3.2522 \(\int x^{-1+3 n} \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=45 \[ \frac{a^2 x^{3 n}}{3 n}+\frac{a b x^{4 n}}{2 n}+\frac{b^2 x^{5 n}}{5 n} \]

[Out]

(a^2*x^(3*n))/(3*n) + (a*b*x^(4*n))/(2*n) + (b^2*x^(5*n))/(5*n)

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Rubi [A]  time = 0.0553094, 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 \[ \frac{a^2 x^{3 n}}{3 n}+\frac{a b x^{4 n}}{2 n}+\frac{b^2 x^{5 n}}{5 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)*(a + b*x^n)^2,x]

[Out]

(a^2*x^(3*n))/(3*n) + (a*b*x^(4*n))/(2*n) + (b^2*x^(5*n))/(5*n)

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Rubi in Sympy [A]  time = 8.59623, size = 34, normalized size = 0.76 \[ \frac{a^{2} x^{3 n}}{3 n} + \frac{a b x^{4 n}}{2 n} + \frac{b^{2} x^{5 n}}{5 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)*(a+b*x**n)**2,x)

[Out]

a**2*x**(3*n)/(3*n) + a*b*x**(4*n)/(2*n) + b**2*x**(5*n)/(5*n)

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Mathematica [A]  time = 0.0190441, size = 35, normalized size = 0.78 \[ \frac{x^{3 n} \left (10 a^2+15 a b x^n+6 b^2 x^{2 n}\right )}{30 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)*(a + b*x^n)^2,x]

[Out]

(x^(3*n)*(10*a^2 + 15*a*b*x^n + 6*b^2*x^(2*n)))/(30*n)

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Maple [A]  time = 0.027, size = 46, normalized size = 1. \[{\frac{{a}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,n}}+{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{5\,n}}+{\frac{ab \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{2\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)*(a+b*x^n)^2,x)

[Out]

1/3*a^2/n*exp(n*ln(x))^3+1/5*b^2/n*exp(n*ln(x))^5+1/2*a*b/n*exp(n*ln(x))^4

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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^(3*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223234, size = 47, normalized size = 1.04 \[ \frac{6 \, b^{2} x^{5 \, n} + 15 \, a b x^{4 \, n} + 10 \, a^{2} x^{3 \, n}}{30 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(3*n - 1),x, algorithm="fricas")

[Out]

1/30*(6*b^2*x^(5*n) + 15*a*b*x^(4*n) + 10*a^2*x^(3*n))/n

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Sympy [A]  time = 37.11, size = 42, normalized size = 0.93 \[ \begin{cases} \frac{a^{2} x^{3 n}}{3 n} + \frac{a b x^{4 n}}{2 n} + \frac{b^{2} x^{5 n}}{5 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+3*n)*(a+b*x**n)**2,x)

[Out]

Piecewise((a**2*x**(3*n)/(3*n) + a*b*x**(4*n)/(2*n) + b**2*x**(5*n)/(5*n), Ne(n,
0)), ((a + b)**2*log(x), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{2} x^{3 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*x^(3*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^2*x^(3*n - 1), x)