3.2451 \(\int x^3 (a+b x^n) \, dx\)

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 = 0.0090356, 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, Rules used = {14} \[ \frac{a x^4}{4}+\frac{b x^{n+4}}{n+4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^n),x]

[Out]

(a*x^4)/4 + (b*x^(4 + n))/(4 + n)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^3 \left (a+b x^n\right ) \, dx &=\int \left (a x^3+b x^{3+n}\right ) \, dx\\ &=\frac{a x^4}{4}+\frac{b x^{4+n}}{4+n}\\ \end{align*}

Mathematica [A]  time = 0.0148781, 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 = 0.008, size = 23, normalized size = 1.1 \begin{align*}{\frac{b{x}^{4}{{\rm e}^{n\ln \left ( x \right ) }}}{4+n}}+{\frac{a{x}^{4}}{4}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56592, size = 61, normalized size = 2.9 \begin{align*} \frac{4 \, b x^{4} x^{n} +{\left (a n + 4 \, a\right )} x^{4}}{4 \,{\left (n + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^n),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 = 0.431633, size = 51, normalized size = 2.43 \begin{align*} \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} \end{align*}

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 [A]  time = 1.1261, size = 39, normalized size = 1.86 \begin{align*} \frac{4 \, b x^{4} x^{n} + a n x^{4} + 4 \, a x^{4}}{4 \,{\left (n + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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