3.348 \(\int x^p (a x^n+b x^{1+13 n+p})^{12} \, dx\)

Optimal. Leaf size=29 \[ \frac{\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \]

[Out]

(a + b*x^(1 + 12*n + p))^13/(13*b*(1 + 12*n + p))

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Rubi [A]  time = 0.0172572, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1584, 261} \[ \frac{\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^p*(a*x^n + b*x^(1 + 13*n + p))^12,x]

[Out]

(a + b*x^(1 + 12*n + p))^13/(13*b*(1 + 12*n + p))

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^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx &=\int x^{12 n+p} \left (a+b x^{1+12 n+p}\right )^{12} \, dx\\ &=\frac{\left (a+b x^{1+12 n+p}\right )^{13}}{13 b (1+12 n+p)}\\ \end{align*}

Mathematica [A]  time = 0.0155429, size = 29, normalized size = 1. \[ \frac{\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^p*(a*x^n + b*x^(1 + 13*n + p))^12,x]

[Out]

(a + b*x^(1 + 12*n + p))^13/(13*b*(1 + 12*n + p))

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Maple [B]  time = 0.168, size = 363, normalized size = 12.5 \begin{align*}{\frac{ \left ({x}^{n} \right ) ^{156}{x}^{13}{b}^{12} \left ({x}^{p} \right ) ^{13}}{13+156\,n+13\,p}}+{\frac{ \left ({x}^{n} \right ) ^{144}{x}^{12}a{b}^{11} \left ({x}^{p} \right ) ^{12}}{1+12\,n+p}}+6\,{\frac{ \left ({x}^{n} \right ) ^{132}{x}^{11}{a}^{2}{b}^{10} \left ({x}^{p} \right ) ^{11}}{1+12\,n+p}}+22\,{\frac{ \left ({x}^{n} \right ) ^{120}{x}^{10}{a}^{3}{b}^{9} \left ({x}^{p} \right ) ^{10}}{1+12\,n+p}}+55\,{\frac{ \left ({x}^{n} \right ) ^{108}{x}^{9}{a}^{4}{b}^{8} \left ({x}^{p} \right ) ^{9}}{1+12\,n+p}}+99\,{\frac{ \left ({x}^{n} \right ) ^{96}{x}^{8}{a}^{5}{b}^{7} \left ({x}^{p} \right ) ^{8}}{1+12\,n+p}}+132\,{\frac{ \left ({x}^{n} \right ) ^{84}{x}^{7}{a}^{6}{b}^{6} \left ({x}^{p} \right ) ^{7}}{1+12\,n+p}}+132\,{\frac{ \left ({x}^{n} \right ) ^{72}{x}^{6}{a}^{7}{b}^{5} \left ({x}^{p} \right ) ^{6}}{1+12\,n+p}}+99\,{\frac{ \left ({x}^{n} \right ) ^{60}{x}^{5}{a}^{8}{b}^{4} \left ({x}^{p} \right ) ^{5}}{1+12\,n+p}}+55\,{\frac{ \left ({x}^{n} \right ) ^{48}{x}^{4}{a}^{9}{b}^{3} \left ({x}^{p} \right ) ^{4}}{1+12\,n+p}}+22\,{\frac{ \left ({x}^{n} \right ) ^{36}{x}^{3}{a}^{10}{b}^{2} \left ({x}^{p} \right ) ^{3}}{1+12\,n+p}}+6\,{\frac{ \left ({x}^{n} \right ) ^{24}{x}^{2}{a}^{11}b \left ({x}^{p} \right ) ^{2}}{1+12\,n+p}}+{\frac{{a}^{12}x \left ({x}^{n} \right ) ^{12}{x}^{p}}{1+12\,n+p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^p*(a*x^n+b*x^(1+13*n+p))^12,x)

[Out]

1/13*b^12*x^13*(x^n)^156/(1+12*n+p)*(x^p)^13+a*b^11*x^12*(x^n)^144/(1+12*n+p)*(x^p)^12+6*a^2*b^10*x^11*(x^n)^1
32/(1+12*n+p)*(x^p)^11+22*a^3*b^9*x^10*(x^n)^120/(1+12*n+p)*(x^p)^10+55*a^4*b^8*x^9*(x^n)^108/(1+12*n+p)*(x^p)
^9+99*a^5*b^7*x^8*(x^n)^96/(1+12*n+p)*(x^p)^8+132*a^6*b^6*x^7*(x^n)^84/(1+12*n+p)*(x^p)^7+132*a^7*b^5*x^6*(x^n
)^72/(1+12*n+p)*(x^p)^6+99*a^8*b^4*x^5*(x^n)^60/(1+12*n+p)*(x^p)^5+55*a^9*b^3*x^4*(x^n)^48/(1+12*n+p)*(x^p)^4+
22*a^10*b^2*x^3*(x^n)^36/(1+12*n+p)*(x^p)^3+6*a^11*b*x^2*(x^n)^24/(1+12*n+p)*(x^p)^2+a^12/(1+12*n+p)*x*(x^n)^1
2*x^p

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^p*(a*x^n+b*x^(1+13*n+p))^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.853519, size = 753, normalized size = 25.97 \begin{align*} \frac{78 \, a^{2} b^{10} x^{2 \, n} x^{143 \, n + 11 \, p + 11} + 286 \, a^{3} b^{9} x^{3 \, n} x^{130 \, n + 10 \, p + 10} + 715 \, a^{4} b^{8} x^{4 \, n} x^{117 \, n + 9 \, p + 9} + 1287 \, a^{5} b^{7} x^{5 \, n} x^{104 \, n + 8 \, p + 8} + 1716 \, a^{6} b^{6} x^{6 \, n} x^{91 \, n + 7 \, p + 7} + 1716 \, a^{7} b^{5} x^{7 \, n} x^{78 \, n + 6 \, p + 6} + 1287 \, a^{8} b^{4} x^{8 \, n} x^{65 \, n + 5 \, p + 5} + 715 \, a^{9} b^{3} x^{9 \, n} x^{52 \, n + 4 \, p + 4} + 286 \, a^{10} b^{2} x^{10 \, n} x^{39 \, n + 3 \, p + 3} + 78 \, a^{11} b x^{11 \, n} x^{26 \, n + 2 \, p + 2} + 13 \, a^{12} x^{12 \, n} x^{13 \, n + p + 1} + 13 \, a b^{11} x^{156 \, n + 12 \, p + 12} x^{n} + b^{12} x^{169 \, n + 13 \, p + 13}}{13 \,{\left (12 \, n + p + 1\right )} x^{13 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^p*(a*x^n+b*x^(1+13*n+p))^12,x, algorithm="fricas")

[Out]

1/13*(78*a^2*b^10*x^(2*n)*x^(143*n + 11*p + 11) + 286*a^3*b^9*x^(3*n)*x^(130*n + 10*p + 10) + 715*a^4*b^8*x^(4
*n)*x^(117*n + 9*p + 9) + 1287*a^5*b^7*x^(5*n)*x^(104*n + 8*p + 8) + 1716*a^6*b^6*x^(6*n)*x^(91*n + 7*p + 7) +
1716*a^7*b^5*x^(7*n)*x^(78*n + 6*p + 6) + 1287*a^8*b^4*x^(8*n)*x^(65*n + 5*p + 5) + 715*a^9*b^3*x^(9*n)*x^(52
*n + 4*p + 4) + 286*a^10*b^2*x^(10*n)*x^(39*n + 3*p + 3) + 78*a^11*b*x^(11*n)*x^(26*n + 2*p + 2) + 13*a^12*x^(
12*n)*x^(13*n + p + 1) + 13*a*b^11*x^(156*n + 12*p + 12)*x^n + b^12*x^(169*n + 13*p + 13))/((12*n + p + 1)*x^(
13*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**p*(a*x**n+b*x**(1+13*n+p))**12,x)

[Out]

Timed out

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Giac [B]  time = 4.31688, size = 363, normalized size = 12.52 \begin{align*} \frac{b^{12} x^{13} x^{156 \, n} x^{13 \, p} + 13 \, a b^{11} x^{12} x^{144 \, n} x^{12 \, p} + 78 \, a^{2} b^{10} x^{11} x^{132 \, n} x^{11 \, p} + 286 \, a^{3} b^{9} x^{10} x^{120 \, n} x^{10 \, p} + 715 \, a^{4} b^{8} x^{9} x^{108 \, n} x^{9 \, p} + 1287 \, a^{5} b^{7} x^{8} x^{96 \, n} x^{8 \, p} + 1716 \, a^{6} b^{6} x^{7} x^{84 \, n} x^{7 \, p} + 1716 \, a^{7} b^{5} x^{6} x^{72 \, n} x^{6 \, p} + 1287 \, a^{8} b^{4} x^{5} x^{60 \, n} x^{5 \, p} + 715 \, a^{9} b^{3} x^{4} x^{48 \, n} x^{4 \, p} + 286 \, a^{10} b^{2} x^{3} x^{36 \, n} x^{3 \, p} + 78 \, a^{11} b x^{2} x^{24 \, n} x^{2 \, p} + 13 \, a^{12} x x^{12 \, n} x^{p}}{13 \,{\left (12 \, n + p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^p*(a*x^n+b*x^(1+13*n+p))^12,x, algorithm="giac")

[Out]

1/13*(b^12*x^13*x^(156*n)*x^(13*p) + 13*a*b^11*x^12*x^(144*n)*x^(12*p) + 78*a^2*b^10*x^11*x^(132*n)*x^(11*p) +
286*a^3*b^9*x^10*x^(120*n)*x^(10*p) + 715*a^4*b^8*x^9*x^(108*n)*x^(9*p) + 1287*a^5*b^7*x^8*x^(96*n)*x^(8*p) +
1716*a^6*b^6*x^7*x^(84*n)*x^(7*p) + 1716*a^7*b^5*x^6*x^(72*n)*x^(6*p) + 1287*a^8*b^4*x^5*x^(60*n)*x^(5*p) + 7
15*a^9*b^3*x^4*x^(48*n)*x^(4*p) + 286*a^10*b^2*x^3*x^(36*n)*x^(3*p) + 78*a^11*b*x^2*x^(24*n)*x^(2*p) + 13*a^12
*x*x^(12*n)*x^p)/(12*n + p + 1)