3.2800 \(\int (c x)^{-1-3 n-n p} (a+b x^n)^p \, dx\)

Optimal. Leaf size=127 \[ \frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+3}}{a^3 c n (p+1) (p+2) (p+3)}-\frac{(c x)^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

[Out]

-((a + b*x^n)^(1 + p)/(a*c*n*(1 + p)*(c*x)^(n*(3 + p)))) + (2*(a + b*x^n)^(2 + p))/(a^2*c*n*(1 + p)*(2 + p)*(c
*x)^(n*(3 + p))) - (2*(a + b*x^n)^(3 + p))/(a^3*c*n*(1 + p)*(2 + p)*(3 + p)*(c*x)^(n*(3 + p)))

________________________________________________________________________________________

Rubi [A]  time = 0.073555, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+3}}{a^3 c n (p+1) (p+2) (p+3)}-\frac{(c x)^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - 3*n - n*p)*(a + b*x^n)^p,x]

[Out]

-((a + b*x^n)^(1 + p)/(a*c*n*(1 + p)*(c*x)^(n*(3 + p)))) + (2*(a + b*x^n)^(2 + p))/(a^2*c*n*(1 + p)*(2 + p)*(c
*x)^(n*(3 + p))) - (2*(a + b*x^n)^(3 + p))/(a^3*c*n*(1 + p)*(2 + p)*(3 + p)*(c*x)^(n*(3 + 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)^{-1-3 n-n p} \left (a+b x^n\right )^p \, dx &=-\frac{(c x)^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}-\frac{2 \int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^{1+p} \, dx}{a (1+p)}\\ &=-\frac{(c x)^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}+\frac{2 (c x)^{-n (3+p)} \left (a+b x^n\right )^{2+p}}{a^2 c n (1+p) (2+p)}+\frac{2 \int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^{2+p} \, dx}{a^2 (1+p) (2+p)}\\ &=-\frac{(c x)^{-n (3+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}+\frac{2 (c x)^{-n (3+p)} \left (a+b x^n\right )^{2+p}}{a^2 c n (1+p) (2+p)}-\frac{2 (c x)^{-n (3+p)} \left (a+b x^n\right )^{3+p}}{a^3 c n (1+p) (2+p) (3+p)}\\ \end{align*}

Mathematica [C]  time = 0.0261541, size = 69, normalized size = 0.54 \[ -\frac{x (c x)^{-n (p+3)-1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p-3,-p;-p-2;-\frac{b x^n}{a}\right )}{n (p+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(-1 - 3*n - n*p)*(a + b*x^n)^p,x]

[Out]

-((x*(c*x)^(-1 - n*(3 + p))*(a + b*x^n)^p*Hypergeometric2F1[-3 - p, -p, -2 - p, -((b*x^n)/a)])/(n*(3 + p)*(1 +
(b*x^n)/a)^p))

________________________________________________________________________________________

Maple [F]  time = 0.077, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{-np-3\,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-3*n-1)*(a+b*x^n)^p,x)

[Out]

int((c*x)^(-n*p-3*n-1)*(a+b*x^n)^p,x)

________________________________________________________________________________________

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 - 3 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-n*p-3*n-1)*(a+b*x^n)^p,x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^p*(c*x)^(-n*p - 3*n - 1), x)

________________________________________________________________________________________

Fricas [A]  time = 1.51635, size = 510, normalized size = 4.02 \begin{align*} \frac{{\left (2 \, a b^{2} p x x^{2 \, n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} - 2 \, b^{3} x x^{3 \, n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} -{\left (a^{2} b p^{2} + a^{2} b p\right )} x x^{n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} -{\left (a^{3} p^{2} + 3 \, a^{3} p + 2 \, a^{3}\right )} x e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}}{a^{3} n p^{3} + 6 \, a^{3} n p^{2} + 11 \, a^{3} n p + 6 \, a^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-n*p-3*n-1)*(a+b*x^n)^p,x, algorithm="fricas")

[Out]

(2*a*b^2*p*x*x^(2*n)*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)) - 2*b^3*x*x^(3*n)*e^(-(n*p + 3*n + 1
)*log(c) - (n*p + 3*n + 1)*log(x)) - (a^2*b*p^2 + a^2*b*p)*x*x^n*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*
log(x)) - (a^3*p^2 + 3*a^3*p + 2*a^3)*x*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)))*(b*x^n + a)^p/(a
^3*n*p^3 + 6*a^3*n*p^2 + 11*a^3*n*p + 6*a^3*n)

________________________________________________________________________________________

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-3*n-1)*(a+b*x**n)**p,x)

[Out]

Timed out

________________________________________________________________________________________

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 - 3 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-n*p-3*n-1)*(a+b*x^n)^p,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^p*(c*x)^(-n*p - 3*n - 1), x)