3.334 \(\int (a x^3+b x^{40})^{12} \, dx\)

Optimal. Leaf size=16 \[ \frac {\left (a+b x^{37}\right )^{13}}{481 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1593, 261} \[ \frac {\left (a+b x^{37}\right )^{13}}{481 b} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^3 + b*x^40)^12,x]

[Out]

(a + b*x^37)^13/(481*b)

Rule 261

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 1593

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

Rubi steps

\begin {align*} \int \left (a x^3+b x^{40}\right )^{12} \, dx &=\int x^{36} \left (a+b x^{37}\right )^{12} \, dx\\ &=\frac {\left (a+b x^{37}\right )^{13}}{481 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 160, normalized size = 10.00 \[ \frac {a^{12} x^{37}}{37}+\frac {6}{37} a^{11} b x^{74}+\frac {22}{37} a^{10} b^2 x^{111}+\frac {55}{37} a^9 b^3 x^{148}+\frac {99}{37} a^8 b^4 x^{185}+\frac {132}{37} a^7 b^5 x^{222}+\frac {132}{37} a^6 b^6 x^{259}+\frac {99}{37} a^5 b^7 x^{296}+\frac {55}{37} a^4 b^8 x^{333}+\frac {22}{37} a^3 b^9 x^{370}+\frac {6}{37} a^2 b^{10} x^{407}+\frac {1}{37} a b^{11} x^{444}+\frac {b^{12} x^{481}}{481} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^3 + b*x^40)^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

________________________________________________________________________________________

fricas [B]  time = 0.34, size = 134, normalized size = 8.38 \[ \frac {1}{481} x^{481} b^{12} + \frac {1}{37} x^{444} b^{11} a + \frac {6}{37} x^{407} b^{10} a^{2} + \frac {22}{37} x^{370} b^{9} a^{3} + \frac {55}{37} x^{333} b^{8} a^{4} + \frac {99}{37} x^{296} b^{7} a^{5} + \frac {132}{37} x^{259} b^{6} a^{6} + \frac {132}{37} x^{222} b^{5} a^{7} + \frac {99}{37} x^{185} b^{4} a^{8} + \frac {55}{37} x^{148} b^{3} a^{9} + \frac {22}{37} x^{111} b^{2} a^{10} + \frac {6}{37} x^{74} b a^{11} + \frac {1}{37} x^{37} a^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^40+a*x^3)^12,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.16, size = 134, normalized size = 8.38 \[ \frac {1}{481} \, b^{12} x^{481} + \frac {1}{37} \, a b^{11} x^{444} + \frac {6}{37} \, a^{2} b^{10} x^{407} + \frac {22}{37} \, a^{3} b^{9} x^{370} + \frac {55}{37} \, a^{4} b^{8} x^{333} + \frac {99}{37} \, a^{5} b^{7} x^{296} + \frac {132}{37} \, a^{6} b^{6} x^{259} + \frac {132}{37} \, a^{7} b^{5} x^{222} + \frac {99}{37} \, a^{8} b^{4} x^{185} + \frac {55}{37} \, a^{9} b^{3} x^{148} + \frac {22}{37} \, a^{10} b^{2} x^{111} + \frac {6}{37} \, a^{11} b x^{74} + \frac {1}{37} \, a^{12} x^{37} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^40+a*x^3)^12,x, algorithm="giac")

[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

________________________________________________________________________________________

maple [B]  time = 0.04, size = 135, normalized size = 8.44 \[ \frac {1}{481} b^{12} x^{481}+\frac {1}{37} a \,b^{11} x^{444}+\frac {6}{37} a^{2} b^{10} x^{407}+\frac {22}{37} a^{3} b^{9} x^{370}+\frac {55}{37} a^{4} b^{8} x^{333}+\frac {99}{37} a^{5} b^{7} x^{296}+\frac {132}{37} a^{6} b^{6} x^{259}+\frac {132}{37} a^{7} b^{5} x^{222}+\frac {99}{37} a^{8} b^{4} x^{185}+\frac {55}{37} a^{9} b^{3} x^{148}+\frac {22}{37} a^{10} b^{2} x^{111}+\frac {6}{37} a^{11} b \,x^{74}+\frac {1}{37} a^{12} x^{37} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^40+a*x^3)^12,x)

[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

________________________________________________________________________________________

maxima [B]  time = 1.35, size = 134, normalized size = 8.38 \[ \frac {1}{481} \, b^{12} x^{481} + \frac {1}{37} \, a b^{11} x^{444} + \frac {6}{37} \, a^{2} b^{10} x^{407} + \frac {22}{37} \, a^{3} b^{9} x^{370} + \frac {55}{37} \, a^{4} b^{8} x^{333} + \frac {99}{37} \, a^{5} b^{7} x^{296} + \frac {132}{37} \, a^{6} b^{6} x^{259} + \frac {132}{37} \, a^{7} b^{5} x^{222} + \frac {99}{37} \, a^{8} b^{4} x^{185} + \frac {55}{37} \, a^{9} b^{3} x^{148} + \frac {22}{37} \, a^{10} b^{2} x^{111} + \frac {6}{37} \, a^{11} b x^{74} + \frac {1}{37} \, a^{12} x^{37} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^40+a*x^3)^12,x, algorithm="maxima")

[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

________________________________________________________________________________________

mupad [B]  time = 5.16, size = 134, normalized size = 8.38 \[ \frac {a^{12}\,x^{37}}{37}+\frac {6\,a^{11}\,b\,x^{74}}{37}+\frac {22\,a^{10}\,b^2\,x^{111}}{37}+\frac {55\,a^9\,b^3\,x^{148}}{37}+\frac {99\,a^8\,b^4\,x^{185}}{37}+\frac {132\,a^7\,b^5\,x^{222}}{37}+\frac {132\,a^6\,b^6\,x^{259}}{37}+\frac {99\,a^5\,b^7\,x^{296}}{37}+\frac {55\,a^4\,b^8\,x^{333}}{37}+\frac {22\,a^3\,b^9\,x^{370}}{37}+\frac {6\,a^2\,b^{10}\,x^{407}}{37}+\frac {a\,b^{11}\,x^{444}}{37}+\frac {b^{12}\,x^{481}}{481} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3 + b*x^40)^12,x)

[Out]

(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)/3
7 + (55*a^4*b^8*x^333)/37 + (22*a^3*b^9*x^370)/37 + (6*a^2*b^10*x^407)/37

________________________________________________________________________________________

sympy [B]  time = 0.13, size = 160, normalized size = 10.00 \[ \frac {a^{12} x^{37}}{37} + \frac {6 a^{11} b x^{74}}{37} + \frac {22 a^{10} b^{2} x^{111}}{37} + \frac {55 a^{9} b^{3} x^{148}}{37} + \frac {99 a^{8} b^{4} x^{185}}{37} + \frac {132 a^{7} b^{5} x^{222}}{37} + \frac {132 a^{6} b^{6} x^{259}}{37} + \frac {99 a^{5} b^{7} x^{296}}{37} + \frac {55 a^{4} b^{8} x^{333}}{37} + \frac {22 a^{3} b^{9} x^{370}}{37} + \frac {6 a^{2} b^{10} x^{407}}{37} + \frac {a b^{11} x^{444}}{37} + \frac {b^{12} x^{481}}{481} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**40+a*x**3)**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

________________________________________________________________________________________