3.2721 \(\int x^{-1+2 n} (a+b x^n)^p \, dx\)

Optimal. Leaf size=49 \[ \frac {\left (a+b x^n\right )^{p+2}}{b^2 n (p+2)}-\frac {a \left (a+b x^n\right )^{p+1}}{b^2 n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac {\left (a+b x^n\right )^{p+2}}{b^2 n (p+2)}-\frac {a \left (a+b x^n\right )^{p+1}}{b^2 n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)*(a + b*x^n)^p,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.82 \[ \frac {\left (a+b x^n\right )^{p+1} \left (b (p+1) x^n-a\right )}{b^2 n (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 62, normalized size = 1.27 \[ \frac {{\left (a b p x^{n} - a^{2} + {\left (b^{2} p + b^{2}\right )} x^{2 \, n}\right )} {\left (b x^{n} + a\right )}^{p}}{b^{2} n p^{2} + 3 \, b^{2} n p + 2 \, b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} x^{2 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 61, normalized size = 1.24 \[ -\frac {\left (-a b p \,x^{n}-b^{2} p \,x^{2 n}-b^{2} x^{2 n}+a^{2}\right ) \left (b \,x^{n}+a \right )^{p}}{\left (p +1\right ) \left (p +2\right ) b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 51, normalized size = 1.04 \[ \frac {{\left (b^{2} {\left (p + 1\right )} x^{2 \, n} + a b p x^{n} - a^{2}\right )} {\left (b x^{n} + a\right )}^{p}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{2\,n-1}\,{\left (a+b\,x^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2*n - 1)*(a + b*x^n)^p,x)

[Out]

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

[Out]

Timed out

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