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 = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 37, normalized size = 1.00 \[ -\frac {x (c x)^{-n (p+1)-1} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]

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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fricas [A]  time = 0.57, size = 75, normalized size = 2.03 \[ -\frac {{\left (b x x^{n} e^{\left (-{\left (n p + n + 1\right )} \log \relax (c) - {\left (n p + n + 1\right )} \log \relax (x)\right )} + a x e^{\left (-{\left (n p + n + 1\right )} \log \relax (c) - {\left (n p + n + 1\right )} \log \relax (x)\right )}\right )} {\left (b x^{n} + a\right )}^{p}}{a n p + a n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1}\,{d x} \]

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)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{-n p -n -1} \left (b \,x^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c\,x\right )}^{n+n\,p+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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