### 3.140 $$\int x^{-1+n} (b+2 c x^n) (b x^n+c x^{2 n})^p \, dx$$

Optimal. Leaf size=26 $\frac{\left (b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)}$

[Out]

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

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Rubi [A]  time = 0.0789861, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {2034, 629} $\frac{\left (b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 2034

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n
, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x]
/; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && Integ
erQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (b x^n+c x^{2 n}\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int (b+2 c x) \left (b x+c x^2\right )^p \, dx,x,x^n\right )}{n}\\ &=\frac{\left (b x^n+c x^{2 n}\right )^{1+p}}{n (1+p)}\\ \end{align*}

Mathematica [C]  time = 0.130146, size = 111, normalized size = 4.27 $\frac{x^{-n p} \left (x^n \left (b+c x^n\right )\right )^p \left (\frac{c x^n}{b}+1\right )^{-p} \left (b (p+2) x^{n (p+1)} \, _2F_1\left (-p,p+1;p+2;-\frac{c x^n}{b}\right )+2 c (p+1) x^{n (p+2)} \, _2F_1\left (-p,p+2;p+3;-\frac{c x^n}{b}\right )\right )}{n (p+1) (p+2)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((x^n*(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^(n*p)*(1 + (c*x^n)/b)^p
)

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Maple [C]  time = 0.098, size = 155, normalized size = 6. \begin{align*}{\frac{{x}^{n} \left ( b+c{x}^{n} \right ) }{n \left ( 1+p \right ) }{{\rm e}^{-{\frac{p \left ( i\pi \, \left ({\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ) \right ) ^{2}{\it csgn} \left ( i \left ( b+c{x}^{n} \right ) \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \left ( b+c{x}^{n} \right ) \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( i \left ( b+c{x}^{n} \right ) \right ) -2\,\ln \left ({x}^{n} \right ) -2\,\ln \left ( b+c{x}^{n} \right ) \right ) }{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^n*(b+c*x^n)/n/(1+p)*exp(-1/2*p*(I*Pi*csgn(I*x^n*(b+c*x^n))^3-I*Pi*csgn(I*x^n*(b+c*x^n))^2*csgn(I*x^n)-I*Pi*c
sgn(I*x^n*(b+c*x^n))^2*csgn(I*(b+c*x^n))+I*Pi*csgn(I*x^n*(b+c*x^n))*csgn(I*x^n)*csgn(I*(b+c*x^n))-2*ln(x^n)-2*
ln(b+c*x^n)))

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Maxima [A]  time = 1.32212, size = 54, normalized size = 2.08 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (p \log \left (c x^{n} + b\right ) + p \log \left (x^{n}\right )\right )}}{n{\left (p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.29047, size = 72, normalized size = 2.77 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n}\right )}{\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{n p + n} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^(2*n) + b*x^n)*(c*x^(2*n) + b*x^n)^p/(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)*(b+2*c*x**n)*(b*x**n+c*x**(2*n))**p,x)

[Out]

Timed out

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Giac [A]  time = 1.14458, size = 35, normalized size = 1.35 \begin{align*} \frac{{\left (c x^{2 \, n} + b x^{n}\right )}^{p + 1}}{n{\left (p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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