3.1.43 \(\int x^3 (a+b x) \, dx\) [43]

Optimal. Leaf size=17 \[ \frac {a x^4}{4}+\frac {b x^5}{5} \]

[Out]

1/4*a*x^4+1/5*b*x^5

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {a x^4}{4}+\frac {b x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x),x]

[Out]

(a*x^4)/4 + (b*x^5)/5

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int x^3 (a+b x) \, dx &=\int \left (a x^3+b x^4\right ) \, dx\\ &=\frac {a x^4}{4}+\frac {b x^5}{5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^4}{4}+\frac {b x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x),x]

[Out]

(a*x^4)/4 + (b*x^5)/5

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
gosper	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)
default	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)
norman	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)
risch	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4+1/5*b*x^5

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/4*a*x^4

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Fricas [A]
time = 0.85, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/4*a*x^4

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Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{4}}{4} + \frac {b x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a),x)

[Out]

a*x**4/4 + b*x**5/5

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Giac [A]
time = 1.21, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/4*a*x^4

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x^4\,\left (5\,a+4\,b\,x\right )}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x),x)

[Out]

(x^4*(5*a + 4*b*x))/20

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