\(\int x^{(-1+n) (1+p)} (b+2 c x^n) (b x+c x^{1+n})^p \, dx\) [176]

Optimal result

Integrand size = 31, antiderivative size = 36 \[ \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx=\frac {x^{-((1-n) (1+p))} \left (b x+c x^{1+n}\right )^{1+p}}{n (1+p)} \]

[Out]

(b*x+c*x^(1+n))^(p+1)/n/(p+1)/(x^((1-n)*(p+1)))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2061} \[ \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx=\frac {x^{-((1-n) (p+1))} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]

[In]

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

[Out]

(b*x + c*x^(1 + n))^(1 + p)/(n*(1 + p)*x^((1 - n)*(1 + p)))

Rule 2061

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] /; FreeQ[{a, b, c, d, e,
j, m, n, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && EqQ[a*d*(m + j*p + 1) - b*c*(m + n
+ p*(j + n) + 1), 0] && (GtQ[e, 0] || IntegersQ[j]) && NeQ[m + j*p + 1, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.00 \[ \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx=\frac {x^{-p} \left (x \left (b+c x^n\right )\right )^p \left (1+\frac {c x^n}{b}\right )^{-p} \left (b (2+p) x^{n (1+p)} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,-\frac {c x^n}{b}\right )+2 c (1+p) x^{n (2+p)} \operatorname {Hypergeometric2F1}\left (-p,2+p,3+p,-\frac {c x^n}{b}\right )\right )}{n (1+p) (2+p)} \]

[In]

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

[Out]

((x*(b + c*x^n))^p*(b*(2 + p)*x^(n*(1 + p))*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^n)/b)] + 2*c*(1 + p)*x^
(n*(2 + p))*Hypergeometric2F1[-p, 2 + p, 3 + p, -((c*x^n)/b)]))/(n*(1 + p)*(2 + p)*x^p*(1 + (c*x^n)/b)^p)

Maple [F]

\[\int x^{\left (-1+n \right ) \left (1+p \right )} \left (b +2 c \,x^{n}\right ) \left (b x +c \,x^{1+n}\right )^{p}d x\]

[In]

int(x^((-1+n)*(1+p))*(b+2*c*x^n)*(b*x+c*x^(1+n))^p,x)

[Out]

int(x^((-1+n)*(1+p))*(b+2*c*x^n)*(b*x+c*x^(1+n))^p,x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx=\frac {{\left (b x + c x^{n + 1}\right )} {\left (b x + c x^{n + 1}\right )}^{p} x^{{\left (n - 1\right )} p + n - 1}}{n p + n} \]

[In]

integrate(x^((-1+n)*(1+p))*(b+2*c*x^n)*(b*x+c*x^(1+n))^p,x, algorithm="fricas")

[Out]

(b*x + c*x^(n + 1))*(b*x + c*x^(n + 1))^p*x^((n - 1)*p + n - 1)/(n*p + n)

Sympy [F(-1)]

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

[In]

integrate(x**((-1+n)*(1+p))*(b+2*c*x**n)*(b*x+c*x**(1+n))**p,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx=\frac {{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (n p \log \left (x\right ) + p \log \left (c x^{n} + b\right )\right )}}{n {\left (p + 1\right )}} \]

[In]

integrate(x^((-1+n)*(1+p))*(b+2*c*x^n)*(b*x+c*x^(1+n))^p,x, algorithm="maxima")

[Out]

(c*x^(2*n) + b*x^n)*e^(n*p*log(x) + p*log(c*x^n + b))/(n*(p + 1))

Giac [F]

\[ \int x^{(-1+n) (1+p)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^p \, dx=\int { {\left (2 \, c x^{n} + b\right )} {\left (b x + c x^{n + 1}\right )}^{p} x^{{\left (n - 1\right )} {\left (p + 1\right )}} \,d x } \]

[In]

integrate(x^((-1+n)*(1+p))*(b+2*c*x^n)*(b*x+c*x^(1+n))^p,x, algorithm="giac")

[Out]

integrate((2*c*x^n + b)*(b*x + c*x^(n + 1))^p*x^((n - 1)*(p + 1)), x)

Mupad [F(-1)]

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

[In]

int(x^((n - 1)*(p + 1))*(b*x + c*x^(n + 1))^p*(b + 2*c*x^n),x)

[Out]

int(x^((n - 1)*(p + 1))*(b*x + c*x^(n + 1))^p*(b + 2*c*x^n), x)