3.11.81 \(\int x^{-1+n (1+p)} (b+c x^n)^p (b+2 c x^n) \, dx\) [1081]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {460} \begin {gather*} \frac {x^{n (p+1)} \left (b+c x^n\right )^{p+1}}{n (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 460

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

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{n+n p} \left (b+c x^n\right )^{1+p}}{n+n p} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int x^{-1+n \left (1+p \right )} \left (b +c \,x^{n}\right )^{p} \left (b +2 c \,x^{n}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.36, size = 39, normalized size = 1.44 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n*(1+p))*(b+c*x^n)^p*(b+2*c*x^n),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 = 2.86, size = 35, normalized size = 1.30 \begin {gather*} \frac {{\left (c x x^{n} + b x\right )} {\left (c x^{n} + b\right )}^{p} x^{n p + n - 1}}{n p + n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
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*(1+p))*(b+c*x**n)**p*(b+2*c*x**n),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).
time = 0.65, size = 66, normalized size = 2.44 \begin {gather*} \frac {{\left (c x^{n} + b\right )}^{p} c x x^{n} e^{\left (n p \log \left (x\right ) + n \log \left (x\right ) - \log \left (x\right )\right )} + {\left (c x^{n} + b\right )}^{p} b x e^{\left (n p \log \left (x\right ) + n \log \left (x\right ) - \log \left (x\right )\right )}}{n p + n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.86, size = 54, normalized size = 2.00 \begin {gather*} \left (\frac {b\,x\,x^{n\,\left (p+1\right )-1}}{n\,\left (p+1\right )}+\frac {c\,x\,x^n\,x^{n\,\left (p+1\right )-1}}{n\,\left (p+1\right )}\right )\,{\left (b+c\,x^n\right )}^p \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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