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

Optimal. Leaf size=38 \[ -\frac{x^{-2 n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{b n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0389941, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {2014} \[ -\frac{x^{-2 n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

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

Mathematica [A]  time = 0.0205104, size = 43, normalized size = 1.13 \[ -\frac{x^{-n (2 p+1)} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-n \left ( 1+2\,p \right ) } \left ( b{x}^{n}+c{x}^{2\,n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n{\left (2 \, p + 1\right )} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.56053, size = 124, normalized size = 3.26 \begin{align*} -\frac{{\left (c x x^{-2 \, n p - n - 1} x^{n} + b x x^{-2 \, n p - n - 1}\right )}{\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{b n p + b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2 \, n} + b x^{n}\right )}^{p} x^{-n{\left (2 \, p + 1\right )} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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