3.353 \(\int x^{24} (a x+b x^{38})^{12} \, dx\)

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

[Out]

(a + b*x^37)^13/(481*b)

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Rubi [A]  time = 0.0083321, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1584, 261} \[ \frac{\left (a+b x^{37}\right )^{13}}{481 b} \]

Antiderivative was successfully verified.

[In]

Int[x^24*(a*x + b*x^38)^12,x]

[Out]

(a + b*x^37)^13/(481*b)

Rule 1584

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]

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]

Rubi steps

\begin{align*} \int x^{24} \left (a x+b x^{38}\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 = 0.0057769, size = 160, normalized size = 10. \[ \frac{6}{37} a^2 b^{10} x^{407}+\frac{22}{37} a^3 b^9 x^{370}+\frac{55}{37} a^4 b^8 x^{333}+\frac{99}{37} a^5 b^7 x^{296}+\frac{132}{37} a^6 b^6 x^{259}+\frac{132}{37} a^7 b^5 x^{222}+\frac{99}{37} a^8 b^4 x^{185}+\frac{55}{37} a^9 b^3 x^{148}+\frac{22}{37} a^{10} b^2 x^{111}+\frac{6}{37} a^{11} b x^{74}+\frac{a^{12} x^{37}}{37}+\frac{1}{37} a b^{11} x^{444}+\frac{b^{12} x^{481}}{481} \]

Antiderivative was successfully verified.

[In]

Integrate[x^24*(a*x + b*x^38)^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

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Maple [B]  time = 0.002, size = 135, normalized size = 8.4 \begin{align*}{\frac{{b}^{12}{x}^{481}}{481}}+{\frac{{b}^{11}a{x}^{444}}{37}}+{\frac{6\,{b}^{10}{a}^{2}{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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24*(b*x^38+a*x)^12,x)

[Out]

1/481*b^12*x^481+1/37*b^11*a*x^444+6/37*b^10*a^2*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

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Maxima [B]  time = 1.01604, size = 181, normalized size = 11.31 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24*(b*x^38+a*x)^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

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Fricas [B]  time = 0.644098, size = 371, normalized size = 23.19 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24*(b*x^38+a*x)^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

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Sympy [B]  time = 0.145064, size = 160, normalized size = 10. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**24*(b*x**38+a*x)**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

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Giac [B]  time = 1.17673, size = 181, normalized size = 11.31 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24*(b*x^38+a*x)^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