3.3.34 \(\int x^{12 (-1+m)} (a x+b x^{2+12 m})^{12} \, dx\)

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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1584, 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[x^(12*(-1 + m))*(a*x + b*x^(2 + 12*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 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^{12 (-1+m)} \left (a x+b x^{2+12 m}\right )^{12} \, dx &=\int x^{12+12 (-1+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 = 0.03, 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[x^(12*(-1 + m))*(a*x + b*x^(2 + 12*m))^12,x]

[Out]

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

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IntegrateAlgebraic [B]  time = 0.07, 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[x^(12*(-1 + m))*(a*x + b*x^(2 + 12*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 = 0.44, size = 231, normalized size = 8.56 \begin {gather*} \frac {13 \, a^{12} x^{12} x^{12 \, m + 2} + 78 \, a^{11} b x^{11} x^{24 \, m + 4} + 286 \, a^{10} b^{2} x^{10} x^{36 \, m + 6} + 715 \, a^{9} b^{3} x^{9} x^{48 \, m + 8} + 1287 \, a^{8} b^{4} x^{8} x^{60 \, m + 10} + 1716 \, a^{7} b^{5} x^{7} x^{72 \, m + 12} + 1716 \, a^{6} b^{6} x^{6} x^{84 \, m + 14} + 1287 \, a^{5} b^{7} x^{5} x^{96 \, m + 16} + 715 \, a^{4} b^{8} x^{4} x^{108 \, m + 18} + 286 \, a^{3} b^{9} x^{3} x^{120 \, m + 20} + 78 \, a^{2} b^{10} x^{2} x^{132 \, m + 22} + 13 \, a b^{11} x x^{144 \, m + 24} + b^{12} x^{156 \, m + 26}}{13 \, {\left (12 \, m + 1\right )} x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-12+12*m)*(a*x+b*x^(2+12*m))^12,x, algorithm="fricas")

[Out]

1/13*(13*a^12*x^12*x^(12*m + 2) + 78*a^11*b*x^11*x^(24*m + 4) + 286*a^10*b^2*x^10*x^(36*m + 6) + 715*a^9*b^3*x
^9*x^(48*m + 8) + 1287*a^8*b^4*x^8*x^(60*m + 10) + 1716*a^7*b^5*x^7*x^(72*m + 12) + 1716*a^6*b^6*x^6*x^(84*m +
14) + 1287*a^5*b^7*x^5*x^(96*m + 16) + 715*a^4*b^8*x^4*x^(108*m + 18) + 286*a^3*b^9*x^3*x^(120*m + 20) + 78*a
^2*b^10*x^2*x^(132*m + 22) + 13*a*b^11*x*x^(144*m + 24) + b^12*x^(156*m + 26))/((12*m + 1)*x^13)

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giac [B]  time = 0.51, size = 285, normalized size = 10.56 \begin {gather*} \frac {13 \, a^{12} x^{12} e^{\left (12 \, m \log \relax (x) + 2 \, \log \relax (x)\right )} + 78 \, a^{11} b x^{11} e^{\left (24 \, m \log \relax (x) + 4 \, \log \relax (x)\right )} + 286 \, a^{10} b^{2} x^{10} e^{\left (36 \, m \log \relax (x) + 6 \, \log \relax (x)\right )} + 715 \, a^{9} b^{3} x^{9} e^{\left (48 \, m \log \relax (x) + 8 \, \log \relax (x)\right )} + 1287 \, a^{8} b^{4} x^{8} e^{\left (60 \, m \log \relax (x) + 10 \, \log \relax (x)\right )} + 1716 \, a^{7} b^{5} x^{7} e^{\left (72 \, m \log \relax (x) + 12 \, \log \relax (x)\right )} + 1716 \, a^{6} b^{6} x^{6} e^{\left (84 \, m \log \relax (x) + 14 \, \log \relax (x)\right )} + 1287 \, a^{5} b^{7} x^{5} e^{\left (96 \, m \log \relax (x) + 16 \, \log \relax (x)\right )} + 715 \, a^{4} b^{8} x^{4} e^{\left (108 \, m \log \relax (x) + 18 \, \log \relax (x)\right )} + 286 \, a^{3} b^{9} x^{3} e^{\left (120 \, m \log \relax (x) + 20 \, \log \relax (x)\right )} + 78 \, a^{2} b^{10} x^{2} e^{\left (132 \, m \log \relax (x) + 22 \, \log \relax (x)\right )} + 13 \, a b^{11} x e^{\left (144 \, m \log \relax (x) + 24 \, \log \relax (x)\right )} + b^{12} e^{\left (156 \, m \log \relax (x) + 26 \, \log \relax (x)\right )}}{13 \, {\left (12 \, m x^{13} + x^{13}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-12+12*m)*(a*x+b*x^(2+12*m))^12,x, algorithm="giac")

[Out]

1/13*(13*a^12*x^12*e^(12*m*log(x) + 2*log(x)) + 78*a^11*b*x^11*e^(24*m*log(x) + 4*log(x)) + 286*a^10*b^2*x^10*
e^(36*m*log(x) + 6*log(x)) + 715*a^9*b^3*x^9*e^(48*m*log(x) + 8*log(x)) + 1287*a^8*b^4*x^8*e^(60*m*log(x) + 10
*log(x)) + 1716*a^7*b^5*x^7*e^(72*m*log(x) + 12*log(x)) + 1716*a^6*b^6*x^6*e^(84*m*log(x) + 14*log(x)) + 1287*
a^5*b^7*x^5*e^(96*m*log(x) + 16*log(x)) + 715*a^4*b^8*x^4*e^(108*m*log(x) + 18*log(x)) + 286*a^3*b^9*x^3*e^(12
0*m*log(x) + 20*log(x)) + 78*a^2*b^10*x^2*e^(132*m*log(x) + 22*log(x)) + 13*a*b^11*x*e^(144*m*log(x) + 24*log(
x)) + b^12*e^(156*m*log(x) + 26*log(x)))/(12*m*x^13 + x^13)

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maple [B]  time = 0.11, size = 339, normalized size = 12.56 \begin {gather*} \frac {a^{12} x^{12 m +2}}{\left (12 m +1\right ) x}+\frac {6 a^{11} b \,x^{24 m +4}}{\left (12 m +1\right ) x^{2}}+\frac {22 a^{10} b^{2} x^{36 m +6}}{\left (12 m +1\right ) x^{3}}+\frac {55 a^{9} b^{3} x^{48 m +8}}{\left (12 m +1\right ) x^{4}}+\frac {99 a^{8} b^{4} x^{60 m +10}}{\left (12 m +1\right ) x^{5}}+\frac {132 a^{7} b^{5} x^{72 m +12}}{\left (12 m +1\right ) x^{6}}+\frac {132 a^{6} b^{6} x^{84 m +14}}{\left (12 m +1\right ) x^{7}}+\frac {99 a^{5} b^{7} x^{96 m +16}}{\left (12 m +1\right ) x^{8}}+\frac {55 a^{4} b^{8} x^{108 m +18}}{\left (12 m +1\right ) x^{9}}+\frac {22 a^{3} b^{9} x^{120 m +20}}{\left (12 m +1\right ) x^{10}}+\frac {6 a^{2} b^{10} x^{132 m +22}}{\left (12 m +1\right ) x^{11}}+\frac {a \,b^{11} x^{144 m +24}}{\left (12 m +1\right ) x^{12}}+\frac {b^{12} x^{156 m +26}}{13 \left (12 m +1\right ) x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(12*m-12)*(a*x+b*x^(2+12*m))^12,x)

[Out]

1/13/(12*m+1)*b^12/x^13*(x^(2+12*m))^13+1/(12*m+1)*a*b^11/x^12*(x^(2+12*m))^12+6/(12*m+1)*a^2*b^10/x^11*(x^(2+
12*m))^11+22/(12*m+1)*a^3*b^9/x^10*(x^(2+12*m))^10+55/(12*m+1)*a^4*b^8/x^9*(x^(2+12*m))^9+99/(12*m+1)*a^5*b^7/
x^8*(x^(2+12*m))^8+132/(12*m+1)*a^6*b^6/x^7*(x^(2+12*m))^7+132/(12*m+1)*a^7*b^5/x^6*(x^(2+12*m))^6+99/(12*m+1)
*a^8*b^4/x^5*(x^(2+12*m))^5+55/(12*m+1)*a^9*b^3/x^4*(x^(2+12*m))^4+22/(12*m+1)*a^10*b^2/x^3*(x^(2+12*m))^3+6/(
12*m+1)*a^11*b/x^2*(x^(2+12*m))^2+1/(12*m+1)*a^12/x*x^(2+12*m)

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maxima [B]  time = 1.43, 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(x^(-12+12*m)*(a*x+b*x^(2+12*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 = 5.87, size = 287, normalized size = 10.63 \begin {gather*} \frac {b^{12}\,x^{156\,m}\,x^{13}}{156\,m+13}+\frac {13\,a^{12}\,x\,x^{12\,m}}{156\,m+13}+\frac {78\,a^{11}\,b\,x^{24\,m}\,x^2}{156\,m+13}+\frac {13\,a\,b^{11}\,x^{144\,m}\,x^{12}}{156\,m+13}+\frac {286\,a^{10}\,b^2\,x^{36\,m}\,x^3}{156\,m+13}+\frac {715\,a^9\,b^3\,x^{48\,m}\,x^4}{156\,m+13}+\frac {1287\,a^8\,b^4\,x^{60\,m}\,x^5}{156\,m+13}+\frac {1716\,a^7\,b^5\,x^{72\,m}\,x^6}{156\,m+13}+\frac {1716\,a^6\,b^6\,x^{84\,m}\,x^7}{156\,m+13}+\frac {1287\,a^5\,b^7\,x^{96\,m}\,x^8}{156\,m+13}+\frac {715\,a^4\,b^8\,x^{108\,m}\,x^9}{156\,m+13}+\frac {286\,a^3\,b^9\,x^{120\,m}\,x^{10}}{156\,m+13}+\frac {78\,a^2\,b^{10}\,x^{132\,m}\,x^{11}}{156\,m+13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(12*m - 12)*(a*x + b*x^(12*m + 2))^12,x)

[Out]

(b^12*x^(156*m)*x^13)/(156*m + 13) + (13*a^12*x*x^(12*m))/(156*m + 13) + (78*a^11*b*x^(24*m)*x^2)/(156*m + 13)
+ (13*a*b^11*x^(144*m)*x^12)/(156*m + 13) + (286*a^10*b^2*x^(36*m)*x^3)/(156*m + 13) + (715*a^9*b^3*x^(48*m)*
x^4)/(156*m + 13) + (1287*a^8*b^4*x^(60*m)*x^5)/(156*m + 13) + (1716*a^7*b^5*x^(72*m)*x^6)/(156*m + 13) + (171
6*a^6*b^6*x^(84*m)*x^7)/(156*m + 13) + (1287*a^5*b^7*x^(96*m)*x^8)/(156*m + 13) + (715*a^4*b^8*x^(108*m)*x^9)/
(156*m + 13) + (286*a^3*b^9*x^(120*m)*x^10)/(156*m + 13) + (78*a^2*b^10*x^(132*m)*x^11)/(156*m + 13)

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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**(-12+12*m)*(a*x+b*x**(2+12*m))**12,x)

[Out]

Timed out

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