3.25.51 \(\int x^3 (a+b x^n) \, dx\) [2451]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a x^4}{4}+\frac {b x^{n+4}}{n+4} \end {gather*}

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*}

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Mathematica [A]
time = 0.03, size = 21, normalized size = 1.00 \begin {gather*} \frac {a x^4}{4}+\frac {b x^{4+n}}{4+n} \end {gather*}

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 = 0.02, size = 21, normalized size = 1.00

method	result	size
risch	\(\frac {b \,x^{4} x^{n}}{4+n}+\frac {a \,x^{4}}{4}\)	\(21\)
norman	\(\frac {b \,x^{4} {\mathrm e}^{n \ln \left (x \right )}}{4+n}+\frac {a \,x^{4}}{4}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {b x^{n + 4}}{n + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4 + b*x^(n + 4)/(n + 4)

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Fricas [A]
time = 0.49, size = 28, normalized size = 1.33 \begin {gather*} \frac {4 \, b x^{4} x^{n} + {\left (a n + 4 \, a\right )} x^{4}}{4 \, {\left (n + 4\right )}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (15) = 30\).
time = 0.14, size = 51, normalized size = 2.43 \begin {gather*} \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 {gather*}

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 = 1.04, size = 29, normalized size = 1.38 \begin {gather*} \frac {4 \, b x^{4} x^{n} + a n x^{4} + 4 \, a x^{4}}{4 \, {\left (n + 4\right )}} \end {gather*}

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)

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Mupad [B]
time = 1.33, size = 20, normalized size = 0.95 \begin {gather*} \frac {a\,x^4}{4}+\frac {b\,x^n\,x^4}{n+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^n),x)

[Out]

(a*x^4)/4 + (b*x^n*x^4)/(n + 4)

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