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

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

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Rubi [A]  time = 0.08, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \end {gather*}

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 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]

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]

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*}

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Mathematica [C]  time = 0.13, size = 111, normalized size = 4.27 \begin {gather*} \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)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (b x^n+c x^{2 n}\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.88, size = 36, normalized size = 1.38 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n}\right )} {\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{n p + n} \end {gather*}

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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giac [A]  time = 0.83, size = 26, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n}\right )}^{p + 1}}{n {\left (p + 1\right )}} \end {gather*}

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))

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maple [C]  time = 0.11, size = 155, normalized size = 5.96 \begin {gather*} \frac {\left (c \,x^{n}+b \right ) x^{n} {\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right )\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (c \,x^{n}+b \right )\right ) \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (c \,x^{n}+b \right ) x^{n}\right )^{3}+2 \ln \left (x^{n}\right )+2 \ln \left (c \,x^{n}+b \right )\right ) p}{2}}}{\left (p +1\right ) n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.77, size = 40, normalized size = 1.54 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 2.13, size = 34, normalized size = 1.31 \begin {gather*} \frac {x^n\,\left (b+c\,x^n\right )\,{\left (b\,x^n+c\,x^{2\,n}\right )}^p}{n\,\left (p+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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