3.2542 \(\int x^{-1+n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=19 \[ \frac{\left (a+b x^n\right )^4}{4 b n} \]

[Out]

(a + b*x^n)^4/(4*b*n)

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Rubi [A]  time = 0.0045087, antiderivative size = 19, 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} \[ \frac{\left (a+b x^n\right )^4}{4 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*(a + b*x^n)^3,x]

[Out]

(a + b*x^n)^4/(4*b*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^{-1+n} \left (a+b x^n\right )^3 \, dx &=\frac{\left (a+b x^n\right )^4}{4 b n}\\ \end{align*}

Mathematica [A]  time = 0.0036553, size = 19, normalized size = 1. \[ \frac{\left (a+b x^n\right )^4}{4 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*(a + b*x^n)^3,x]

[Out]

(a + b*x^n)^4/(4*b*n)

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Maple [B]  time = 0.014, size = 60, normalized size = 3.2 \begin{align*}{\frac{{a}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,n}}+{\frac{3\,b{a}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*(a+b*x^n)^3,x)

[Out]

a^3/n*exp(n*ln(x))+b^2*a/n*exp(n*ln(x))^3+1/4*b^3/n*exp(n*ln(x))^4+3/2*b*a^2/n*exp(n*ln(x))^2

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Maxima [A]  time = 0.95788, size = 23, normalized size = 1.21 \begin{align*} \frac{{\left (b x^{n} + a\right )}^{4}}{4 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^3,x, algorithm="maxima")

[Out]

1/4*(b*x^n + a)^4/(b*n)

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Fricas [B]  time = 1.27201, size = 93, normalized size = 4.89 \begin{align*} \frac{b^{3} x^{4 \, n} + 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} + 4 \, a^{3} x^{n}}{4 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

1/4*(b^3*x^(4*n) + 4*a*b^2*x^(3*n) + 6*a^2*b*x^(2*n) + 4*a^3*x^n)/n

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Sympy [A]  time = 4.69126, size = 54, normalized size = 2.84 \begin{align*} \begin{cases} \frac{a^{3} x^{n}}{n} + \frac{3 a^{2} b x^{2 n}}{2 n} + \frac{a b^{2} x^{3 n}}{n} + \frac{b^{3} x^{4 n}}{4 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*(a+b*x**n)**3,x)

[Out]

Piecewise((a**3*x**n/n + 3*a**2*b*x**(2*n)/(2*n) + a*b**2*x**(3*n)/n + b**3*x**(4*n)/(4*n), Ne(n, 0)), ((a + b
)**3*log(x), True))

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Giac [B]  time = 1.20417, size = 61, normalized size = 3.21 \begin{align*} \frac{b^{3} x^{4 \, n} + 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} + 4 \, a^{3} x^{n}}{4 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

1/4*(b^3*x^(4*n) + 4*a*b^2*x^(3*n) + 6*a^2*b*x^(2*n) + 4*a^3*x^n)/n