3.2523 \(\int x^{-1+2 n} \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=45 \[ \frac{a^2 x^{2 n}}{2 n}+\frac{2 a b x^{3 n}}{3 n}+\frac{b^2 x^{4 n}}{4 n} \]

[Out]

(a^2*x^(2*n))/(2*n) + (2*a*b*x^(3*n))/(3*n) + (b^2*x^(4*n))/(4*n)

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Rubi [A]  time = 0.0532772, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^2 x^{2 n}}{2 n}+\frac{2 a b x^{3 n}}{3 n}+\frac{b^2 x^{4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)*(a + b*x^n)^2,x]

[Out]

(a^2*x^(2*n))/(2*n) + (2*a*b*x^(3*n))/(3*n) + (b^2*x^(4*n))/(4*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \int ^{x^{n}} x\, dx}{n} + \frac{2 a b x^{3 n}}{3 n} + \frac{b^{2} x^{4 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)*(a+b*x**n)**2,x)

[Out]

a**2*Integral(x, (x, x**n))/n + 2*a*b*x**(3*n)/(3*n) + b**2*x**(4*n)/(4*n)

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Mathematica [A]  time = 0.0189312, size = 35, normalized size = 0.78 \[ \frac{x^{2 n} \left (6 a^2+8 a b x^n+3 b^2 x^{2 n}\right )}{12 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)*(a + b*x^n)^2,x]

[Out]

(x^(2*n)*(6*a^2 + 8*a*b*x^n + 3*b^2*x^(2*n)))/(12*n)

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Maple [A]  time = 0.025, size = 46, normalized size = 1. \[{\frac{{a}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,n}}+{\frac{2\,ab \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a+b*x^n)^2,x)

[Out]

1/2*a^2/n*exp(n*ln(x))^2+1/4*b^2/n*exp(n*ln(x))^4+2/3*a*b/n*exp(n*ln(x))^3

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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*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22452, size = 47, normalized size = 1.04 \[ \frac{3 \, b^{2} x^{4 \, n} + 8 \, a b x^{3 \, n} + 6 \, a^{2} x^{2 \, n}}{12 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(2*n - 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 37.535, size = 44, normalized size = 0.98 \[ \begin{cases} \frac{a^{2} x^{2 n}}{2 n} + \frac{2 a b x^{3 n}}{3 n} + \frac{b^{2} x^{4 n}}{4 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*n)*(a+b*x**n)**2,x)

[Out]

Piecewise((a**2*x**(2*n)/(2*n) + 2*a*b*x**(3*n)/(3*n) + b**2*x**(4*n)/(4*n), Ne(
n, 0)), ((a + b)**2*log(x), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{2} x^{2 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(2*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^2*x^(2*n - 1), x)