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

Optimal. Leaf size=20 \[ x \left (a+b x^n+c x^{2 n}\right )^{p+1} \]

[Out]

x*(a + b*x^n + c*x^(2*n))^(1 + p)

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Rubi [A]  time = 0.023506, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {1775} \[ x \left (a+b x^n+c x^{2 n}\right )^{p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(a + b*x^n + c*x^(2*n))^(1 + p)

Rule 1775

Int[((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.) + (f_.)*(x_)^(n2_.)), x_Symbo
l] :> Simp[(d*x*(a + b*x^n + c*x^(2*n))^(p + 1))/a, x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] &
& EqQ[a*e - b*d*(n*(p + 1) + 1), 0] && EqQ[a*f - c*d*(2*n*(p + 1) + 1), 0]

Rubi steps

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

Mathematica [A]  time = 0.354304, size = 19, normalized size = 0.95 \[ x \left (a+x^n \left (b+c x^n\right )\right )^{p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(a + x^n*(b + c*x^n))^(1 + p)

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Maple [A]  time = 0.046, size = 33, normalized size = 1.7 \begin{align*} x \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.4414, size = 47, normalized size = 2.35 \begin{align*}{\left (c x x^{2 \, n} + b x x^{n} + a x\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78385, size = 77, normalized size = 3.85 \begin{align*}{\left (c x x^{2 \, n} + b x x^{n} + a x\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [B]  time = 1.15811, size = 89, normalized size = 4.45 \begin{align*}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} c x x^{2 \, n} +{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} b x x^{n} +{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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