3.2461 \(\int \left (a+b x^n\right )^3 \, dx\)

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 = 0.0595546, 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 \[ a^3 x+\frac{3 a^2 b x^{n+1}}{n+1}+\frac{3 a b^2 x^{2 n+1}}{2 n+1}+\frac{b^3 x^{3 n+1}}{3 n+1} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} b x^{n + 1}}{n + 1} + \frac{3 a b^{2} x^{2 n + 1}}{2 n + 1} + \frac{b^{3} x^{3 n + 1}}{3 n + 1} + \int a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0396337, size = 54, normalized size = 0.9 \[ x \left (a^3+\frac{3 a^2 b x^n}{n+1}+\frac{3 a b^2 x^{2 n}}{2 n+1}+\frac{b^3 x^{3 n}}{3 n+1}\right ) \]

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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Maple [A]  time = 0.015, size = 64, normalized size = 1.1 \[{a}^{3}x+{\frac{{b}^{3}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+3\,{\frac{a{b}^{2}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+3\,{\frac{{a}^{2}bx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238675, size = 176, normalized size = 2.93 \[ \frac{{\left (2 \, b^{3} n^{2} + 3 \, b^{3} n + b^{3}\right )} x x^{3 \, n} + 3 \,{\left (3 \, a b^{2} n^{2} + 4 \, a b^{2} n + a b^{2}\right )} x x^{2 \, n} + 3 \,{\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} +{\left (6 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 6 \, a^{3} n + a^{3}\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^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 = 2.16393, size = 469, normalized size = 7.82 \[ \begin{cases} a^{3} x + 3 a^{2} b \log{\left (x \right )} - \frac{3 a b^{2}}{x} - \frac{b^{3}}{2 x^{2}} & \text{for}\: n = -1 \\a^{3} x + 6 a^{2} b \sqrt{x} + 3 a b^{2} \log{\left (x \right )} - \frac{2 b^{3}}{\sqrt{x}} & \text{for}\: n = - \frac{1}{2} \\a^{3} x + \frac{9 a^{2} b x^{\frac{2}{3}}}{2} + 9 a b^{2} \sqrt [3]{x} + b^{3} \log{\left (x \right )} & \text{for}\: n = - \frac{1}{3} \\\frac{6 a^{3} n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{11 a^{3} n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a^{3} n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{18 a^{2} b n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{15 a^{2} b n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 a^{2} b x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{9 a b^{2} n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{12 a b^{2} n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 a b^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 b^{3} n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 b^{3} n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b^{3} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.216177, size = 231, normalized size = 3.85 \[ \frac{6 \, a^{3} n^{3} x + 2 \, b^{3} n^{2} x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 9 \, a b^{2} n^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 18 \, a^{2} b n^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 11 \, a^{3} n^{2} x + 3 \, b^{3} n x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 12 \, a b^{2} n x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 15 \, a^{2} b n x e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a^{3} n x + b^{3} x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 3 \, a b^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 3 \, a^{2} b x e^{\left (n{\rm ln}\left (x\right )\right )} + a^{3} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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