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

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

[Out]

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

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Rubi [A]  time = 0.0147372, antiderivative size = 27, normalized size of antiderivative = 1., 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} \[ \frac{\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \]

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 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^{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*}

Mathematica [A]  time = 0.013241, 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[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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Maple [B]  time = 0.106, size = 339, normalized size = 12.6 \begin{align*}{\frac{{b}^{12} \left ({x}^{2+12\,m} \right ) ^{13}}{ \left ( 13+156\,m \right ){x}^{13}}}+{\frac{a{b}^{11} \left ({x}^{2+12\,m} \right ) ^{12}}{ \left ( 1+12\,m \right ){x}^{12}}}+6\,{\frac{{a}^{2}{b}^{10} \left ({x}^{2+12\,m} \right ) ^{11}}{ \left ( 1+12\,m \right ){x}^{11}}}+22\,{\frac{{a}^{3}{b}^{9} \left ({x}^{2+12\,m} \right ) ^{10}}{ \left ( 1+12\,m \right ){x}^{10}}}+55\,{\frac{{a}^{4}{b}^{8} \left ({x}^{2+12\,m} \right ) ^{9}}{ \left ( 1+12\,m \right ){x}^{9}}}+99\,{\frac{{a}^{5}{b}^{7} \left ({x}^{2+12\,m} \right ) ^{8}}{ \left ( 1+12\,m \right ){x}^{8}}}+132\,{\frac{{a}^{6}{b}^{6} \left ({x}^{2+12\,m} \right ) ^{7}}{ \left ( 1+12\,m \right ){x}^{7}}}+132\,{\frac{{a}^{7}{b}^{5} \left ({x}^{2+12\,m} \right ) ^{6}}{ \left ( 1+12\,m \right ){x}^{6}}}+99\,{\frac{{a}^{8}{b}^{4} \left ({x}^{2+12\,m} \right ) ^{5}}{ \left ( 1+12\,m \right ){x}^{5}}}+55\,{\frac{{a}^{9}{b}^{3} \left ({x}^{2+12\,m} \right ) ^{4}}{ \left ( 1+12\,m \right ){x}^{4}}}+22\,{\frac{{a}^{10}{b}^{2} \left ({x}^{2+12\,m} \right ) ^{3}}{ \left ( 1+12\,m \right ){x}^{3}}}+6\,{\frac{{a}^{11}b \left ({x}^{2+12\,m} \right ) ^{2}}{ \left ( 1+12\,m \right ){x}^{2}}}+{\frac{{a}^{12}{x}^{2+12\,m}}{ \left ( 1+12\,m \right ) x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.723535, size = 578, normalized size = 21.41 \begin{align*} \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{align*}

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a x + b x^{12 \, m + 2}\right )}^{12} x^{12 \, m - 12}\,{d x} \end{align*}

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]

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