3.2799 \(\int (c x)^{-1-2 n-n p} (a+b x^n)^p \, dx\)

Optimal. Leaf size=79 \[ \frac{(c x)^{-n (p+2)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac{(c x)^{-n (p+2)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0393983, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac{(c x)^{-n (p+2)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac{(c x)^{-n (p+2)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 273

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

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

Mathematica [C]  time = 0.0253821, size = 69, normalized size = 0.87 \[ -\frac{x (c x)^{-n (p+2)-1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p-2,-p;-p-1;-\frac{b x^n}{a}\right )}{n (p+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(c*x)^(-1 - n*(2 + p))*(a + b*x^n)^p*Hypergeometric2F1[-2 - p, -p, -1 - p, -((b*x^n)/a)])/(n*(2 + p)*(1 +
(b*x^n)/a)^p))

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{-np-2\,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-2*n-1)*(a+b*x^n)^p,x)

[Out]

int((c*x)^(-n*p-2*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 - 2 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.39999, size = 347, normalized size = 4.39 \begin{align*} -\frac{{\left (a b p x x^{n} e^{\left (-{\left (n p + 2 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 2 \, n + 1\right )} \log \left (x\right )\right )} - b^{2} x x^{2 \, n} e^{\left (-{\left (n p + 2 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 2 \, n + 1\right )} \log \left (x\right )\right )} +{\left (a^{2} p + a^{2}\right )} x e^{\left (-{\left (n p + 2 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 2 \, n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}}{a^{2} n p^{2} + 3 \, a^{2} n p + 2 \, a^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*b*p*x*x^n*e^(-(n*p + 2*n + 1)*log(c) - (n*p + 2*n + 1)*log(x)) - b^2*x*x^(2*n)*e^(-(n*p + 2*n + 1)*log(c)
- (n*p + 2*n + 1)*log(x)) + (a^2*p + a^2)*x*e^(-(n*p + 2*n + 1)*log(c) - (n*p + 2*n + 1)*log(x)))*(b*x^n + a)^
p/(a^2*n*p^2 + 3*a^2*n*p + 2*a^2*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-2*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 - 2 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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