\(\int x^{12} (a x^2+b x^{39})^{12} \, dx\) [330]

Optimal result

Integrand size = 17, antiderivative size = 16 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {\left (a+b x^{37}\right )^{13}}{481 b} \] Output:

1/481*(b*x^37+a)^13/b
 

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 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {a^{12} x^{37}}{37}+\frac {6}{37} a^{11} b x^{74}+\frac {22}{37} a^{10} b^2 x^{111}+\frac {55}{37} a^9 b^3 x^{148}+\frac {99}{37} a^8 b^4 x^{185}+\frac {132}{37} a^7 b^5 x^{222}+\frac {132}{37} a^6 b^6 x^{259}+\frac {99}{37} a^5 b^7 x^{296}+\frac {55}{37} a^4 b^8 x^{333}+\frac {22}{37} a^3 b^9 x^{370}+\frac {6}{37} a^2 b^{10} x^{407}+\frac {1}{37} a b^{11} x^{444}+\frac {b^{12} x^{481}}{481} \] Input:

Integrate[x^12*(a*x^2 + b*x^39)^12,x]
 

Output:

(a^12*x^37)/37 + (6*a^11*b*x^74)/37 + (22*a^10*b^2*x^111)/37 + (55*a^9*b^3 
*x^148)/37 + (99*a^8*b^4*x^185)/37 + (132*a^7*b^5*x^222)/37 + (132*a^6*b^6 
*x^259)/37 + (99*a^5*b^7*x^296)/37 + (55*a^4*b^8*x^333)/37 + (22*a^3*b^9*x 
^370)/37 + (6*a^2*b^10*x^407)/37 + (a*b^11*x^444)/37 + (b^12*x^481)/481
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {9, 793}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int x^{36} \left (a+b x^{37}\right )^{12}dx\)

\(\Big \downarrow \) 793

\(\displaystyle \frac {\left (a+b x^{37}\right )^{13}}{481 b}\)

Input:

Int[x^12*(a*x^2 + b*x^39)^12,x]
 

Output:

(a + b*x^37)^13/(481*b)
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

rule 793

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]
 

Maple [B] (verified)

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

Time = 1.75 (sec) , antiderivative size = 135, normalized size of antiderivative = 8.44

method	result	size
default	\(\frac {99}{37} a^{8} b^{4} x^{185}+\frac {22}{37} a^{3} b^{9} x^{370}+\frac {6}{37} a^{11} b \,x^{74}+\frac {1}{37} a \,b^{11} x^{444}+\frac {132}{37} a^{6} b^{6} x^{259}+\frac {22}{37} a^{10} b^{2} x^{111}+\frac {55}{37} a^{4} b^{8} x^{333}+\frac {1}{481} b^{12} x^{481}+\frac {132}{37} a^{7} b^{5} x^{222}+\frac {55}{37} a^{9} b^{3} x^{148}+\frac {99}{37} a^{5} b^{7} x^{296}+\frac {1}{37} a^{12} x^{37}+\frac {6}{37} a^{2} b^{10} x^{407}\)	\(135\)
parallelrisch	\(\frac {99}{37} a^{8} b^{4} x^{185}+\frac {22}{37} a^{3} b^{9} x^{370}+\frac {6}{37} a^{11} b \,x^{74}+\frac {1}{37} a \,b^{11} x^{444}+\frac {132}{37} a^{6} b^{6} x^{259}+\frac {22}{37} a^{10} b^{2} x^{111}+\frac {55}{37} a^{4} b^{8} x^{333}+\frac {1}{481} b^{12} x^{481}+\frac {132}{37} a^{7} b^{5} x^{222}+\frac {55}{37} a^{9} b^{3} x^{148}+\frac {99}{37} a^{5} b^{7} x^{296}+\frac {1}{37} a^{12} x^{37}+\frac {6}{37} a^{2} b^{10} x^{407}\)	\(135\)
gosper	\(\frac {x^{37} \left (b^{12} x^{444}+13 a \,b^{11} x^{407}+78 a^{2} b^{10} x^{370}+286 a^{3} b^{9} x^{333}+715 a^{4} b^{8} x^{296}+1287 a^{5} b^{7} x^{259}+1716 a^{6} b^{6} x^{222}+1716 a^{7} b^{5} x^{185}+1287 a^{8} b^{4} x^{148}+715 a^{9} b^{3} x^{111}+286 a^{10} b^{2} x^{74}+78 a^{11} b \,x^{37}+13 a^{12}\right )}{481}\)	\(136\)
risch	\(\frac {b^{12} x^{481}}{481}+\frac {a \,b^{11} x^{444}}{37}+\frac {6 a^{2} b^{10} x^{407}}{37}+\frac {22 a^{3} b^{9} x^{370}}{37}+\frac {55 a^{4} b^{8} x^{333}}{37}+\frac {99 a^{5} b^{7} x^{296}}{37}+\frac {132 a^{6} b^{6} x^{259}}{37}+\frac {132 a^{7} b^{5} x^{222}}{37}+\frac {99 a^{8} b^{4} x^{185}}{37}+\frac {55 a^{9} b^{3} x^{148}}{37}+\frac {22 a^{10} b^{2} x^{111}}{37}+\frac {6 a^{11} b \,x^{74}}{37}+\frac {a^{12} x^{37}}{37}+\frac {a^{13}}{481 b}\)	\(143\)
orering	\(\frac {x^{13} \left (b^{12} x^{444}+13 a \,b^{11} x^{407}+78 a^{2} b^{10} x^{370}+286 a^{3} b^{9} x^{333}+715 a^{4} b^{8} x^{296}+1287 a^{5} b^{7} x^{259}+1716 a^{6} b^{6} x^{222}+1716 a^{7} b^{5} x^{185}+1287 a^{8} b^{4} x^{148}+715 a^{9} b^{3} x^{111}+286 a^{10} b^{2} x^{74}+78 a^{11} b \,x^{37}+13 a^{12}\right ) \left (b \,x^{39}+a \,x^{2}\right )^{12}}{481 \left (b \,x^{37}+a \right )^{12}}\)	\(158\)

Input:

int(x^12*(b*x^39+a*x^2)^12,x,method=_RETURNVERBOSE)
 

Output:

99/37*a^8*b^4*x^185+22/37*a^3*b^9*x^370+6/37*a^11*b*x^74+1/37*a*b^11*x^444 
+132/37*a^6*b^6*x^259+22/37*a^10*b^2*x^111+55/37*a^4*b^8*x^333+1/481*b^12* 
x^481+132/37*a^7*b^5*x^222+55/37*a^9*b^3*x^148+99/37*a^5*b^7*x^296+1/37*a^ 
12*x^37+6/37*a^2*b^10*x^407
 

Fricas [B] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 8.38 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {1}{481} \, b^{12} x^{481} + \frac {1}{37} \, a b^{11} x^{444} + \frac {6}{37} \, a^{2} b^{10} x^{407} + \frac {22}{37} \, a^{3} b^{9} x^{370} + \frac {55}{37} \, a^{4} b^{8} x^{333} + \frac {99}{37} \, a^{5} b^{7} x^{296} + \frac {132}{37} \, a^{6} b^{6} x^{259} + \frac {132}{37} \, a^{7} b^{5} x^{222} + \frac {99}{37} \, a^{8} b^{4} x^{185} + \frac {55}{37} \, a^{9} b^{3} x^{148} + \frac {22}{37} \, a^{10} b^{2} x^{111} + \frac {6}{37} \, a^{11} b x^{74} + \frac {1}{37} \, a^{12} x^{37} \] Input:

integrate(x^12*(b*x^39+a*x^2)^12,x, algorithm="fricas")
 

Output:

1/481*b^12*x^481 + 1/37*a*b^11*x^444 + 6/37*a^2*b^10*x^407 + 22/37*a^3*b^9 
*x^370 + 55/37*a^4*b^8*x^333 + 99/37*a^5*b^7*x^296 + 132/37*a^6*b^6*x^259 
+ 132/37*a^7*b^5*x^222 + 99/37*a^8*b^4*x^185 + 55/37*a^9*b^3*x^148 + 22/37 
*a^10*b^2*x^111 + 6/37*a^11*b*x^74 + 1/37*a^12*x^37
 

Sympy [B] (verification not implemented)

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

Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 10.00 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {a^{12} x^{37}}{37} + \frac {6 a^{11} b x^{74}}{37} + \frac {22 a^{10} b^{2} x^{111}}{37} + \frac {55 a^{9} b^{3} x^{148}}{37} + \frac {99 a^{8} b^{4} x^{185}}{37} + \frac {132 a^{7} b^{5} x^{222}}{37} + \frac {132 a^{6} b^{6} x^{259}}{37} + \frac {99 a^{5} b^{7} x^{296}}{37} + \frac {55 a^{4} b^{8} x^{333}}{37} + \frac {22 a^{3} b^{9} x^{370}}{37} + \frac {6 a^{2} b^{10} x^{407}}{37} + \frac {a b^{11} x^{444}}{37} + \frac {b^{12} x^{481}}{481} \] Input:

integrate(x**12*(b*x**39+a*x**2)**12,x)
 

Output:

a**12*x**37/37 + 6*a**11*b*x**74/37 + 22*a**10*b**2*x**111/37 + 55*a**9*b* 
*3*x**148/37 + 99*a**8*b**4*x**185/37 + 132*a**7*b**5*x**222/37 + 132*a**6 
*b**6*x**259/37 + 99*a**5*b**7*x**296/37 + 55*a**4*b**8*x**333/37 + 22*a** 
3*b**9*x**370/37 + 6*a**2*b**10*x**407/37 + a*b**11*x**444/37 + b**12*x**4 
81/481
 

Maxima [B] (verification not implemented)

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

Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 8.38 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {1}{481} \, b^{12} x^{481} + \frac {1}{37} \, a b^{11} x^{444} + \frac {6}{37} \, a^{2} b^{10} x^{407} + \frac {22}{37} \, a^{3} b^{9} x^{370} + \frac {55}{37} \, a^{4} b^{8} x^{333} + \frac {99}{37} \, a^{5} b^{7} x^{296} + \frac {132}{37} \, a^{6} b^{6} x^{259} + \frac {132}{37} \, a^{7} b^{5} x^{222} + \frac {99}{37} \, a^{8} b^{4} x^{185} + \frac {55}{37} \, a^{9} b^{3} x^{148} + \frac {22}{37} \, a^{10} b^{2} x^{111} + \frac {6}{37} \, a^{11} b x^{74} + \frac {1}{37} \, a^{12} x^{37} \] Input:

integrate(x^12*(b*x^39+a*x^2)^12,x, algorithm="maxima")
 

Output:

1/481*b^12*x^481 + 1/37*a*b^11*x^444 + 6/37*a^2*b^10*x^407 + 22/37*a^3*b^9 
*x^370 + 55/37*a^4*b^8*x^333 + 99/37*a^5*b^7*x^296 + 132/37*a^6*b^6*x^259 
+ 132/37*a^7*b^5*x^222 + 99/37*a^8*b^4*x^185 + 55/37*a^9*b^3*x^148 + 22/37 
*a^10*b^2*x^111 + 6/37*a^11*b*x^74 + 1/37*a^12*x^37
 

Giac [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 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {1}{481} \, b^{12} x^{481} + \frac {1}{37} \, a b^{11} x^{444} + \frac {6}{37} \, a^{2} b^{10} x^{407} + \frac {22}{37} \, a^{3} b^{9} x^{370} + \frac {55}{37} \, a^{4} b^{8} x^{333} + \frac {99}{37} \, a^{5} b^{7} x^{296} + \frac {132}{37} \, a^{6} b^{6} x^{259} + \frac {132}{37} \, a^{7} b^{5} x^{222} + \frac {99}{37} \, a^{8} b^{4} x^{185} + \frac {55}{37} \, a^{9} b^{3} x^{148} + \frac {22}{37} \, a^{10} b^{2} x^{111} + \frac {6}{37} \, a^{11} b x^{74} + \frac {1}{37} \, a^{12} x^{37} \] Input:

integrate(x^12*(b*x^39+a*x^2)^12,x, algorithm="giac")
 

Output:

1/481*b^12*x^481 + 1/37*a*b^11*x^444 + 6/37*a^2*b^10*x^407 + 22/37*a^3*b^9 
*x^370 + 55/37*a^4*b^8*x^333 + 99/37*a^5*b^7*x^296 + 132/37*a^6*b^6*x^259 
+ 132/37*a^7*b^5*x^222 + 99/37*a^8*b^4*x^185 + 55/37*a^9*b^3*x^148 + 22/37 
*a^10*b^2*x^111 + 6/37*a^11*b*x^74 + 1/37*a^12*x^37
 

Mupad [B] (verification not implemented)

Time = 9.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 8.38 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {a^{12}\,x^{37}}{37}+\frac {6\,a^{11}\,b\,x^{74}}{37}+\frac {22\,a^{10}\,b^2\,x^{111}}{37}+\frac {55\,a^9\,b^3\,x^{148}}{37}+\frac {99\,a^8\,b^4\,x^{185}}{37}+\frac {132\,a^7\,b^5\,x^{222}}{37}+\frac {132\,a^6\,b^6\,x^{259}}{37}+\frac {99\,a^5\,b^7\,x^{296}}{37}+\frac {55\,a^4\,b^8\,x^{333}}{37}+\frac {22\,a^3\,b^9\,x^{370}}{37}+\frac {6\,a^2\,b^{10}\,x^{407}}{37}+\frac {a\,b^{11}\,x^{444}}{37}+\frac {b^{12}\,x^{481}}{481} \] Input:

int(x^12*(a*x^2 + b*x^39)^12,x)
 

Output:

(a^12*x^37)/37 + (b^12*x^481)/481 + (6*a^11*b*x^74)/37 + (a*b^11*x^444)/37 
 + (22*a^10*b^2*x^111)/37 + (55*a^9*b^3*x^148)/37 + (99*a^8*b^4*x^185)/37 
+ (132*a^7*b^5*x^222)/37 + (132*a^6*b^6*x^259)/37 + (99*a^5*b^7*x^296)/37 
+ (55*a^4*b^8*x^333)/37 + (22*a^3*b^9*x^370)/37 + (6*a^2*b^10*x^407)/37
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 8.44 \[ \int x^{12} \left (a x^2+b x^{39}\right )^{12} \, dx=\frac {x^{37} \left (b^{12} x^{444}+13 a \,b^{11} x^{407}+78 a^{2} b^{10} x^{370}+286 a^{3} b^{9} x^{333}+715 a^{4} b^{8} x^{296}+1287 a^{5} b^{7} x^{259}+1716 a^{6} b^{6} x^{222}+1716 a^{7} b^{5} x^{185}+1287 a^{8} b^{4} x^{148}+715 a^{9} b^{3} x^{111}+286 a^{10} b^{2} x^{74}+78 a^{11} b \,x^{37}+13 a^{12}\right )}{481} \] Input:

int(x^12*(b*x^39+a*x^2)^12,x)
 

Output:

(x**37*(13*a**12 + 78*a**11*b*x**37 + 286*a**10*b**2*x**74 + 715*a**9*b**3 
*x**111 + 1287*a**8*b**4*x**148 + 1716*a**7*b**5*x**185 + 1716*a**6*b**6*x 
**222 + 1287*a**5*b**7*x**259 + 715*a**4*b**8*x**296 + 286*a**3*b**9*x**33 
3 + 78*a**2*b**10*x**370 + 13*a*b**11*x**407 + b**12*x**444))/481