3.2514 \(\int x^{-1+n} \left (a+b x^n\right ) \, dx\)

Optimal. Leaf size=22 \[ \frac{a x^n}{n}+\frac{b x^{2 n}}{2 n} \]

[Out]

(a*x^n)/n + (b*x^(2*n))/(2*n)

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Rubi [A]  time = 0.020589, antiderivative size = 22, 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 \[ \frac{a x^n}{n}+\frac{b x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)*(a + b*x^n),x]

[Out]

(a*x^n)/n + (b*x^(2*n))/(2*n)

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Rubi in Sympy [A]  time = 3.79068, size = 15, normalized size = 0.68 \[ \frac{a x^{n}}{n} + \frac{b x^{2 n}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**n/n + b*x**(2*n)/(2*n)

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Mathematica [A]  time = 0.00790198, size = 19, normalized size = 0.86 \[ \frac{x^n \left (2 a+b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

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

[Out]

(x^n*(2*a + b*x^n))/(2*n)

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Maple [A]  time = 0.02, size = 25, normalized size = 1.1 \[{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)*(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.43313, size = 23, normalized size = 1.05 \[ \frac{{\left (b x^{n} + a\right )}^{2}}{2 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^n + a)^2/(b*n)

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Fricas [A]  time = 0.221738, size = 26, normalized size = 1.18 \[ \frac{b x^{2 \, n} + 2 \, a x^{n}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^(2*n) + 2*a*x^n)/n

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Sympy [A]  time = 4.06205, size = 22, normalized size = 1. \[ \begin{cases} \frac{a x^{n}}{n} + \frac{b x^{2 n}}{2 n} & \text{for}\: n \neq 0 \\\left (a + b\right ) \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211288, size = 26, normalized size = 1.18 \[ \frac{b x^{2 \, n} + 2 \, a x^{n}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^(2*n) + 2*a*x^n)/n