3.1.44 \(\int x^2 (a+b x) \, dx\)

Optimal. Leaf size=17 \[ \frac {a x^3}{3}+\frac {b x^4}{4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, 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 = {43} \begin {gather*} \frac {a x^3}{3}+\frac {b x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x),x]

[Out]

(a*x^3)/3 + (b*x^4)/4

Rule 43

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^2 (a+b x) \, dx &=\int \left (a x^2+b x^3\right ) \, dx\\ &=\frac {a x^3}{3}+\frac {b x^4}{4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^3}{3}+\frac {b x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x),x]

[Out]

(a*x^3)/3 + (b*x^4)/4

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (a+b x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^2*(a + b*x),x]

[Out]

IntegrateAlgebraic[x^2*(a + b*x), x]

________________________________________________________________________________________

fricas [A]  time = 1.45, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} x^{4} b + \frac {1}{3} x^{3} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a),x, algorithm="fricas")

[Out]

1/4*x^4*b + 1/3*x^3*a

________________________________________________________________________________________

giac [A]  time = 1.74, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a),x, algorithm="giac")

[Out]

1/4*b*x^4 + 1/3*a*x^3

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} \frac {1}{4} b \,x^{4}+\frac {1}{3} a \,x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a),x)

[Out]

1/3*a*x^3+1/4*b*x^4

________________________________________________________________________________________

maxima [A]  time = 0.86, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a),x, algorithm="maxima")

[Out]

1/4*b*x^4 + 1/3*a*x^3

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x^3\,\left (4\,a+3\,b\,x\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x),x)

[Out]

(x^3*(4*a + 3*b*x))/12

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{3}}{3} + \frac {b x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a),x)

[Out]

a*x**3/3 + b*x**4/4

________________________________________________________________________________________