3.2513 \(\int x^{-1+2 n} \left (a+b x^n\right ) \, dx\)

Optimal. Leaf size=27 \[ \frac{a x^{2 n}}{2 n}+\frac{b x^{3 n}}{3 n} \]

[Out]

(a*x^(2*n))/(2*n) + (b*x^(3*n))/(3*n)

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Rubi [A]  time = 0.0223623, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{a x^{2 n}}{2 n}+\frac{b x^{3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)*(a + b*x^n),x]

[Out]

(a*x^(2*n))/(2*n) + (b*x^(3*n))/(3*n)

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Rubi in Sympy [A]  time = 4.01799, size = 19, normalized size = 0.7 \[ \frac{a x^{2 n}}{2 n} + \frac{b x^{3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**(2*n)/(2*n) + b*x**(3*n)/(3*n)

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Mathematica [A]  time = 0.00965357, size = 22, normalized size = 0.81 \[ \frac{x^{2 n} \left (3 a+2 b x^n\right )}{6 n} \]

Antiderivative was successfully verified.

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

[Out]

(x^(2*n)*(3*a + 2*b*x^n))/(6*n)

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Maple [A]  time = 0.022, size = 28, normalized size = 1. \[{\frac{a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}+{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a+b*x^n),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223527, size = 30, normalized size = 1.11 \[ \frac{2 \, b x^{3 \, n} + 3 \, a x^{2 \, n}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x^(3*n) + 3*a*x^(2*n))/n

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Sympy [A]  time = 14.2686, size = 26, normalized size = 0.96 \[ \begin{cases} \frac{a x^{2 n}}{2 n} + \frac{b x^{3 n}}{3 n} & \text{for}\: n \neq 0 \\\left (a + b\right ) \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)*x^(2*n - 1), x)