\(\int x^{36} (a+b x^{37})^{12} \, dx\) [354]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {\left (a+b x^{37}\right )^{13}}{481 b} \]

[Out]

1/481*(b*x^37+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} \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {\left (a+b x^{37}\right )^{13}}{481 b} \]

[In]

Int[x^36*(a + b*x^37)^12,x]

[Out]

(a + b*x^37)^13/(481*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^{37}\right )^{13}}{481 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 \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {a^{12} x^{37}}{37}+\frac {6}{37} a^{11} b x^{74}+\frac {22}{37} a^{10} b^2 x^{111}+\frac {55}{37} a^9 b^3 x^{148}+\frac {99}{37} a^8 b^4 x^{185}+\frac {132}{37} a^7 b^5 x^{222}+\frac {132}{37} a^6 b^6 x^{259}+\frac {99}{37} a^5 b^7 x^{296}+\frac {55}{37} a^4 b^8 x^{333}+\frac {22}{37} a^3 b^9 x^{370}+\frac {6}{37} a^2 b^{10} x^{407}+\frac {1}{37} a b^{11} x^{444}+\frac {b^{12} x^{481}}{481} \]

[In]

Integrate[x^36*(a + b*x^37)^12,x]

[Out]

(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

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (b \,x^{37}+a \right )^{13}}{481 b}\)	\(15\)
gosper	\(\frac {6}{37} b \,a^{11} x^{74}+\frac {1}{481} b^{12} x^{481}+\frac {55}{37} a^{9} b^{3} x^{148}+\frac {99}{37} a^{5} b^{7} x^{296}+\frac {132}{37} a^{7} b^{5} x^{222}+\frac {132}{37} a^{6} b^{6} x^{259}+\frac {22}{37} a^{3} b^{9} x^{370}+\frac {1}{37} a^{12} x^{37}+\frac {99}{37} a^{8} b^{4} x^{185}+\frac {22}{37} a^{10} b^{2} x^{111}+\frac {55}{37} a^{4} b^{8} x^{333}+\frac {1}{37} a \,b^{11} x^{444}+\frac {6}{37} a^{2} b^{10} x^{407}\)	\(135\)
parallelrisch	\(\frac {6}{37} b \,a^{11} x^{74}+\frac {1}{481} b^{12} x^{481}+\frac {55}{37} a^{9} b^{3} x^{148}+\frac {99}{37} a^{5} b^{7} x^{296}+\frac {132}{37} a^{7} b^{5} x^{222}+\frac {132}{37} a^{6} b^{6} x^{259}+\frac {22}{37} a^{3} b^{9} x^{370}+\frac {1}{37} a^{12} x^{37}+\frac {99}{37} a^{8} b^{4} x^{185}+\frac {22}{37} a^{10} b^{2} x^{111}+\frac {55}{37} a^{4} b^{8} x^{333}+\frac {1}{37} a \,b^{11} x^{444}+\frac {6}{37} a^{2} b^{10} x^{407}\)	\(135\)
risch	\(\frac {b^{12} x^{481}}{481}+\frac {a \,b^{11} x^{444}}{37}+\frac {6 a^{2} b^{10} x^{407}}{37}+\frac {22 a^{3} b^{9} x^{370}}{37}+\frac {55 a^{4} b^{8} x^{333}}{37}+\frac {99 a^{5} b^{7} x^{296}}{37}+\frac {132 a^{6} b^{6} x^{259}}{37}+\frac {132 a^{7} b^{5} x^{222}}{37}+\frac {99 a^{8} b^{4} x^{185}}{37}+\frac {55 a^{9} b^{3} x^{148}}{37}+\frac {22 a^{10} b^{2} x^{111}}{37}+\frac {6 b \,a^{11} x^{74}}{37}+\frac {a^{12} x^{37}}{37}+\frac {a^{13}}{481 b}\)	\(143\)

[In]

int(x^36*(b*x^37+a)^12,x,method=_RETURNVERBOSE)

[Out]

1/481*(b*x^37+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 = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 8.38 \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {1}{481} \, b^{12} x^{481} + \frac {1}{37} \, a b^{11} x^{444} + \frac {6}{37} \, a^{2} b^{10} x^{407} + \frac {22}{37} \, a^{3} b^{9} x^{370} + \frac {55}{37} \, a^{4} b^{8} x^{333} + \frac {99}{37} \, a^{5} b^{7} x^{296} + \frac {132}{37} \, a^{6} b^{6} x^{259} + \frac {132}{37} \, a^{7} b^{5} x^{222} + \frac {99}{37} \, a^{8} b^{4} x^{185} + \frac {55}{37} \, a^{9} b^{3} x^{148} + \frac {22}{37} \, a^{10} b^{2} x^{111} + \frac {6}{37} \, a^{11} b x^{74} + \frac {1}{37} \, a^{12} x^{37} \]

[In]

integrate(x^36*(b*x^37+a)^12,x, algorithm="fricas")

[Out]

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 \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {a^{12} x^{37}}{37} + \frac {6 a^{11} b x^{74}}{37} + \frac {22 a^{10} b^{2} x^{111}}{37} + \frac {55 a^{9} b^{3} x^{148}}{37} + \frac {99 a^{8} b^{4} x^{185}}{37} + \frac {132 a^{7} b^{5} x^{222}}{37} + \frac {132 a^{6} b^{6} x^{259}}{37} + \frac {99 a^{5} b^{7} x^{296}}{37} + \frac {55 a^{4} b^{8} x^{333}}{37} + \frac {22 a^{3} b^{9} x^{370}}{37} + \frac {6 a^{2} b^{10} x^{407}}{37} + \frac {a b^{11} x^{444}}{37} + \frac {b^{12} x^{481}}{481} \]

[In]

integrate(x**36*(b*x**37+a)**12,x)

[Out]

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**185/3
7 + 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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {{\left (b x^{37} + a\right )}^{13}}{481 \, b} \]

[In]

integrate(x^36*(b*x^37+a)^12,x, algorithm="maxima")

[Out]

1/481*(b*x^37 + a)^13/b

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {{\left (b x^{37} + a\right )}^{13}}{481 \, b} \]

[In]

integrate(x^36*(b*x^37+a)^12,x, algorithm="giac")

[Out]

1/481*(b*x^37 + a)^13/b

Mupad [B] (verification not implemented)

Time = 8.99 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^{36} \left (a+b x^{37}\right )^{12} \, dx=\frac {{\left (b\,x^{37}+a\right )}^{13}}{481\,b} \]

[In]

int(x^36*(a + b*x^37)^12,x)

[Out]

(a + b*x^37)^13/(481*b)