3.1.52 \(\int x^3 (a+b x)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 30, 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 = {43} \begin {gather*} \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^2,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

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^3 (a+b x)^2 \, dx &=\int \left (a^2 x^3+2 a b x^4+b^2 x^5\right ) \, dx\\ &=\frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {b^2 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^2,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + (b^2*x^6)/6

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 (a+b x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(a + b*x)^2,x]

[Out]

IntegrateAlgebraic[x^3*(a + b*x)^2, x]

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{6} x^{6} b^{2} + \frac {2}{5} x^{5} b a + \frac {1}{4} x^{4} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2,x, algorithm="fricas")

[Out]

1/6*x^6*b^2 + 2/5*x^5*b*a + 1/4*x^4*a^2

________________________________________________________________________________________

giac [A]  time = 1.61, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2,x, algorithm="giac")

[Out]

1/6*b^2*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{6} b^{2} x^{6}+\frac {2}{5} a b \,x^{5}+\frac {1}{4} a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^2,x)

[Out]

1/4*a^2*x^4+2/5*a*b*x^5+1/6*b^2*x^6

________________________________________________________________________________________

maxima [A]  time = 1.20, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2,x, algorithm="maxima")

[Out]

1/6*b^2*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {2\,a\,b\,x^5}{5}+\frac {b^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^2,x)

[Out]

(a^2*x^4)/4 + (b^2*x^6)/6 + (2*a*b*x^5)/5

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 26, normalized size = 0.87 \begin {gather*} \frac {a^{2} x^{4}}{4} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**2,x)

[Out]

a**2*x**4/4 + 2*a*b*x**5/5 + b**2*x**6/6

________________________________________________________________________________________