\(\int (c x)^{-1-n-n p} (a+b x^n)^p \, dx\) [2798]

Optimal result

Integrand size = 23, antiderivative size = 37 \[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=-\frac {(c x)^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {270} \[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=-\frac {(c x)^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

[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 270

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*} \text {integral}& = -\frac {(c x)^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=-\frac {x (c x)^{-1-n (1+p)} \left (a+b x^n\right )^{1+p}}{a n (1+p)} \]

[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)))

Maple [A] (verified)

Time = 4.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.00

method	result	size
parallelrisch	\(-\frac {x \,x^{n} \left (a +b \,x^{n}\right )^{p} \left (c x \right )^{-n p -n -1} b^{2}+x \left (a +b \,x^{n}\right )^{p} \left (c x \right )^{-n p -n -1} a b}{b n \left (1+p \right ) a}\)	\(74\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.03 \[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=-\frac {{\left (b x x^{n} e^{\left (-{\left (n p + n + 1\right )} \log \left (c\right ) - {\left (n p + n + 1\right )} \log \left (x\right )\right )} + a x e^{\left (-{\left (n p + n + 1\right )} \log \left (c\right ) - {\left (n p + n + 1\right )} \log \left (x\right )\right )}\right )} {\left (b x^{n} + a\right )}^{p}}{a n p + a n} \]

[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)

Sympy [A] (verification not implemented)

Time = 4.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=\frac {a^{p} a^{- p - 1} b^{p + 1} c^{- n p - n - 1} \left (\frac {a x^{- n}}{b} + 1\right )^{p + 1} \Gamma \left (- p - 1\right )}{n \Gamma \left (- p\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1} \,d x } \]

[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)

Giac [F]

\[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c\,x\right )}^{n+n\,p+1}} \,d x \]

[In]

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

[Out]

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