3.2592 \(\int x^{12} (a+b x^{13})^{12} \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^{13}\right )^{13}}{169 b} \]

[Out]

(a + b*x^13)^13/(169*b)

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Rubi [A]  time = 0.0031126, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{\left (a+b x^{13}\right )^{13}}{169 b} \]

Antiderivative was successfully verified.

[In]

Int[x^12*(a + b*x^13)^12,x]

[Out]

(a + b*x^13)^13/(169*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]

Rubi steps

\begin{align*} \int x^{12} \left (a+b x^{13}\right )^{12} \, dx &=\frac{\left (a+b x^{13}\right )^{13}}{169 b}\\ \end{align*}

Mathematica [B]  time = 0.0049, size = 160, normalized size = 10. \[ \frac{6}{13} a^2 b^{10} x^{143}+\frac{22}{13} a^3 b^9 x^{130}+\frac{55}{13} a^4 b^8 x^{117}+\frac{99}{13} a^5 b^7 x^{104}+\frac{132}{13} a^6 b^6 x^{91}+\frac{132}{13} a^7 b^5 x^{78}+\frac{99}{13} a^8 b^4 x^{65}+\frac{55}{13} a^9 b^3 x^{52}+\frac{22}{13} a^{10} b^2 x^{39}+\frac{6}{13} a^{11} b x^{26}+\frac{a^{12} x^{13}}{13}+\frac{1}{13} a b^{11} x^{156}+\frac{b^{12} x^{169}}{169} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12*(a + b*x^13)^12,x]

[Out]

(a^12*x^13)/13 + (6*a^11*b*x^26)/13 + (22*a^10*b^2*x^39)/13 + (55*a^9*b^3*x^52)/13 + (99*a^8*b^4*x^65)/13 + (1
32*a^7*b^5*x^78)/13 + (132*a^6*b^6*x^91)/13 + (99*a^5*b^7*x^104)/13 + (55*a^4*b^8*x^117)/13 + (22*a^3*b^9*x^13
0)/13 + (6*a^2*b^10*x^143)/13 + (a*b^11*x^156)/13 + (b^12*x^169)/169

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Maple [B]  time = 0.003, size = 135, normalized size = 8.4 \begin{align*}{\frac{{b}^{12}{x}^{169}}{169}}+{\frac{{b}^{11}a{x}^{156}}{13}}+{\frac{6\,{b}^{10}{a}^{2}{x}^{143}}{13}}+{\frac{22\,{a}^{3}{b}^{9}{x}^{130}}{13}}+{\frac{55\,{a}^{4}{b}^{8}{x}^{117}}{13}}+{\frac{99\,{a}^{5}{b}^{7}{x}^{104}}{13}}+{\frac{132\,{a}^{6}{b}^{6}{x}^{91}}{13}}+{\frac{132\,{a}^{7}{b}^{5}{x}^{78}}{13}}+{\frac{99\,{a}^{8}{b}^{4}{x}^{65}}{13}}+{\frac{55\,{a}^{9}{b}^{3}{x}^{52}}{13}}+{\frac{22\,{a}^{10}{b}^{2}{x}^{39}}{13}}+{\frac{6\,{a}^{11}b{x}^{26}}{13}}+{\frac{{a}^{12}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12*(b*x^13+a)^12,x)

[Out]

1/169*b^12*x^169+1/13*b^11*a*x^156+6/13*b^10*a^2*x^143+22/13*a^3*b^9*x^130+55/13*a^4*b^8*x^117+99/13*a^5*b^7*x
^104+132/13*a^6*b^6*x^91+132/13*a^7*b^5*x^78+99/13*a^8*b^4*x^65+55/13*a^9*b^3*x^52+22/13*a^10*b^2*x^39+6/13*a^
11*b*x^26+1/13*a^12*x^13

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Maxima [A]  time = 0.947921, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{13} + a\right )}^{13}}{169 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^13+a)^12,x, algorithm="maxima")

[Out]

1/169*(b*x^13 + a)^13/b

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Fricas [B]  time = 0.787081, size = 365, normalized size = 22.81 \begin{align*} \frac{1}{169} x^{169} b^{12} + \frac{1}{13} x^{156} b^{11} a + \frac{6}{13} x^{143} b^{10} a^{2} + \frac{22}{13} x^{130} b^{9} a^{3} + \frac{55}{13} x^{117} b^{8} a^{4} + \frac{99}{13} x^{104} b^{7} a^{5} + \frac{132}{13} x^{91} b^{6} a^{6} + \frac{132}{13} x^{78} b^{5} a^{7} + \frac{99}{13} x^{65} b^{4} a^{8} + \frac{55}{13} x^{52} b^{3} a^{9} + \frac{22}{13} x^{39} b^{2} a^{10} + \frac{6}{13} x^{26} b a^{11} + \frac{1}{13} x^{13} a^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^13+a)^12,x, algorithm="fricas")

[Out]

1/169*x^169*b^12 + 1/13*x^156*b^11*a + 6/13*x^143*b^10*a^2 + 22/13*x^130*b^9*a^3 + 55/13*x^117*b^8*a^4 + 99/13
*x^104*b^7*a^5 + 132/13*x^91*b^6*a^6 + 132/13*x^78*b^5*a^7 + 99/13*x^65*b^4*a^8 + 55/13*x^52*b^3*a^9 + 22/13*x
^39*b^2*a^10 + 6/13*x^26*b*a^11 + 1/13*x^13*a^12

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Sympy [B]  time = 0.101562, size = 160, normalized size = 10. \begin{align*} \frac{a^{12} x^{13}}{13} + \frac{6 a^{11} b x^{26}}{13} + \frac{22 a^{10} b^{2} x^{39}}{13} + \frac{55 a^{9} b^{3} x^{52}}{13} + \frac{99 a^{8} b^{4} x^{65}}{13} + \frac{132 a^{7} b^{5} x^{78}}{13} + \frac{132 a^{6} b^{6} x^{91}}{13} + \frac{99 a^{5} b^{7} x^{104}}{13} + \frac{55 a^{4} b^{8} x^{117}}{13} + \frac{22 a^{3} b^{9} x^{130}}{13} + \frac{6 a^{2} b^{10} x^{143}}{13} + \frac{a b^{11} x^{156}}{13} + \frac{b^{12} x^{169}}{169} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12*(b*x**13+a)**12,x)

[Out]

a**12*x**13/13 + 6*a**11*b*x**26/13 + 22*a**10*b**2*x**39/13 + 55*a**9*b**3*x**52/13 + 99*a**8*b**4*x**65/13 +
132*a**7*b**5*x**78/13 + 132*a**6*b**6*x**91/13 + 99*a**5*b**7*x**104/13 + 55*a**4*b**8*x**117/13 + 22*a**3*b
**9*x**130/13 + 6*a**2*b**10*x**143/13 + a*b**11*x**156/13 + b**12*x**169/169

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Giac [A]  time = 1.22622, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{13} + a\right )}^{13}}{169 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^13+a)^12,x, algorithm="giac")

[Out]

1/169*(b*x^13 + a)^13/b