3.3.38 \(\int (a x^m+b x^{1+13 m})^{12} \, dx\)

Optimal. Leaf size=27 \[ \frac {\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1593, 261} \begin {gather*} \frac {\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^m + b*x^(1 + 13*m))^12,x]

[Out]

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

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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 24, normalized size = 0.89 \begin {gather*} \frac {\left (a+b x^{12 m+1}\right )^{13}}{156 b m+13 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x^(1 + 12*m))^13/(13*b + 156*b*m)

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IntegrateAlgebraic [B]  time = 0.06, size = 196, normalized size = 7.26 \begin {gather*} \frac {x^{12 m+1} \left (13 a^{12}+78 a^{11} b x^{12 m+1}+286 a^{10} b^2 x^{24 m+2}+715 a^9 b^3 x^{36 m+3}+1287 a^8 b^4 x^{48 m+4}+1716 a^7 b^5 x^{60 m+5}+1716 a^6 b^6 x^{72 m+6}+1287 a^5 b^7 x^{84 m+7}+715 a^4 b^8 x^{96 m+8}+286 a^3 b^9 x^{108 m+9}+78 a^2 b^{10} x^{120 m+10}+13 a b^{11} x^{132 m+11}+b^{12} x^{144 m+12}\right )}{13 (12 m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 205, normalized size = 7.59 \begin {gather*} \frac {b^{12} x^{13} x^{156 \, m} + 13 \, a b^{11} x^{12} x^{144 \, m} + 78 \, a^{2} b^{10} x^{11} x^{132 \, m} + 286 \, a^{3} b^{9} x^{10} x^{120 \, m} + 715 \, a^{4} b^{8} x^{9} x^{108 \, m} + 1287 \, a^{5} b^{7} x^{8} x^{96 \, m} + 1716 \, a^{6} b^{6} x^{7} x^{84 \, m} + 1716 \, a^{7} b^{5} x^{6} x^{72 \, m} + 1287 \, a^{8} b^{4} x^{5} x^{60 \, m} + 715 \, a^{9} b^{3} x^{4} x^{48 \, m} + 286 \, a^{10} b^{2} x^{3} x^{36 \, m} + 78 \, a^{11} b x^{2} x^{24 \, m} + 13 \, a^{12} x x^{12 \, m}}{13 \, {\left (12 \, m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.27, size = 205, normalized size = 7.59 \begin {gather*} \frac {b^{12} x^{13} x^{156 \, m} + 13 \, a b^{11} x^{12} x^{144 \, m} + 78 \, a^{2} b^{10} x^{11} x^{132 \, m} + 286 \, a^{3} b^{9} x^{10} x^{120 \, m} + 715 \, a^{4} b^{8} x^{9} x^{108 \, m} + 1287 \, a^{5} b^{7} x^{8} x^{96 \, m} + 1716 \, a^{6} b^{6} x^{7} x^{84 \, m} + 1716 \, a^{7} b^{5} x^{6} x^{72 \, m} + 1287 \, a^{8} b^{4} x^{5} x^{60 \, m} + 715 \, a^{9} b^{3} x^{4} x^{48 \, m} + 286 \, a^{10} b^{2} x^{3} x^{36 \, m} + 78 \, a^{11} b x^{2} x^{24 \, m} + 13 \, a^{12} x x^{12 \, m}}{13 \, {\left (12 \, m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 287, normalized size = 10.63 \begin {gather*} \frac {b^{12} x^{13} x^{156 m}}{13+156 m}+\frac {a \,b^{11} x^{12} x^{144 m}}{12 m +1}+\frac {6 a^{2} b^{10} x^{11} x^{132 m}}{12 m +1}+\frac {22 a^{3} b^{9} x^{10} x^{120 m}}{12 m +1}+\frac {55 a^{4} b^{8} x^{9} x^{108 m}}{12 m +1}+\frac {99 a^{5} b^{7} x^{8} x^{96 m}}{12 m +1}+\frac {132 a^{6} b^{6} x^{7} x^{84 m}}{12 m +1}+\frac {132 a^{7} b^{5} x^{6} x^{72 m}}{12 m +1}+\frac {99 a^{8} b^{4} x^{5} x^{60 m}}{12 m +1}+\frac {55 a^{9} b^{3} x^{4} x^{48 m}}{12 m +1}+\frac {22 a^{10} b^{2} x^{3} x^{36 m}}{12 m +1}+\frac {6 a^{11} b \,x^{2} x^{24 m}}{12 m +1}+\frac {a^{12} x \,x^{12 m}}{12 m +1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.48, size = 275, normalized size = 10.19 \begin {gather*} \frac {b^{12} x^{156 \, m + 13}}{13 \, {\left (12 \, m + 1\right )}} + \frac {a b^{11} x^{144 \, m + 12}}{12 \, m + 1} + \frac {6 \, a^{2} b^{10} x^{132 \, m + 11}}{12 \, m + 1} + \frac {22 \, a^{3} b^{9} x^{120 \, m + 10}}{12 \, m + 1} + \frac {55 \, a^{4} b^{8} x^{108 \, m + 9}}{12 \, m + 1} + \frac {99 \, a^{5} b^{7} x^{96 \, m + 8}}{12 \, m + 1} + \frac {132 \, a^{6} b^{6} x^{84 \, m + 7}}{12 \, m + 1} + \frac {132 \, a^{7} b^{5} x^{72 \, m + 6}}{12 \, m + 1} + \frac {99 \, a^{8} b^{4} x^{60 \, m + 5}}{12 \, m + 1} + \frac {55 \, a^{9} b^{3} x^{48 \, m + 4}}{12 \, m + 1} + \frac {22 \, a^{10} b^{2} x^{36 \, m + 3}}{12 \, m + 1} + \frac {6 \, a^{11} b x^{24 \, m + 2}}{12 \, m + 1} + \frac {a^{12} x^{12 \, m + 1}}{12 \, m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.95, size = 285, normalized size = 10.56 \begin {gather*} \frac {b^{12}\,x^{156\,m}\,x^{13}}{156\,m+13}+\frac {a^{12}\,x\,x^{12\,m}}{12\,m+1}+\frac {6\,a^{11}\,b\,x^{24\,m}\,x^2}{12\,m+1}+\frac {a\,b^{11}\,x^{144\,m}\,x^{12}}{12\,m+1}+\frac {22\,a^{10}\,b^2\,x^{36\,m}\,x^3}{12\,m+1}+\frac {55\,a^9\,b^3\,x^{48\,m}\,x^4}{12\,m+1}+\frac {99\,a^8\,b^4\,x^{60\,m}\,x^5}{12\,m+1}+\frac {132\,a^7\,b^5\,x^{72\,m}\,x^6}{12\,m+1}+\frac {132\,a^6\,b^6\,x^{84\,m}\,x^7}{12\,m+1}+\frac {99\,a^5\,b^7\,x^{96\,m}\,x^8}{12\,m+1}+\frac {55\,a^4\,b^8\,x^{108\,m}\,x^9}{12\,m+1}+\frac {22\,a^3\,b^9\,x^{120\,m}\,x^{10}}{12\,m+1}+\frac {6\,a^2\,b^{10}\,x^{132\,m}\,x^{11}}{12\,m+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^m + b*x^(13*m + 1))^12,x)

[Out]

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

[Out]

Timed out

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