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]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {\left (a+b x^n\right )^4}{4 b n} \]

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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fricas [B]  time = 0.66, size = 45, normalized size = 2.37 \[ \frac {b^{3} x^{4 \, n} + 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} + 4 \, a^{3} x^{n}}{4 \, n} \]

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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giac [B]  time = 0.15, size = 45, normalized size = 2.37 \[ \frac {b^{3} x^{4 \, n} + 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} + 4 \, a^{3} x^{n}}{4 \, n} \]

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

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maple [B]  time = 0.02, size = 60, normalized size = 3.16 \[ \frac {a^{3} {\mathrm e}^{n \ln \relax (x )}}{n}+\frac {3 a^{2} b \,{\mathrm e}^{2 n \ln \relax (x )}}{2 n}+\frac {a \,b^{2} {\mathrm e}^{3 n \ln \relax (x )}}{n}+\frac {b^{3} {\mathrm e}^{4 n \ln \relax (x )}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 17, normalized size = 0.89 \[ \frac {{\left (b x^{n} + a\right )}^{4}}{4 \, b n} \]

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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mupad [B]  time = 1.30, size = 51, normalized size = 2.68 \[ \frac {a^3\,x^n}{n}+\frac {b^3\,x^{4\,n}}{4\,n}+\frac {3\,a^2\,b\,x^{2\,n}}{2\,n}+\frac {a\,b^2\,x^{3\,n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)*(a + b*x^n)^3,x)

[Out]

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

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sympy [A]  time = 5.48, size = 54, normalized size = 2.84 \[ \begin {cases} \frac {a^{3} x^{n}}{n} + \frac {3 a^{2} b x^{2 n}}{2 n} + \frac {a b^{2} x^{3 n}}{n} + \frac {b^{3} x^{4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\relax (x )} & \text {otherwise} \end {cases} \]

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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