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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0077874, antiderivative size = 27, normalized size of antiderivative = 1., 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} \[ \frac{\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \]

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 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]

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 \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*}

Mathematica [A]  time = 0.0040226, size = 24, normalized size = 0.89 \[ \frac{\left (a+b x^{12 m+1}\right )^{13}}{156 b m+13 b} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.041, size = 287, normalized size = 10.6 \begin{align*}{\frac{{b}^{12}{x}^{13} \left ({x}^{m} \right ) ^{156}}{13+156\,m}}+{\frac{a{b}^{11}{x}^{12} \left ({x}^{m} \right ) ^{144}}{1+12\,m}}+6\,{\frac{{a}^{2}{b}^{10}{x}^{11} \left ({x}^{m} \right ) ^{132}}{1+12\,m}}+22\,{\frac{{a}^{3}{b}^{9}{x}^{10} \left ({x}^{m} \right ) ^{120}}{1+12\,m}}+55\,{\frac{{a}^{4}{b}^{8}{x}^{9} \left ({x}^{m} \right ) ^{108}}{1+12\,m}}+99\,{\frac{{a}^{5}{b}^{7}{x}^{8} \left ({x}^{m} \right ) ^{96}}{1+12\,m}}+132\,{\frac{{a}^{6}{b}^{6}{x}^{7} \left ({x}^{m} \right ) ^{84}}{1+12\,m}}+132\,{\frac{{a}^{7}{b}^{5}{x}^{6} \left ({x}^{m} \right ) ^{72}}{1+12\,m}}+99\,{\frac{{a}^{8}{b}^{4}{x}^{5} \left ({x}^{m} \right ) ^{60}}{1+12\,m}}+55\,{\frac{{a}^{9}{b}^{3}{x}^{4} \left ({x}^{m} \right ) ^{48}}{1+12\,m}}+22\,{\frac{{a}^{10}{b}^{2}{x}^{3} \left ({x}^{m} \right ) ^{36}}{1+12\,m}}+6\,{\frac{{a}^{11}b{x}^{2} \left ({x}^{m} \right ) ^{24}}{1+12\,m}}+{\frac{{a}^{12}x \left ({x}^{m} \right ) ^{12}}{1+12\,m}} \end{align*}

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 0.862526, size = 493, normalized size = 18.26 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.25509, size = 277, normalized size = 10.26 \begin{align*} \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{align*}

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)