 m} x^{144 m + 12}}{156 m + 13} & \text {for}\: m \neq - \frac {1}{12} \\\left (a + b\right )^{12} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(12*m)*(a+b*x**(1+12*m))**12,x)

[Out]

Piecewise((13*a**12*x*x**(12*m)/(156*m + 13) + 78*a**11*b*x*x**(12*m)*x**(12*m + 1)/(156*m + 13) + 286*a**10*b
**2*x*x**(12*m)*x**(24*m + 2)/(156*m + 13) + 715*a**9*b**3*x*x**(12*m)*x**(36*m + 3)/(156*m + 13) + 1287*a**8*
b**4*x*x**(12*m)*x**(48*m + 4)/(156*m + 13) + 1716*a**7*b**5*x*x**(12*m)*x**(60*m + 5)/(156*m + 13) + 1716*a**
6*b**6*x*x**(12*m)*x**(72*m + 6)/(156*m + 13) + 1287*a**5*b**7*x*x**(12*m)*x**(84*m + 7)/(156*m + 13) + 715*a*
*4*b**8*x*x**(12*m)*x**(96*m + 8)/(156*m + 13) + 286*a**3*b**9*x*x**(12*m)*x**(108*m + 9)/(156*m + 13) + 78*a*
*2*b**10*x*x**(12*m)*x**(120*m + 10)/(156*m + 13) + 13*a*b**11*x*x**(12*m)*x**(132*m + 11)/(156*m + 13) + b**1
2*x*x**(12*m)*x**(144*m + 12)/(156*m + 13), Ne(m, -1/12)), ((a + b)**12*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {{\left (b x^{12 \, m + 1} + a\right )}^{13}}{13 \, b {\left (12 \, m + 1\right )}} \]

[In]

integrate(x^(12*m)*(a+b*x^(1+12*m))^12,x, algorithm="maxima")

[Out]

1/13*(b*x^(12*m + 1) + a)^13/(b*(12*m + 1))

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {{\left (b x^{12 \, m + 1} + a\right )}^{13}}{13 \, b {\left (12 \, m + 1\right )}} \]

[In]

integrate(x^(12*m)*(a+b*x^(1+12*m))^12,x, algorithm="giac")

[Out]

1/13*(b*x^(12*m + 1) + a)^13/(b*(12*m + 1))

Mupad [B] (verification not implemented)

Time = 6.58 (sec) , antiderivative size = 285, normalized size of antiderivative = 10.56 \[ \int x^{12 m} \left (a+b x^{1+12 m}\right )^{12} \, dx=\frac {b^{12}\,x^{156\,m}\,x^{13}}{156\,m+13}+\frac {a^{12}\,x\,x^{12\,m}}{12\,m+1}+\frac {6\,a^{11}\,b\,x^{24\,m}\,x^2}{12\,m+1}+\frac {a\,b^{11}\,x^{144\,m}\,x^{12}}{12\,m+1}+\frac {22\,a^{10}\,b^2\,x^{36\,m}\,x^3}{12\,m+1}+\frac {55\,a^9\,b^3\,x^{48\,m}\,x^4}{12\,m+1}+\frac {99\,a^8\,b^4\,x^{60\,m}\,x^5}{12\,m+1}+\frac {132\,a^7\,b^5\,x^{72\,m}\,x^6}{12\,m+1}+\frac {132\,a^6\,b^6\,x^{84\,m}\,x^7}{12\,m+1}+\frac {99\,a^5\,b^7\,x^{96\,m}\,x^8}{12\,m+1}+\frac {55\,a^4\,b^8\,x^{108\,m}\,x^9}{12\,m+1}+\frac {22\,a^3\,b^9\,x^{120\,m}\,x^{10}}{12\,m+1}+\frac {6\,a^2\,b^{10}\,x^{132\,m}\,x^{11}}{12\,m+1} \]

[In]

int(x^(12*m)*(a + b*x^(12*m + 1))^12,x)

[Out]

(b^12*x^(156*m)*x^13)/(156*m + 13) + (a^12*x*x^(12*m))/(12*m + 1) + (6*a^11*b*x^(24*m)*x^2)/(12*m + 1) + (a*b^
11*x^(144*m)*x^12)/(12*m + 1) + (22*a^10*b^2*x^(36*m)*x^3)/(12*m + 1) + (55*a^9*b^3*x^(48*m)*x^4)/(12*m + 1) +
(99*a^8*b^4*x^(60*m)*x^5)/(12*m + 1) + (132*a^7*b^5*x^(72*m)*x^6)/(12*m + 1) + (132*a^6*b^6*x^(84*m)*x^7)/(12
*m + 1) + (99*a^5*b^7*x^(96*m)*x^8)/(12*m + 1) + (55*a^4*b^8*x^(108*m)*x^9)/(12*m + 1) + (22*a^3*b^9*x^(120*m)
*x^10)/(12*m + 1) + (6*a^2*b^10*x^(132*m)*x^11)/(12*m + 1)