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

Optimal. Leaf size=27 \[ \frac {\left (a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1468, 629} \begin {gather*} \frac {\left (a+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)*(a + b*x^n + c*x^(2*n))^p,x]

[Out]

(a + 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 1468

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 26, normalized size = 0.96 \begin {gather*} \frac {\left (a+x^n \left (b+c x^n\right )\right )^{p+1}}{n (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (a+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)*(a + b*x^n + c*x^(2*n))^p,x]

[Out]

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

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fricas [A]  time = 0.96, size = 38, normalized size = 1.41 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (c x^{2 \, n} + b x^{n} + a\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)*(a+b*x^n+c*x^(2*n))^p,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.84, size = 27, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n} + a\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)*(a+b*x^n+c*x^(2*n))^p,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 40, normalized size = 1.48 \begin {gather*} \frac {\left (b \,x^{n}+c \,x^{2 n}+a \right ) \left (b \,x^{n}+c \,x^{2 n}+a \right )^{p}}{\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)+a)^p,x)

[Out]

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

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maxima [A]  time = 0.72, size = 39, normalized size = 1.44 \begin {gather*} \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{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)*(a+b*x^n+c*x^(2*n))^p,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.57, size = 56, normalized size = 2.07 \begin {gather*} {\left (a+b\,x^n+c\,x^{2\,n}\right )}^p\,\left (\frac {a}{n\,\left (p+1\right )}+\frac {b\,x^n}{n\,\left (p+1\right )}+\frac {c\,x^{2\,n}}{n\,\left (p+1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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