3.2446 \(\int x \left (a+b x^n\right ) \, dx\)

Optimal. Leaf size=21 \[ \frac{a x^2}{2}+\frac{b x^{n+2}}{n+2} \]

[Out]

(a*x^2)/2 + (b*x^(2 + n))/(2 + n)

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Rubi [A]  time = 0.023482, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{a x^2}{2}+\frac{b x^{n+2}}{n+2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^n),x]

[Out]

(a*x^2)/2 + (b*x^(2 + n))/(2 + n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.0158452, size = 21, normalized size = 1. \[ \frac{a x^2}{2}+\frac{b x^{n+2}}{n+2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^n),x]

[Out]

(a*x^2)/2 + (b*x^(2 + n))/(2 + n)

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Maple [A]  time = 0.012, size = 23, normalized size = 1.1 \[{\frac{b{x}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{2+n}}+{\frac{a{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^n),x)

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237998, size = 38, normalized size = 1.81 \[ \frac{2 \, b x^{2} x^{n} +{\left (a n + 2 \, a\right )} x^{2}}{2 \,{\left (n + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.713776, size = 51, normalized size = 2.43 \[ \begin{cases} \frac{a n x^{2}}{2 n + 4} + \frac{2 a x^{2}}{2 n + 4} + \frac{2 b x^{2} x^{n}}{2 n + 4} & \text{for}\: n \neq -2 \\\frac{a x^{2}}{2} + b \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b*x**n),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213672, size = 42, normalized size = 2. \[ \frac{a n x^{2} + 2 \, b x^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 2 \, a x^{2}}{2 \,{\left (n + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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