3.2444 \(\int x^3 \left (a+b x^n\right ) \, dx\)

Optimal. Leaf size=21 \[ \frac{a x^4}{4}+\frac{b x^{n+4}}{n+4} \]

[Out]

(a*x^4)/4 + (b*x^(4 + n))/(4 + n)

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Rubi [A]  time = 0.0268645, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a x^4}{4}+\frac{b x^{n+4}}{n+4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^n),x]

[Out]

(a*x^4)/4 + (b*x^(4 + n))/(4 + n)

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Rubi in Sympy [A]  time = 3.57214, size = 15, normalized size = 0.71 \[ \frac{a x^{4}}{4} + \frac{b x^{n + 4}}{n + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4/4 + b*x**(n + 4)/(n + 4)

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Mathematica [A]  time = 0.0172007, size = 21, normalized size = 1. \[ \frac{a x^4}{4}+\frac{b x^{n+4}}{n+4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^n),x]

[Out]

(a*x^4)/4 + (b*x^(4 + n))/(4 + n)

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Maple [A]  time = 0.015, size = 23, normalized size = 1.1 \[{\frac{b{x}^{4}{{\rm e}^{n\ln \left ( x \right ) }}}{4+n}}+{\frac{a{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^n),x)

[Out]

b/(4+n)*x^4*exp(n*ln(x))+1/4*a*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)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239493, size = 38, normalized size = 1.81 \[ \frac{4 \, b x^{4} x^{n} +{\left (a n + 4 \, a\right )} x^{4}}{4 \,{\left (n + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*b*x^4*x^n + (a*n + 4*a)*x^4)/(n + 4)

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Sympy [A]  time = 1.407, size = 51, normalized size = 2.43 \[ \begin{cases} \frac{a n x^{4}}{4 n + 16} + \frac{4 a x^{4}}{4 n + 16} + \frac{4 b x^{4} x^{n}}{4 n + 16} & \text{for}\: n \neq -4 \\\frac{a x^{4}}{4} + b \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*x**n),x)

[Out]

Piecewise((a*n*x**4/(4*n + 16) + 4*a*x**4/(4*n + 16) + 4*b*x**4*x**n/(4*n + 16),
Ne(n, -4)), (a*x**4/4 + b*log(x), True))

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GIAC/XCAS [A]  time = 0.213113, size = 42, normalized size = 2. \[ \frac{a n x^{4} + 4 \, b x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 4 \, a x^{4}}{4 \,{\left (n + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(a*n*x^4 + 4*b*x^4*e^(n*ln(x)) + 4*a*x^4)/(n + 4)