3.2451 \(\int x^3 \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=44 \[ \frac{a^2 x^4}{4}+\frac{2 a b x^{n+4}}{n+4}+\frac{b^2 x^{2 (n+2)}}{2 (n+2)} \]

[Out]

(a^2*x^4)/4 + (b^2*x^(2*(2 + n)))/(2*(2 + n)) + (2*a*b*x^(4 + n))/(4 + n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0566562, antiderivative size = 44, 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 \[ \frac{a^2 x^4}{4}+\frac{2 a b x^{n+4}}{n+4}+\frac{b^2 x^{2 (n+2)}}{2 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^n)^2,x]

[Out]

(a^2*x^4)/4 + (b^2*x^(2*(2 + n)))/(2*(2 + n)) + (2*a*b*x^(4 + n))/(4 + n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.14883, size = 36, normalized size = 0.82 \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{n + 4}}{n + 4} + \frac{b^{2} x^{2 n + 4}}{2 \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b*x**n)**2,x)

[Out]

a**2*x**4/4 + 2*a*b*x**(n + 4)/(n + 4) + b**2*x**(2*n + 4)/(2*(n + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0581358, size = 38, normalized size = 0.86 \[ \frac{1}{4} x^4 \left (a^2+\frac{8 a b x^n}{n+4}+\frac{2 b^2 x^{2 n}}{n+2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^n)^2,x]

[Out]

(x^4*(a^2 + (8*a*b*x^n)/(4 + n) + (2*b^2*x^(2*n))/(2 + n)))/4

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 47, normalized size = 1.1 \[{\frac{{x}^{4}{a}^{2}}{4}}+{\frac{{b}^{2}{x}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{4+2\,n}}+2\,{\frac{ab{x}^{4}{{\rm e}^{n\ln \left ( x \right ) }}}{4+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^n)^2,x)

[Out]

1/4*x^4*a^2+1/2*b^2/(2+n)*x^4*exp(n*ln(x))^2+2*a*b/(4+n)*x^4*exp(n*ln(x))

_______________________________________________________________________________________

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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238972, size = 100, normalized size = 2.27 \[ \frac{2 \,{\left (b^{2} n + 4 \, b^{2}\right )} x^{4} x^{2 \, n} + 8 \,{\left (a b n + 2 \, a b\right )} x^{4} x^{n} +{\left (a^{2} n^{2} + 6 \, a^{2} n + 8 \, a^{2}\right )} x^{4}}{4 \,{\left (n^{2} + 6 \, n + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^3,x, algorithm="fricas")

[Out]

1/4*(2*(b^2*n + 4*b^2)*x^4*x^(2*n) + 8*(a*b*n + 2*a*b)*x^4*x^n + (a^2*n^2 + 6*a^
2*n + 8*a^2)*x^4)/(n^2 + 6*n + 8)

_______________________________________________________________________________________

Sympy [A]  time = 2.74612, size = 202, normalized size = 4.59 \[ \begin{cases} \frac{a^{2} x^{4}}{4} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{4 x^{4}} & \text{for}\: n = -4 \\\frac{a^{2} x^{4}}{4} + a b x^{2} + b^{2} \log{\left (x \right )} & \text{for}\: n = -2 \\\frac{a^{2} n^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac{6 a^{2} n x^{4}}{4 n^{2} + 24 n + 32} + \frac{8 a^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac{8 a b n x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac{16 a b x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac{2 b^{2} n x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} + \frac{8 b^{2} x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*x**n)**2,x)

[Out]

Piecewise((a**2*x**4/4 + 2*a*b*log(x) - b**2/(4*x**4), Eq(n, -4)), (a**2*x**4/4
+ a*b*x**2 + b**2*log(x), Eq(n, -2)), (a**2*n**2*x**4/(4*n**2 + 24*n + 32) + 6*a
**2*n*x**4/(4*n**2 + 24*n + 32) + 8*a**2*x**4/(4*n**2 + 24*n + 32) + 8*a*b*n*x**
4*x**n/(4*n**2 + 24*n + 32) + 16*a*b*x**4*x**n/(4*n**2 + 24*n + 32) + 2*b**2*n*x
**4*x**(2*n)/(4*n**2 + 24*n + 32) + 8*b**2*x**4*x**(2*n)/(4*n**2 + 24*n + 32), T
rue))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216387, size = 127, normalized size = 2.89 \[ \frac{a^{2} n^{2} x^{4} + 2 \, b^{2} n x^{4} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 8 \, a b n x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a^{2} n x^{4} + 8 \, b^{2} x^{4} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 16 \, a b x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 8 \, a^{2} x^{4}}{4 \,{\left (n^{2} + 6 \, n + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^3,x, algorithm="giac")

[Out]

1/4*(a^2*n^2*x^4 + 2*b^2*n*x^4*e^(2*n*ln(x)) + 8*a*b*n*x^4*e^(n*ln(x)) + 6*a^2*n
*x^4 + 8*b^2*x^4*e^(2*n*ln(x)) + 16*a*b*x^4*e^(n*ln(x)) + 8*a^2*x^4)/(n^2 + 6*n
+ 8)