3.2739 \(\int x^m (a+b x^{1+m})^n \, dx\)

Optimal. Leaf size=27 \[ \frac{\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]

[Out]

(a + b*x^(1 + m))^(1 + n)/(b*(1 + m)*(1 + n))

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Rubi [A]  time = 0.0096207, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^(1 + m))^n,x]

[Out]

(a + b*x^(1 + m))^(1 + n)/(b*(1 + m)*(1 + n))

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0095069, size = 27, normalized size = 1. \[ \frac{\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^(1 + m))^n,x]

[Out]

(a + b*x^(1 + m))^(1 + n)/(b*(1 + m)*(1 + n))

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Maple [B]  time = 0.024, size = 60, normalized size = 2.2 \begin{align*}{\frac{a{{\rm e}^{n\ln \left ( a+bx{{\rm e}^{m\ln \left ( x \right ) }} \right ) }}}{b \left ( nm+m+n+1 \right ) }}+{\frac{x{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( a+bx{{\rm e}^{m\ln \left ( x \right ) }} \right ) }}}{nm+m+n+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a+b*x^(1+m))^n,x)

[Out]

a/b/(m*n+m+n+1)*exp(n*ln(a+b*x*exp(m*ln(x))))+1/(m*n+m+n+1)*x*exp(m*ln(x))*exp(n*ln(a+b*x*exp(m*ln(x))))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36694, size = 85, normalized size = 3.15 \begin{align*} \frac{{\left (b x^{m + 1} + a\right )}{\left (b x^{m + 1} + a\right )}^{n}}{b m +{\left (b m + b\right )} n + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^(m + 1) + a)*(b*x^(m + 1) + a)^n/(b*m + (b*m + b)*n + b)

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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(x**m*(a+b*x**(1+m))**n,x)

[Out]

Timed out

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Giac [A]  time = 1.12034, size = 36, normalized size = 1.33 \begin{align*} \frac{{\left (b x^{m + 1} + a\right )}^{n + 1}}{b{\left (m + 1\right )}{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^(m + 1) + a)^(n + 1)/(b*(m + 1)*(n + 1))