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

Optimal. Leaf size=29 \[ \frac{(g x)^{m+1} \left (a+b x^n+c x^{2 n}\right )^{p+1}}{g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0712211, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.018, Rules used = {1747} \[ \frac{(g x)^{m+1} \left (a+b x^n+c x^{2 n}\right )^{p+1}}{g} \]

Antiderivative was successfully verified.

[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 1747

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*} \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}\\ \end{align*}

Mathematica [A]  time = 0.449842, size = 24, normalized size = 0.83 \[ x (g x)^m \left (a+x^n \left (b+c x^n\right )\right )^{p+1} \]

Antiderivative was successfully verified.

[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 [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{m} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{p} \left ( a \left ( 1+m \right ) +b \left ( pn+m+n+1 \right ){x}^{n}+c \left ( 1+m+2\,n \left ( 1+p \right ) \right ){x}^{2\,n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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)

________________________________________________________________________________________

Maxima [B]  time = 1.54354, size = 81, normalized size = 2.79 \begin{align*}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [B]  time = 1.4631, size = 174, normalized size = 6. \begin{align*}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  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((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

________________________________________________________________________________________

Giac [B]  time = 1.18331, size = 130, normalized size = 4.48 \begin{align*}{\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))