3.2460 \(\int x \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=65 \[ \frac{a^3 x^2}{2}+\frac{3 a^2 b x^{n+2}}{n+2}+\frac{3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac{b^3 x^{3 n+2}}{3 n+2} \]

[Out]

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

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Rubi [A]  time = 0.0740201, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^3 x^2}{2}+\frac{3 a^2 b x^{n+2}}{n+2}+\frac{3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac{b^3 x^{3 n+2}}{3 n+2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^n)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b*x**n)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0608854, size = 58, normalized size = 0.89 \[ \frac{1}{2} x^2 \left (a^3+\frac{6 a^2 b x^n}{n+2}+\frac{3 a b^2 x^{2 n}}{n+1}+\frac{2 b^3 x^{3 n}}{3 n+2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^n)^3,x]

[Out]

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

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Maple [A]  time = 0.018, size = 71, normalized size = 1.1 \[{\frac{{b}^{3}{x}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2+3\,n}}+{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{3\,a{b}^{2}{x}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2+2\,n}}+3\,{\frac{{a}^{2}b{x}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{2+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^n)^3,x)

[Out]

b^3/(2+3*n)*x^2*exp(n*ln(x))^3+1/2*x^2*a^3+3/2*a*b^2/(1+n)*x^2*exp(n*ln(x))^2+3*
a^2*b/(2+n)*x^2*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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242931, size = 196, normalized size = 3.02 \[ \frac{2 \,{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{2} x^{3 \, n} + 3 \,{\left (3 \, a b^{2} n^{2} + 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2} x^{2 \, n} + 6 \,{\left (3 \, a^{2} b n^{2} + 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{2} x^{n} +{\left (3 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 12 \, a^{3} n + 4 \, a^{3}\right )} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*(b^3*n^2 + 3*b^3*n + 2*b^3)*x^2*x^(3*n) + 3*(3*a*b^2*n^2 + 8*a*b^2*n + 4*
a*b^2)*x^2*x^(2*n) + 6*(3*a^2*b*n^2 + 5*a^2*b*n + 2*a^2*b)*x^2*x^n + (3*a^3*n^3
+ 11*a^3*n^2 + 12*a^3*n + 4*a^3)*x^2)/(3*n^3 + 11*n^2 + 12*n + 4)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b*x**n)**3,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.217834, size = 270, normalized size = 4.15 \[ \frac{3 \, a^{3} n^{3} x^{2} + 2 \, b^{3} n^{2} x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 9 \, a b^{2} n^{2} x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 18 \, a^{2} b n^{2} x^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 11 \, a^{3} n^{2} x^{2} + 6 \, b^{3} n x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 24 \, a b^{2} n x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 30 \, a^{2} b n x^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 12 \, a^{3} n x^{2} + 4 \, b^{3} x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 12 \, a b^{2} x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 12 \, a^{2} b x^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 4 \, a^{3} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(3*a^3*n^3*x^2 + 2*b^3*n^2*x^2*e^(3*n*ln(x)) + 9*a*b^2*n^2*x^2*e^(2*n*ln(x))
+ 18*a^2*b*n^2*x^2*e^(n*ln(x)) + 11*a^3*n^2*x^2 + 6*b^3*n*x^2*e^(3*n*ln(x)) + 2
4*a*b^2*n*x^2*e^(2*n*ln(x)) + 30*a^2*b*n*x^2*e^(n*ln(x)) + 12*a^3*n*x^2 + 4*b^3*
x^2*e^(3*n*ln(x)) + 12*a*b^2*x^2*e^(2*n*ln(x)) + 12*a^2*b*x^2*e^(n*ln(x)) + 4*a^
3*x^2)/(3*n^3 + 11*n^2 + 12*n + 4)