3.2468 \(\int (a+b x^n)^3 \, dx\)

Optimal. Leaf size=60 \[ a^3 x+\frac {3 a^2 b x^{n+1}}{n+1}+\frac {3 a b^2 x^{2 n+1}}{2 n+1}+\frac {b^3 x^{3 n+1}}{3 n+1} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {244} \[ \frac {3 a^2 b x^{n+1}}{n+1}+a^3 x+\frac {3 a b^2 x^{2 n+1}}{2 n+1}+\frac {b^3 x^{3 n+1}}{3 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^3,x]

[Out]

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

Rule 244

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n},
x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \left (a+b x^n\right )^3 \, dx &=\int \left (a^3+3 a^2 b x^n+3 a b^2 x^{2 n}+b^3 x^{3 n}\right ) \, dx\\ &=a^3 x+\frac {3 a^2 b x^{1+n}}{1+n}+\frac {3 a b^2 x^{1+2 n}}{1+2 n}+\frac {b^3 x^{1+3 n}}{1+3 n}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 54, normalized size = 0.90 \[ x \left (a^3+\frac {3 a^2 b x^n}{n+1}+\frac {3 a b^2 x^{2 n}}{2 n+1}+\frac {b^3 x^{3 n}}{3 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^3,x]

[Out]

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

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fricas [B]  time = 0.54, size = 130, normalized size = 2.17 \[ \frac {{\left (2 \, b^{3} n^{2} + 3 \, b^{3} n + b^{3}\right )} x x^{3 \, n} + 3 \, {\left (3 \, a b^{2} n^{2} + 4 \, a b^{2} n + a b^{2}\right )} x x^{2 \, n} + 3 \, {\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} + {\left (6 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 6 \, a^{3} n + a^{3}\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b^3*n^2 + 3*b^3*n + b^3)*x*x^(3*n) + 3*(3*a*b^2*n^2 + 4*a*b^2*n + a*b^2)*x*x^(2*n) + 3*(6*a^2*b*n^2 + 5*a^
2*b*n + a^2*b)*x*x^n + (6*a^3*n^3 + 11*a^3*n^2 + 6*a^3*n + a^3)*x)/(6*n^3 + 11*n^2 + 6*n + 1)

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giac [B]  time = 0.16, size = 159, normalized size = 2.65 \[ \frac {6 \, a^{3} n^{3} x + 2 \, b^{3} n^{2} x x^{3 \, n} + 9 \, a b^{2} n^{2} x x^{2 \, n} + 18 \, a^{2} b n^{2} x x^{n} + 11 \, a^{3} n^{2} x + 3 \, b^{3} n x x^{3 \, n} + 12 \, a b^{2} n x x^{2 \, n} + 15 \, a^{2} b n x x^{n} + 6 \, a^{3} n x + b^{3} x x^{3 \, n} + 3 \, a b^{2} x x^{2 \, n} + 3 \, a^{2} b x x^{n} + a^{3} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*a^3*n^3*x + 2*b^3*n^2*x*x^(3*n) + 9*a*b^2*n^2*x*x^(2*n) + 18*a^2*b*n^2*x*x^n + 11*a^3*n^2*x + 3*b^3*n*x*x^(
3*n) + 12*a*b^2*n*x*x^(2*n) + 15*a^2*b*n*x*x^n + 6*a^3*n*x + b^3*x*x^(3*n) + 3*a*b^2*x*x^(2*n) + 3*a^2*b*x*x^n
+ a^3*x)/(6*n^3 + 11*n^2 + 6*n + 1)

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maple [A]  time = 0.01, size = 64, normalized size = 1.07 \[ \frac {3 a^{2} b x \,{\mathrm e}^{n \ln \relax (x )}}{n +1}+\frac {3 a \,b^{2} x \,{\mathrm e}^{2 n \ln \relax (x )}}{2 n +1}+\frac {b^{3} x \,{\mathrm e}^{3 n \ln \relax (x )}}{3 n +1}+a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.46, size = 60, normalized size = 1.00 \[ a^{3} x + \frac {b^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, a b^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a^{2} b x^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.19, size = 57, normalized size = 0.95 \[ a^3\,x+\frac {b^3\,x\,x^{3\,n}}{3\,n+1}+\frac {3\,a\,b^2\,x\,x^{2\,n}}{2\,n+1}+\frac {3\,a^2\,b\,x\,x^n}{n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^3,x)

[Out]

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

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sympy [A]  time = 0.82, size = 469, normalized size = 7.82 \[ \begin {cases} a^{3} x + 3 a^{2} b \log {\relax (x )} - \frac {3 a b^{2}}{x} - \frac {b^{3}}{2 x^{2}} & \text {for}\: n = -1 \\a^{3} x + 6 a^{2} b \sqrt {x} + 3 a b^{2} \log {\relax (x )} - \frac {2 b^{3}}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\a^{3} x + \frac {9 a^{2} b x^{\frac {2}{3}}}{2} + 9 a b^{2} \sqrt [3]{x} + b^{3} \log {\relax (x )} & \text {for}\: n = - \frac {1}{3} \\\frac {6 a^{3} n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {11 a^{3} n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a^{3} n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {18 a^{2} b n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {15 a^{2} b n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 a^{2} b x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {9 a b^{2} n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {12 a b^{2} n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 a b^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 b^{3} n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 b^{3} n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b^{3} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x + 3*a**2*b*log(x) - 3*a*b**2/x - b**3/(2*x**2), Eq(n, -1)), (a**3*x + 6*a**2*b*sqrt(x) + 3*a
*b**2*log(x) - 2*b**3/sqrt(x), Eq(n, -1/2)), (a**3*x + 9*a**2*b*x**(2/3)/2 + 9*a*b**2*x**(1/3) + b**3*log(x),
Eq(n, -1/3)), (6*a**3*n**3*x/(6*n**3 + 11*n**2 + 6*n + 1) + 11*a**3*n**2*x/(6*n**3 + 11*n**2 + 6*n + 1) + 6*a*
*3*n*x/(6*n**3 + 11*n**2 + 6*n + 1) + a**3*x/(6*n**3 + 11*n**2 + 6*n + 1) + 18*a**2*b*n**2*x*x**n/(6*n**3 + 11
*n**2 + 6*n + 1) + 15*a**2*b*n*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 3*a**2*b*x*x**n/(6*n**3 + 11*n**2 + 6*n +
1) + 9*a*b**2*n**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 12*a*b**2*n*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n +
1) + 3*a*b**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 2*b**3*n**2*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1) +
3*b**3*n*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1) + b**3*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1), True))

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