3.2798 \(\int (c x)^{-1-n-n p} (a+b x^n)^p \, dx\)

Optimal. Leaf size=37 \[ -\frac{(c x)^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0123034, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {264} \[ -\frac{(c x)^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 264

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

Rubi steps

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

Mathematica [A]  time = 0.0253412, size = 37, normalized size = 1. \[ -\frac{x (c x)^{-n (p+1)-1} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{-np-n-1} \left ( a+b{x}^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42492, size = 194, normalized size = 5.24 \begin{align*} -\frac{{\left (b x x^{n} e^{\left (-{\left (n p + n + 1\right )} \log \left (c\right ) -{\left (n p + n + 1\right )} \log \left (x\right )\right )} + a x e^{\left (-{\left (n p + n + 1\right )} \log \left (c\right ) -{\left (n p + n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}}{a n p + a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x*x^n*e^(-(n*p + n + 1)*log(c) - (n*p + n + 1)*log(x)) + a*x*e^(-(n*p + n + 1)*log(c) - (n*p + n + 1)*log(
x)))*(b*x^n + a)^p/(a*n*p + a*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((c*x)**(-n*p-n-1)*(a+b*x**n)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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