3.350 \(\int x^{12} (a x+b x^{26})^{12} \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^{25}\right )^{13}}{325 b} \]

[Out]

(a + b*x^25)^13/(325*b)

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Rubi [A]  time = 0.0086186, 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} \[ \frac{\left (a+b x^{25}\right )^{13}}{325 b} \]

Antiderivative was successfully verified.

[In]

Int[x^12*(a*x + b*x^26)^12,x]

[Out]

(a + b*x^25)^13/(325*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^{12} \left (a x+b x^{26}\right )^{12} \, dx &=\int x^{24} \left (a+b x^{25}\right )^{12} \, dx\\ &=\frac{\left (a+b x^{25}\right )^{13}}{325 b}\\ \end{align*}

Mathematica [B]  time = 0.0060576, size = 160, normalized size = 10. \[ \frac{6}{25} a^2 b^{10} x^{275}+\frac{22}{25} a^3 b^9 x^{250}+\frac{11}{5} a^4 b^8 x^{225}+\frac{99}{25} a^5 b^7 x^{200}+\frac{132}{25} a^6 b^6 x^{175}+\frac{132}{25} a^7 b^5 x^{150}+\frac{99}{25} a^8 b^4 x^{125}+\frac{11}{5} a^9 b^3 x^{100}+\frac{22}{25} a^{10} b^2 x^{75}+\frac{6}{25} a^{11} b x^{50}+\frac{a^{12} x^{25}}{25}+\frac{1}{25} a b^{11} x^{300}+\frac{b^{12} x^{325}}{325} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12*(a*x + b*x^26)^12,x]

[Out]

(a^12*x^25)/25 + (6*a^11*b*x^50)/25 + (22*a^10*b^2*x^75)/25 + (11*a^9*b^3*x^100)/5 + (99*a^8*b^4*x^125)/25 + (
132*a^7*b^5*x^150)/25 + (132*a^6*b^6*x^175)/25 + (99*a^5*b^7*x^200)/25 + (11*a^4*b^8*x^225)/5 + (22*a^3*b^9*x^
250)/25 + (6*a^2*b^10*x^275)/25 + (a*b^11*x^300)/25 + (b^12*x^325)/325

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Maple [B]  time = 0., size = 135, normalized size = 8.4 \begin{align*}{\frac{{b}^{12}{x}^{325}}{325}}+{\frac{{b}^{11}a{x}^{300}}{25}}+{\frac{6\,{b}^{10}{a}^{2}{x}^{275}}{25}}+{\frac{22\,{a}^{3}{b}^{9}{x}^{250}}{25}}+{\frac{11\,{a}^{4}{b}^{8}{x}^{225}}{5}}+{\frac{99\,{a}^{5}{b}^{7}{x}^{200}}{25}}+{\frac{132\,{a}^{6}{b}^{6}{x}^{175}}{25}}+{\frac{132\,{a}^{7}{b}^{5}{x}^{150}}{25}}+{\frac{99\,{a}^{8}{b}^{4}{x}^{125}}{25}}+{\frac{11\,{a}^{9}{b}^{3}{x}^{100}}{5}}+{\frac{22\,{a}^{10}{b}^{2}{x}^{75}}{25}}+{\frac{6\,{a}^{11}b{x}^{50}}{25}}+{\frac{{a}^{12}{x}^{25}}{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12*(b*x^26+a*x)^12,x)

[Out]

1/325*b^12*x^325+1/25*b^11*a*x^300+6/25*b^10*a^2*x^275+22/25*a^3*b^9*x^250+11/5*a^4*b^8*x^225+99/25*a^5*b^7*x^
200+132/25*a^6*b^6*x^175+132/25*a^7*b^5*x^150+99/25*a^8*b^4*x^125+11/5*a^9*b^3*x^100+22/25*a^10*b^2*x^75+6/25*
a^11*b*x^50+1/25*a^12*x^25

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Maxima [B]  time = 1.05386, size = 181, normalized size = 11.31 \begin{align*} \frac{1}{325} \, b^{12} x^{325} + \frac{1}{25} \, a b^{11} x^{300} + \frac{6}{25} \, a^{2} b^{10} x^{275} + \frac{22}{25} \, a^{3} b^{9} x^{250} + \frac{11}{5} \, a^{4} b^{8} x^{225} + \frac{99}{25} \, a^{5} b^{7} x^{200} + \frac{132}{25} \, a^{6} b^{6} x^{175} + \frac{132}{25} \, a^{7} b^{5} x^{150} + \frac{99}{25} \, a^{8} b^{4} x^{125} + \frac{11}{5} \, a^{9} b^{3} x^{100} + \frac{22}{25} \, a^{10} b^{2} x^{75} + \frac{6}{25} \, a^{11} b x^{50} + \frac{1}{25} \, a^{12} x^{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^26+a*x)^12,x, algorithm="maxima")

[Out]

1/325*b^12*x^325 + 1/25*a*b^11*x^300 + 6/25*a^2*b^10*x^275 + 22/25*a^3*b^9*x^250 + 11/5*a^4*b^8*x^225 + 99/25*
a^5*b^7*x^200 + 132/25*a^6*b^6*x^175 + 132/25*a^7*b^5*x^150 + 99/25*a^8*b^4*x^125 + 11/5*a^9*b^3*x^100 + 22/25
*a^10*b^2*x^75 + 6/25*a^11*b*x^50 + 1/25*a^12*x^25

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Fricas [B]  time = 0.619714, size = 367, normalized size = 22.94 \begin{align*} \frac{1}{325} x^{325} b^{12} + \frac{1}{25} x^{300} b^{11} a + \frac{6}{25} x^{275} b^{10} a^{2} + \frac{22}{25} x^{250} b^{9} a^{3} + \frac{11}{5} x^{225} b^{8} a^{4} + \frac{99}{25} x^{200} b^{7} a^{5} + \frac{132}{25} x^{175} b^{6} a^{6} + \frac{132}{25} x^{150} b^{5} a^{7} + \frac{99}{25} x^{125} b^{4} a^{8} + \frac{11}{5} x^{100} b^{3} a^{9} + \frac{22}{25} x^{75} b^{2} a^{10} + \frac{6}{25} x^{50} b a^{11} + \frac{1}{25} x^{25} a^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^26+a*x)^12,x, algorithm="fricas")

[Out]

1/325*x^325*b^12 + 1/25*x^300*b^11*a + 6/25*x^275*b^10*a^2 + 22/25*x^250*b^9*a^3 + 11/5*x^225*b^8*a^4 + 99/25*
x^200*b^7*a^5 + 132/25*x^175*b^6*a^6 + 132/25*x^150*b^5*a^7 + 99/25*x^125*b^4*a^8 + 11/5*x^100*b^3*a^9 + 22/25
*x^75*b^2*a^10 + 6/25*x^50*b*a^11 + 1/25*x^25*a^12

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Sympy [B]  time = 0.151775, size = 160, normalized size = 10. \begin{align*} \frac{a^{12} x^{25}}{25} + \frac{6 a^{11} b x^{50}}{25} + \frac{22 a^{10} b^{2} x^{75}}{25} + \frac{11 a^{9} b^{3} x^{100}}{5} + \frac{99 a^{8} b^{4} x^{125}}{25} + \frac{132 a^{7} b^{5} x^{150}}{25} + \frac{132 a^{6} b^{6} x^{175}}{25} + \frac{99 a^{5} b^{7} x^{200}}{25} + \frac{11 a^{4} b^{8} x^{225}}{5} + \frac{22 a^{3} b^{9} x^{250}}{25} + \frac{6 a^{2} b^{10} x^{275}}{25} + \frac{a b^{11} x^{300}}{25} + \frac{b^{12} x^{325}}{325} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12*(b*x**26+a*x)**12,x)

[Out]

a**12*x**25/25 + 6*a**11*b*x**50/25 + 22*a**10*b**2*x**75/25 + 11*a**9*b**3*x**100/5 + 99*a**8*b**4*x**125/25
+ 132*a**7*b**5*x**150/25 + 132*a**6*b**6*x**175/25 + 99*a**5*b**7*x**200/25 + 11*a**4*b**8*x**225/5 + 22*a**3
*b**9*x**250/25 + 6*a**2*b**10*x**275/25 + a*b**11*x**300/25 + b**12*x**325/325

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Giac [B]  time = 1.1787, size = 181, normalized size = 11.31 \begin{align*} \frac{1}{325} \, b^{12} x^{325} + \frac{1}{25} \, a b^{11} x^{300} + \frac{6}{25} \, a^{2} b^{10} x^{275} + \frac{22}{25} \, a^{3} b^{9} x^{250} + \frac{11}{5} \, a^{4} b^{8} x^{225} + \frac{99}{25} \, a^{5} b^{7} x^{200} + \frac{132}{25} \, a^{6} b^{6} x^{175} + \frac{132}{25} \, a^{7} b^{5} x^{150} + \frac{99}{25} \, a^{8} b^{4} x^{125} + \frac{11}{5} \, a^{9} b^{3} x^{100} + \frac{22}{25} \, a^{10} b^{2} x^{75} + \frac{6}{25} \, a^{11} b x^{50} + \frac{1}{25} \, a^{12} x^{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^26+a*x)^12,x, algorithm="giac")

[Out]

1/325*b^12*x^325 + 1/25*a*b^11*x^300 + 6/25*a^2*b^10*x^275 + 22/25*a^3*b^9*x^250 + 11/5*a^4*b^8*x^225 + 99/25*
a^5*b^7*x^200 + 132/25*a^6*b^6*x^175 + 132/25*a^7*b^5*x^150 + 99/25*a^8*b^4*x^125 + 11/5*a^9*b^3*x^100 + 22/25
*a^10*b^2*x^75 + 6/25*a^11*b*x^50 + 1/25*a^12*x^25