3.6.6 \(\int x^{-1-n (-1+p)} (b x^n+c x^{2 n})^p \, dx\) [506]

Optimal. Leaf size=37 \[ \frac {x^{-n (1+p)} \left (b x^n+c x^{2 n}\right )^{1+p}}{c n (1+p)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2039} \begin {gather*} \frac {x^{-n (p+1)} \left (b x^n+c x^{2 n}\right )^{p+1}}{c n (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2039

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

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 38, normalized size = 1.03 \begin {gather*} \frac {x^{-n p} \left (b+c x^n\right ) \left (x^n \left (b+c x^n\right )\right )^p}{c n (1+p)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{-1-n \left (-1+p \right )} \left (b \,x^{n}+c \,x^{2 n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 43, normalized size = 1.16 \begin {gather*} \frac {{\left (c x^{n} + b\right )} e^{\left (-n p \log \left (x\right ) + p \log \left (c x^{n} + b\right ) + p \log \left (x^{n}\right )\right )}}{c n {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.40, size = 59, normalized size = 1.59 \begin {gather*} \frac {{\left (c x x^{-n p + n - 1} x^{n} + b x x^{-n p + n - 1}\right )} {\left (c x^{2 \, n} + b x^{n}\right )}^{p}}{{\left (c n p + c n\right )} x^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (b\,x^n+c\,x^{2\,n}\right )}^p}{x^{n\,\left (p-1\right )+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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