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

Optimal. Leaf size=36 \[ \frac{x^{-(1-n) (p+1)} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.085884, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {2036} \[ \frac{x^{-(1-n) (p+1)} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]

Antiderivative was successfully verified.

[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 2036

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

Mathematica [C]  time = 0.147862, size = 108, normalized size = 3. \[ \frac{x^{-p} \left (x \left (b+c x^n\right )\right )^p \left (\frac{c x^n}{b}+1\right )^{-p} \left (b (p+2) x^{n (p+1)} \, _2F_1\left (-p,p+1;p+2;-\frac{c x^n}{b}\right )+2 c (p+1) x^{n (p+2)} \, _2F_1\left (-p,p+2;p+3;-\frac{c x^n}{b}\right )\right )}{n (p+1) (p+2)} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{x}^{ \left ( -1+n \right ) \left ( 1+p \right ) } \left ( b+2\,c{x}^{n} \right ) \left ( bx+c{x}^{1+n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 1.36925, size = 53, normalized size = 1.47 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]  time = 1.46847, size = 101, normalized size = 2.81 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)