3.2452 \(\int x^2 \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 x^3}{3}+\frac{2 a b x^{n+3}}{n+3}+\frac{b^2 x^{2 n+3}}{2 n+3} \]

[Out]

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

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Rubi [A]  time = 0.0540864, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^2 x^3}{3}+\frac{2 a b x^{n+3}}{n+3}+\frac{b^2 x^{2 n+3}}{2 n+3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^n)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.43175, size = 36, normalized size = 0.84 \[ \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{n + 3}}{n + 3} + \frac{b^{2} x^{2 n + 3}}{2 n + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(a+b*x**n)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0536579, size = 40, normalized size = 0.93 \[ \frac{1}{3} x^3 \left (a^2+\frac{6 a b x^n}{n+3}+\frac{3 b^2 x^{2 n}}{2 n+3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.017, size = 48, normalized size = 1.1 \[{\frac{{b}^{2}{x}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3+2\,n}}+{\frac{{x}^{3}{a}^{2}}{3}}+2\,{\frac{ab{x}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{3+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*x^n)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238887, size = 105, normalized size = 2.44 \[ \frac{3 \,{\left (b^{2} n + 3 \, b^{2}\right )} x^{3} x^{2 \, n} + 6 \,{\left (2 \, a b n + 3 \, a b\right )} x^{3} x^{n} +{\left (2 \, a^{2} n^{2} + 9 \, a^{2} n + 9 \, a^{2}\right )} x^{3}}{3 \,{\left (2 \, n^{2} + 9 \, n + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^2,x, algorithm="fricas")

[Out]

1/3*(3*(b^2*n + 3*b^2)*x^3*x^(2*n) + 6*(2*a*b*n + 3*a*b)*x^3*x^n + (2*a^2*n^2 +
9*a^2*n + 9*a^2)*x^3)/(2*n^2 + 9*n + 9)

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Sympy [A]  time = 6.19677, size = 211, normalized size = 4.91 \[ \begin{cases} \frac{a^{2} x^{3}}{3} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{3 x^{3}} & \text{for}\: n = -3 \\\frac{a^{2} x^{3}}{3} + \frac{4 a b x^{\frac{3}{2}}}{3} + b^{2} \log{\left (x \right )} & \text{for}\: n = - \frac{3}{2} \\\frac{2 a^{2} n^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac{9 a^{2} n x^{3}}{6 n^{2} + 27 n + 27} + \frac{9 a^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac{12 a b n x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac{18 a b x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac{3 b^{2} n x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} + \frac{9 b^{2} x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a+b*x**n)**2,x)

[Out]

Piecewise((a**2*x**3/3 + 2*a*b*log(x) - b**2/(3*x**3), Eq(n, -3)), (a**2*x**3/3
+ 4*a*b*x**(3/2)/3 + b**2*log(x), Eq(n, -3/2)), (2*a**2*n**2*x**3/(6*n**2 + 27*n
+ 27) + 9*a**2*n*x**3/(6*n**2 + 27*n + 27) + 9*a**2*x**3/(6*n**2 + 27*n + 27) +
12*a*b*n*x**3*x**n/(6*n**2 + 27*n + 27) + 18*a*b*x**3*x**n/(6*n**2 + 27*n + 27)
+ 3*b**2*n*x**3*x**(2*n)/(6*n**2 + 27*n + 27) + 9*b**2*x**3*x**(2*n)/(6*n**2 +
27*n + 27), True))

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GIAC/XCAS [A]  time = 0.213767, size = 131, normalized size = 3.05 \[ \frac{2 \, a^{2} n^{2} x^{3} + 3 \, b^{2} n x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 12 \, a b n x^{3} e^{\left (n{\rm ln}\left (x\right )\right )} + 9 \, a^{2} n x^{3} + 9 \, b^{2} x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 18 \, a b x^{3} e^{\left (n{\rm ln}\left (x\right )\right )} + 9 \, a^{2} x^{3}}{3 \,{\left (2 \, n^{2} + 9 \, n + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^2,x, algorithm="giac")

[Out]

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