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

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

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \end {gather*}

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 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 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]

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*}

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Mathematica [A]  time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.09, size = 232, normalized size = 8.00 \begin {gather*} \frac {x^{12 n+p+1} \left (13 a^{12}+78 a^{11} b x^{12 n+p+1}+286 a^{10} b^2 x^{24 n+2 p+2}+715 a^9 b^3 x^{36 n+3 p+3}+1287 a^8 b^4 x^{48 n+4 p+4}+1716 a^7 b^5 x^{60 n+5 p+5}+1716 a^6 b^6 x^{72 n+6 p+6}+1287 a^5 b^7 x^{84 n+7 p+7}+715 a^4 b^8 x^{96 n+8 p+8}+286 a^3 b^9 x^{108 n+9 p+9}+78 a^2 b^{10} x^{120 n+10 p+10}+13 a b^{11} x^{132 n+11 p+11}+b^{12} x^{144 n+12 p+12}\right )}{13 (12 n+p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 297, normalized size = 10.24 \begin {gather*} \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 {gather*}

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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giac [B]  time = 2.47, size = 269, normalized size = 9.28 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.21, size = 363, normalized size = 12.52 \begin {gather*} \frac {b^{12} x^{13} x^{156 n} x^{13 p}}{13+156 n +13 p}+\frac {a \,b^{11} x^{12} x^{144 n} x^{12 p}}{12 n +p +1}+\frac {6 a^{2} b^{10} x^{11} x^{132 n} x^{11 p}}{12 n +p +1}+\frac {22 a^{3} b^{9} x^{10} x^{120 n} x^{10 p}}{12 n +p +1}+\frac {55 a^{4} b^{8} x^{9} x^{108 n} x^{9 p}}{12 n +p +1}+\frac {99 a^{5} b^{7} x^{8} x^{96 n} x^{8 p}}{12 n +p +1}+\frac {132 a^{6} b^{6} x^{7} x^{84 n} x^{7 p}}{12 n +p +1}+\frac {132 a^{7} b^{5} x^{6} x^{72 n} x^{6 p}}{12 n +p +1}+\frac {99 a^{8} b^{4} x^{5} x^{60 n} x^{5 p}}{12 n +p +1}+\frac {55 a^{9} b^{3} x^{4} x^{48 n} x^{4 p}}{12 n +p +1}+\frac {22 a^{10} b^{2} x^{3} x^{36 n} x^{3 p}}{12 n +p +1}+\frac {6 a^{11} b \,x^{2} x^{24 n} x^{2 p}}{12 n +p +1}+\frac {a^{12} x \,x^{p} x^{12 n}}{12 n +p +1} \end {gather*}

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 [B]  time = 1.48, size = 325, normalized size = 11.21 \begin {gather*} \frac {b^{12} x^{156 \, n + 13 \, p + 13}}{13 \, {\left (12 \, n + p + 1\right )}} + \frac {a b^{11} x^{144 \, n + 12 \, p + 12}}{12 \, n + p + 1} + \frac {6 \, a^{2} b^{10} x^{132 \, n + 11 \, p + 11}}{12 \, n + p + 1} + \frac {22 \, a^{3} b^{9} x^{120 \, n + 10 \, p + 10}}{12 \, n + p + 1} + \frac {55 \, a^{4} b^{8} x^{108 \, n + 9 \, p + 9}}{12 \, n + p + 1} + \frac {99 \, a^{5} b^{7} x^{96 \, n + 8 \, p + 8}}{12 \, n + p + 1} + \frac {132 \, a^{6} b^{6} x^{84 \, n + 7 \, p + 7}}{12 \, n + p + 1} + \frac {132 \, a^{7} b^{5} x^{72 \, n + 6 \, p + 6}}{12 \, n + p + 1} + \frac {99 \, a^{8} b^{4} x^{60 \, n + 5 \, p + 5}}{12 \, n + p + 1} + \frac {55 \, a^{9} b^{3} x^{48 \, n + 4 \, p + 4}}{12 \, n + p + 1} + \frac {22 \, a^{10} b^{2} x^{36 \, n + 3 \, p + 3}}{12 \, n + p + 1} + \frac {6 \, a^{11} b x^{24 \, n + 2 \, p + 2}}{12 \, n + p + 1} + \frac {a^{12} x^{12 \, n + p + 1}}{12 \, n + p + 1} \end {gather*}

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]

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

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mupad [B]  time = 6.78, size = 363, normalized size = 12.52 \begin {gather*} \frac {a^{12}\,x\,x^p\,x^{12\,n}}{12\,n+p+1}+\frac {b^{12}\,x^{156\,n}\,x^{13\,p}\,x^{13}}{156\,n+13\,p+13}+\frac {22\,a^{10}\,b^2\,x^{36\,n}\,x^{3\,p}\,x^3}{12\,n+p+1}+\frac {55\,a^9\,b^3\,x^{48\,n}\,x^{4\,p}\,x^4}{12\,n+p+1}+\frac {99\,a^8\,b^4\,x^{60\,n}\,x^{5\,p}\,x^5}{12\,n+p+1}+\frac {132\,a^7\,b^5\,x^{72\,n}\,x^{6\,p}\,x^6}{12\,n+p+1}+\frac {132\,a^6\,b^6\,x^{84\,n}\,x^{7\,p}\,x^7}{12\,n+p+1}+\frac {99\,a^5\,b^7\,x^{96\,n}\,x^{8\,p}\,x^8}{12\,n+p+1}+\frac {55\,a^4\,b^8\,x^{108\,n}\,x^{9\,p}\,x^9}{12\,n+p+1}+\frac {22\,a^3\,b^9\,x^{120\,n}\,x^{10\,p}\,x^{10}}{12\,n+p+1}+\frac {6\,a^2\,b^{10}\,x^{132\,n}\,x^{11\,p}\,x^{11}}{12\,n+p+1}+\frac {6\,a^{11}\,b\,x^{24\,n}\,x^{2\,p}\,x^2}{12\,n+p+1}+\frac {a\,b^{11}\,x^{144\,n}\,x^{12\,p}\,x^{12}}{12\,n+p+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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