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

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

[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 = 0.021777, antiderivative size = 60, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0322624, size = 54, normalized size = 0.9 \[ x \left (\frac{3 a^2 b x^n}{n+1}+a^3+\frac{3 a b^2 x^{2 n}}{2 n+1}+\frac{b^3 x^{3 n}}{3 n+1}\right ) \]

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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Maple [A]  time = 0.006, size = 64, normalized size = 1.1 \begin{align*}{a}^{3}x+{\frac{{b}^{3}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+3\,{\frac{{a}^{2}bx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+3\,{\frac{xa{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}} \end{align*}

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

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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((a+b*x^n)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.04128, size = 277, normalized size = 4.62 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 0.963732, size = 469, normalized size = 7.82 \begin{align*} \begin{cases} a^{3} x + 3 a^{2} b \log{\left (x \right )} - \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{\left (x \right )} - \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{\left (x \right )} & \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} \end{align*}

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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Giac [B]  time = 1.17196, size = 215, normalized size = 3.58 \begin{align*} \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} \end{align*}

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)