\(\int x^{12 (-1+m)} (a x+b x^{2+12 m})^{12} \, dx\) [331]

Optimal result

Integrand size = 23, antiderivative size = 27 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b (1+12 m)} \]

[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 = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1598, 267} \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \]

[In]

Int[x^(12*(-1 + m))*(a*x + b*x^(2 + 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]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int x^{12+12 (-1+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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b+156 b m} \]

[In]

Integrate[x^(12*(-1 + m))*(a*x + b*x^(2 + 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. \(337\) vs. \(2(25)=50\).

Time = 53.75 (sec) , antiderivative size = 338, normalized size of antiderivative = 12.52

method	result	size
parallelrisch	\(\frac {13 a^{12} x^{-12+12 m} x^{13}+78 b \,a^{11} x^{-12+12 m} x^{2+12 m} x^{12}+286 a^{10} b^{2} x^{-12+12 m} x^{4+24 m} x^{11}+715 a^{9} b^{3} x^{-12+12 m} x^{6+36 m} x^{10}+1287 a^{8} b^{4} x^{-12+12 m} x^{8+48 m} x^{9}+1716 a^{7} b^{5} x^{-12+12 m} x^{10+60 m} x^{8}+1716 a^{6} b^{6} x^{-12+12 m} x^{12+72 m} x^{7}+1287 a^{5} b^{7} x^{-12+12 m} x^{14+84 m} x^{6}+715 a^{4} b^{8} x^{-12+12 m} x^{16+96 m} x^{5}+286 a^{3} b^{9} x^{-12+12 m} x^{18+108 m} x^{4}+78 a^{2} b^{10} x^{-12+12 m} x^{20+120 m} x^{3}+13 a \,b^{11} x^{-12+12 m} x^{22+132 m} x^{2}+b^{12} x^{-12+12 m} x^{24+144 m} x}{13+156 m}\)	\(338\)
risch	\(\frac {b^{12} x^{26+156 m}}{13 \left (1+12 m \right ) x^{13}}+\frac {a \,b^{11} x^{24+144 m}}{\left (1+12 m \right ) x^{12}}+\frac {6 a^{2} b^{10} x^{22+132 m}}{\left (1+12 m \right ) x^{11}}+\frac {22 a^{3} b^{9} x^{20+120 m}}{\left (1+12 m \right ) x^{10}}+\frac {55 a^{4} b^{8} x^{18+108 m}}{\left (1+12 m \right ) x^{9}}+\frac {99 a^{5} b^{7} x^{16+96 m}}{\left (1+12 m \right ) x^{8}}+\frac {132 a^{6} b^{6} x^{14+84 m}}{\left (1+12 m \right ) x^{7}}+\frac {132 a^{7} b^{5} x^{12+72 m}}{\left (1+12 m \right ) x^{6}}+\frac {99 a^{8} b^{4} x^{10+60 m}}{\left (1+12 m \right ) x^{5}}+\frac {55 a^{9} b^{3} x^{8+48 m}}{\left (1+12 m \right ) x^{4}}+\frac {22 a^{10} b^{2} x^{6+36 m}}{\left (1+12 m \right ) x^{3}}+\frac {6 b \,a^{11} x^{4+24 m}}{\left (1+12 m \right ) x^{2}}+\frac {a^{12} x^{2+12 m}}{\left (1+12 m \right ) x}\)	\(339\)

[In]

int(x^(-12+12*m)*(a*x+b*x^(2+12*m))^12,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 231, normalized size of antiderivative = 8.56 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {13 \, a^{12} x^{12} x^{12 \, m + 2} + 78 \, a^{11} b x^{11} x^{24 \, m + 4} + 286 \, a^{10} b^{2} x^{10} x^{36 \, m + 6} + 715 \, a^{9} b^{3} x^{9} x^{48 \, m + 8} + 1287 \, a^{8} b^{4} x^{8} x^{60 \, m + 10} + 1716 \, a^{7} b^{5} x^{7} x^{72 \, m + 12} + 1716 \, a^{6} b^{6} x^{6} x^{84 \, m + 14} + 1287 \, a^{5} b^{7} x^{5} x^{96 \, m + 16} + 715 \, a^{4} b^{8} x^{4} x^{108 \, m + 18} + 286 \, a^{3} b^{9} x^{3} x^{120 \, m + 20} + 78 \, a^{2} b^{10} x^{2} x^{132 \, m + 22} + 13 \, a b^{11} x x^{144 \, m + 24} + b^{12} x^{156 \, m + 26}}{13 \, {\left (12 \, m + 1\right )} x^{13}} \]

[In]

integrate(x^(-12+12*m)*(a*x+b*x^(2+12*m))^12,x, algorithm="fricas")

[Out]

1/13*(13*a^12*x^12*x^(12*m + 2) + 78*a^11*b*x^11*x^(24*m + 4) + 286*a^10*b^2*x^10*x^(36*m + 6) + 715*a^9*b^3*x
^9*x^(48*m + 8) + 1287*a^8*b^4*x^8*x^(60*m + 10) + 1716*a^7*b^5*x^7*x^(72*m + 12) + 1716*a^6*b^6*x^6*x^(84*m +
14) + 1287*a^5*b^7*x^5*x^(96*m + 16) + 715*a^4*b^8*x^4*x^(108*m + 18) + 286*a^3*b^9*x^3*x^(120*m + 20) + 78*a
^2*b^10*x^2*x^(132*m + 22) + 13*a*b^11*x*x^(144*m + 24) + b^12*x^(156*m + 26))/((12*m + 1)*x^13)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (19) = 38\).

Time = 6.30 (sec) , antiderivative size = 520, normalized size of antiderivative = 19.26 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\begin {cases} \frac {13 a^{12} x^{13} x^{12 m - 12}}{156 m + 13} + \frac {78 a^{11} b x^{12} x^{12 m - 12} x^{12 m + 2}}{156 m + 13} + \frac {286 a^{10} b^{2} x^{11} x^{12 m - 12} x^{24 m + 4}}{156 m + 13} + \frac {715 a^{9} b^{3} x^{10} x^{12 m - 12} x^{36 m + 6}}{156 m + 13} + \frac {1287 a^{8} b^{4} x^{9} x^{12 m - 12} x^{48 m + 8}}{156 m + 13} + \frac {1716 a^{7} b^{5} x^{8} x^{12 m - 12} x^{60 m + 10}}{156 m + 13} + \frac {1716 a^{6} b^{6} x^{7} x^{12 m - 12} x^{72 m + 12}}{156 m + 13} + \frac {1287 a^{5} b^{7} x^{6} x^{12 m - 12} x^{84 m + 14}}{156 m + 13} + \frac {715 a^{4} b^{8} x^{5} x^{12 m - 12} x^{96 m + 16}}{156 m + 13} + \frac {286 a^{3} b^{9} x^{4} x^{12 m - 12} x^{108 m + 18}}{156 m + 13} + \frac {78 a^{2} b^{10} x^{3} x^{12 m - 12} x^{120 m + 20}}{156 m + 13} + \frac {13 a b^{11} x^{2} x^{12 m - 12} x^{132 m + 22}}{156 m + 13} + \frac {b^{12} x x^{12 m - 12} x^{144 m + 24}}{156 m + 13} & \text {for}\: m \neq - \frac {1}{12} \\a^{12} \log {\left (x \right )} + 12 a^{11} b \log {\left (x \right )} + 66 a^{10} b^{2} \log {\left (x \right )} + 220 a^{9} b^{3} \log {\left (x \right )} + 495 a^{8} b^{4} \log {\left (x \right )} + 792 a^{7} b^{5} \log {\left (x \right )} + 924 a^{6} b^{6} \log {\left (x \right )} + 792 a^{5} b^{7} \log {\left (x \right )} + 495 a^{4} b^{8} \log {\left (x \right )} + 220 a^{3} b^{9} \log {\left (x \right )} + 66 a^{2} b^{10} \log {\left (x \right )} + 12 a b^{11} \log {\left (x \right )} + b^{12} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-12+12*m)*(a*x+b*x**(2+12*m))**12,x)

[Out]

Piecewise((13*a**12*x**13*x**(12*m - 12)/(156*m + 13) + 78*a**11*b*x**12*x**(12*m - 12)*x**(12*m + 2)/(156*m +
13) + 286*a**10*b**2*x**11*x**(12*m - 12)*x**(24*m + 4)/(156*m + 13) + 715*a**9*b**3*x**10*x**(12*m - 12)*x**
(36*m + 6)/(156*m + 13) + 1287*a**8*b**4*x**9*x**(12*m - 12)*x**(48*m + 8)/(156*m + 13) + 1716*a**7*b**5*x**8*
x**(12*m - 12)*x**(60*m + 10)/(156*m + 13) + 1716*a**6*b**6*x**7*x**(12*m - 12)*x**(72*m + 12)/(156*m + 13) +
1287*a**5*b**7*x**6*x**(12*m - 12)*x**(84*m + 14)/(156*m + 13) + 715*a**4*b**8*x**5*x**(12*m - 12)*x**(96*m +
16)/(156*m + 13) + 286*a**3*b**9*x**4*x**(12*m - 12)*x**(108*m + 18)/(156*m + 13) + 78*a**2*b**10*x**3*x**(12*
m - 12)*x**(120*m + 20)/(156*m + 13) + 13*a*b**11*x**2*x**(12*m - 12)*x**(132*m + 22)/(156*m + 13) + b**12*x*x
**(12*m - 12)*x**(144*m + 24)/(156*m + 13), Ne(m, -1/12)), (a**12*log(x) + 12*a**11*b*log(x) + 66*a**10*b**2*l
og(x) + 220*a**9*b**3*log(x) + 495*a**8*b**4*log(x) + 792*a**7*b**5*log(x) + 924*a**6*b**6*log(x) + 792*a**5*b
**7*log(x) + 495*a**4*b**8*log(x) + 220*a**3*b**9*log(x) + 66*a**2*b**10*log(x) + 12*a*b**11*log(x) + b**12*lo
g(x), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (25) = 50\).

Time = 0.23 (sec) , antiderivative size = 275, normalized size of antiderivative = 10.19 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {b^{12} x^{156 \, m + 13}}{13 \, {\left (12 \, m + 1\right )}} + \frac {a b^{11} x^{144 \, m + 12}}{12 \, m + 1} + \frac {6 \, a^{2} b^{10} x^{132 \, m + 11}}{12 \, m + 1} + \frac {22 \, a^{3} b^{9} x^{120 \, m + 10}}{12 \, m + 1} + \frac {55 \, a^{4} b^{8} x^{108 \, m + 9}}{12 \, m + 1} + \frac {99 \, a^{5} b^{7} x^{96 \, m + 8}}{12 \, m + 1} + \frac {132 \, a^{6} b^{6} x^{84 \, m + 7}}{12 \, m + 1} + \frac {132 \, a^{7} b^{5} x^{72 \, m + 6}}{12 \, m + 1} + \frac {99 \, a^{8} b^{4} x^{60 \, m + 5}}{12 \, m + 1} + \frac {55 \, a^{9} b^{3} x^{48 \, m + 4}}{12 \, m + 1} + \frac {22 \, a^{10} b^{2} x^{36 \, m + 3}}{12 \, m + 1} + \frac {6 \, a^{11} b x^{24 \, m + 2}}{12 \, m + 1} + \frac {a^{12} x^{12 \, m + 1}}{12 \, m + 1} \]

[In]

integrate(x^(-12+12*m)*(a*x+b*x^(2+12*m))^12,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (25) = 50\).

Time = 0.40 (sec) , antiderivative size = 285, normalized size of antiderivative = 10.56 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {13 \, a^{12} x^{12} e^{\left (12 \, m \log \left (x\right ) + 2 \, \log \left (x\right )\right )} + 78 \, a^{11} b x^{11} e^{\left (24 \, m \log \left (x\right ) + 4 \, \log \left (x\right )\right )} + 286 \, a^{10} b^{2} x^{10} e^{\left (36 \, m \log \left (x\right ) + 6 \, \log \left (x\right )\right )} + 715 \, a^{9} b^{3} x^{9} e^{\left (48 \, m \log \left (x\right ) + 8 \, \log \left (x\right )\right )} + 1287 \, a^{8} b^{4} x^{8} e^{\left (60 \, m \log \left (x\right ) + 10 \, \log \left (x\right )\right )} + 1716 \, a^{7} b^{5} x^{7} e^{\left (72 \, m \log \left (x\right ) + 12 \, \log \left (x\right )\right )} + 1716 \, a^{6} b^{6} x^{6} e^{\left (84 \, m \log \left (x\right ) + 14 \, \log \left (x\right )\right )} + 1287 \, a^{5} b^{7} x^{5} e^{\left (96 \, m \log \left (x\right ) + 16 \, \log \left (x\right )\right )} + 715 \, a^{4} b^{8} x^{4} e^{\left (108 \, m \log \left (x\right ) + 18 \, \log \left (x\right )\right )} + 286 \, a^{3} b^{9} x^{3} e^{\left (120 \, m \log \left (x\right ) + 20 \, \log \left (x\right )\right )} + 78 \, a^{2} b^{10} x^{2} e^{\left (132 \, m \log \left (x\right ) + 22 \, \log \left (x\right )\right )} + 13 \, a b^{11} x e^{\left (144 \, m \log \left (x\right ) + 24 \, \log \left (x\right )\right )} + b^{12} e^{\left (156 \, m \log \left (x\right ) + 26 \, \log \left (x\right )\right )}}{13 \, {\left (12 \, m x^{13} + x^{13}\right )}} \]

[In]

integrate(x^(-12+12*m)*(a*x+b*x^(2+12*m))^12,x, algorithm="giac")

[Out]

1/13*(13*a^12*x^12*e^(12*m*log(x) + 2*log(x)) + 78*a^11*b*x^11*e^(24*m*log(x) + 4*log(x)) + 286*a^10*b^2*x^10*
e^(36*m*log(x) + 6*log(x)) + 715*a^9*b^3*x^9*e^(48*m*log(x) + 8*log(x)) + 1287*a^8*b^4*x^8*e^(60*m*log(x) + 10
*log(x)) + 1716*a^7*b^5*x^7*e^(72*m*log(x) + 12*log(x)) + 1716*a^6*b^6*x^6*e^(84*m*log(x) + 14*log(x)) + 1287*
a^5*b^7*x^5*e^(96*m*log(x) + 16*log(x)) + 715*a^4*b^8*x^4*e^(108*m*log(x) + 18*log(x)) + 286*a^3*b^9*x^3*e^(12
0*m*log(x) + 20*log(x)) + 78*a^2*b^10*x^2*e^(132*m*log(x) + 22*log(x)) + 13*a*b^11*x*e^(144*m*log(x) + 24*log(
x)) + b^12*e^(156*m*log(x) + 26*log(x)))/(12*m*x^13 + x^13)

Mupad [B] (verification not implemented)

Time = 9.85 (sec) , antiderivative size = 287, normalized size of antiderivative = 10.63 \[ \int x^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx=\frac {b^{12}\,x^{156\,m}\,x^{13}}{156\,m+13}+\frac {13\,a^{12}\,x\,x^{12\,m}}{156\,m+13}+\frac {78\,a^{11}\,b\,x^{24\,m}\,x^2}{156\,m+13}+\frac {13\,a\,b^{11}\,x^{144\,m}\,x^{12}}{156\,m+13}+\frac {286\,a^{10}\,b^2\,x^{36\,m}\,x^3}{156\,m+13}+\frac {715\,a^9\,b^3\,x^{48\,m}\,x^4}{156\,m+13}+\frac {1287\,a^8\,b^4\,x^{60\,m}\,x^5}{156\,m+13}+\frac {1716\,a^7\,b^5\,x^{72\,m}\,x^6}{156\,m+13}+\frac {1716\,a^6\,b^6\,x^{84\,m}\,x^7}{156\,m+13}+\frac {1287\,a^5\,b^7\,x^{96\,m}\,x^8}{156\,m+13}+\frac {715\,a^4\,b^8\,x^{108\,m}\,x^9}{156\,m+13}+\frac {286\,a^3\,b^9\,x^{120\,m}\,x^{10}}{156\,m+13}+\frac {78\,a^2\,b^{10}\,x^{132\,m}\,x^{11}}{156\,m+13} \]

[In]

int(x^(12*m - 12)*(a*x + b*x^(12*m + 2))^12,x)

[Out]

(b^12*x^(156*m)*x^13)/(156*m + 13) + (13*a^12*x*x^(12*m))/(156*m + 13) + (78*a^11*b*x^(24*m)*x^2)/(156*m + 13)
+ (13*a*b^11*x^(144*m)*x^12)/(156*m + 13) + (286*a^10*b^2*x^(36*m)*x^3)/(156*m + 13) + (715*a^9*b^3*x^(48*m)*
x^4)/(156*m + 13) + (1287*a^8*b^4*x^(60*m)*x^5)/(156*m + 13) + (1716*a^7*b^5*x^(72*m)*x^6)/(156*m + 13) + (171
6*a^6*b^6*x^(84*m)*x^7)/(156*m + 13) + (1287*a^5*b^7*x^(96*m)*x^8)/(156*m + 13) + (715*a^4*b^8*x^(108*m)*x^9)/
(156*m + 13) + (286*a^3*b^9*x^(120*m)*x^10)/(156*m + 13) + (78*a^2*b^10*x^(132*m)*x^11)/(156*m + 13)