\(\int (g x)^m (a+b x^n+c x^{2 n})^p (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}) \, dx\) [16]

Optimal result

Integrand size = 56, antiderivative size = 29 \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx=\frac {(g x)^{1+m} \left (a+b x^n+c x^{2 n}\right )^{1+p}}{g} \]

[Out]

(g*x)^(1+m)*(a+b*x^n+c*x^(2*n))^(p+1)/g

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {1761} \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx=\frac {(g x)^{m+1} \left (a+b x^n+c x^{2 n}\right )^{p+1}}{g} \]

[In]

Int[(g*x)^m*(a + b*x^n + c*x^(2*n))^p*(a*(1 + m) + b*(1 + m + n + n*p)*x^n + c*(1 + m + 2*n*(1 + p))*x^(2*n)),
x]

[Out]

((g*x)^(1 + m)*(a + b*x^n + c*x^(2*n))^(1 + p))/g

Rule 1761

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.) + (f_.)*(x
_)^(n2_.)), x_Symbol] :> Simp[d*(g*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*g*(m + 1))), x] /; FreeQ[{a,
b, c, d, e, f, g, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1), 0] && EqQ[a*f*(m
+ 1) - c*d*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(g x)^{1+m} \left (a+b x^n+c x^{2 n}\right )^{1+p}}{g} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx=x (g x)^m \left (a+x^n \left (b+c x^n\right )\right )^{1+p} \]

[In]

Integrate[(g*x)^m*(a + b*x^n + c*x^(2*n))^p*(a*(1 + m) + b*(1 + m + n + n*p)*x^n + c*(1 + m + 2*n*(1 + p))*x^(
2*n)),x]

[Out]

x*(g*x)^m*(a + x^n*(b + c*x^n))^(1 + p)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(29)=58\).

Time = 102.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10

method	result	size
parallelrisch	\(\frac {x \,x^{n} \left (g x \right )^{m} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p} b c +x \,x^{2 n} \left (g x \right )^{m} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p} c^{2}+x \left (g x \right )^{m} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p} a c}{c}\)	\(90\)

[In]

int((g*x)^m*(a+b*x^n+c*x^(2*n))^p*(a*(1+m)+b*(n*p+m+n+1)*x^n+c*(1+m+2*n*(1+p))*x^(2*n)),x,method=_RETURNVERBOS
E)

[Out]

(x*x^n*(g*x)^m*(a+b*x^n+c*x^(2*n))^p*b*c+x*x^(2*n)*(g*x)^m*(a+b*x^n+c*x^(2*n))^p*c^2+x*(g*x)^m*(a+b*x^n+c*x^(2
*n))^p*a*c)/c

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).

Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx={\left (c x x^{2 \, n} e^{\left (m \log \left (g\right ) + m \log \left (x\right )\right )} + b x x^{n} e^{\left (m \log \left (g\right ) + m \log \left (x\right )\right )} + a x e^{\left (m \log \left (g\right ) + m \log \left (x\right )\right )}\right )} {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \]

[In]

integrate((g*x)^m*(a+b*x^n+c*x^(2*n))^p*(a*(1+m)+b*(n*p+m+n+1)*x^n+c*(1+m+2*n*(1+p))*x^(2*n)),x, algorithm="fr
icas")

[Out]

(c*x*x^(2*n)*e^(m*log(g) + m*log(x)) + b*x*x^n*e^(m*log(g) + m*log(x)) + a*x*e^(m*log(g) + m*log(x)))*(c*x^(2*
n) + b*x^n + a)^p

Sympy [F(-1)]

Timed out. \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx=\text {Timed out} \]

[In]

integrate((g*x)**m*(a+b*x**n+c*x**(2*n))**p*(a*(1+m)+b*(n*p+m+n+1)*x**n+c*(1+m+2*n*(1+p))*x**(2*n)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx={\left (a g^{m} x x^{m} + c g^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )} + b g^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}\right )} {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \]

[In]

integrate((g*x)^m*(a+b*x^n+c*x^(2*n))^p*(a*(1+m)+b*(n*p+m+n+1)*x^n+c*(1+m+2*n*(1+p))*x^(2*n)),x, algorithm="ma
xima")

[Out]

(a*g^m*x*x^m + c*g^m*x*e^(m*log(x) + 2*n*log(x)) + b*g^m*x*e^(m*log(x) + n*log(x)))*(c*x^(2*n) + b*x^n + a)^p

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (29) = 58\).

Time = 0.44 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.31 \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx={\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} c x x^{2 \, n} e^{\left (m \log \left (g\right ) + m \log \left (x\right )\right )} + {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} b x x^{n} e^{\left (m \log \left (g\right ) + m \log \left (x\right )\right )} + {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} a x e^{\left (m \log \left (g\right ) + m \log \left (x\right )\right )} \]

[In]

integrate((g*x)^m*(a+b*x^n+c*x^(2*n))^p*(a*(1+m)+b*(n*p+m+n+1)*x^n+c*(1+m+2*n*(1+p))*x^(2*n)),x, algorithm="gi
ac")

[Out]

(c*x^(2*n) + b*x^n + a)^p*c*x*x^(2*n)*e^(m*log(g) + m*log(x)) + (c*x^(2*n) + b*x^n + a)^p*b*x*x^n*e^(m*log(g)
+ m*log(x)) + (c*x^(2*n) + b*x^n + a)^p*a*x*e^(m*log(g) + m*log(x))

Mupad [B] (verification not implemented)

Time = 8.73 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int (g x)^m \left (a+b x^n+c x^{2 n}\right )^p \left (a (1+m)+b (1+m+n+n p) x^n+c (1+m+2 n (1+p)) x^{2 n}\right ) \, dx=\left (a\,x\,{\left (g\,x\right )}^m+b\,x\,x^n\,{\left (g\,x\right )}^m+c\,x\,x^{2\,n}\,{\left (g\,x\right )}^m\right )\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^p \]

[In]

int((g*x)^m*(a + b*x^n + c*x^(2*n))^p*(a*(m + 1) + b*x^n*(m + n + n*p + 1) + c*x^(2*n)*(m + 2*n*(p + 1) + 1)),
x)

[Out]

(a*x*(g*x)^m + b*x*x^n*(g*x)^m + c*x*x^(2*n)*(g*x)^m)*(a + b*x^n + c*x^(2*n))^p