3.1.66 \(\int x^2 (a+b x)^3 \, dx\) [66]

Optimal. Leaf size=43 \[ \frac {a^3 x^3}{3}+\frac {3}{4} a^2 b x^4+\frac {3}{5} a b^2 x^5+\frac {b^3 x^6}{6} \]

[Out]

1/3*a^3*x^3+3/4*a^2*b*x^4+3/5*a*b^2*x^5+1/6*b^3*x^6

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Rubi [A]
time = 0.01, antiderivative size = 43, 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 = {45} \begin {gather*} \frac {a^3 x^3}{3}+\frac {3}{4} a^2 b x^4+\frac {3}{5} a b^2 x^5+\frac {b^3 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^5)/5 + (b^3*x^6)/6

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

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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {a^3 x^3}{3}+\frac {3}{4} a^2 b x^4+\frac {3}{5} a b^2 x^5+\frac {b^3 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^5)/5 + (b^3*x^6)/6

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Maple [A]
time = 0.07, size = 36, normalized size = 0.84

method	result	size
gosper	\(\frac {1}{3} a^{3} x^{3}+\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{6} b^{3} x^{6}\)	\(36\)
default	\(\frac {1}{3} a^{3} x^{3}+\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{6} b^{3} x^{6}\)	\(36\)
norman	\(\frac {1}{3} a^{3} x^{3}+\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{6} b^{3} x^{6}\)	\(36\)
risch	\(\frac {1}{3} a^{3} x^{3}+\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{6} b^{3} x^{6}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3+3/4*a^2*b*x^4+3/5*a*b^2*x^5+1/6*b^3*x^6

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Maxima [A]
time = 0.27, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/5*a*b^2*x^5 + 3/4*a^2*b*x^4 + 1/3*a^3*x^3

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Fricas [A]
time = 0.68, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/5*a*b^2*x^5 + 3/4*a^2*b*x^4 + 1/3*a^3*x^3

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Sympy [A]
time = 0.01, size = 39, normalized size = 0.91 \begin {gather*} \frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + 3*a**2*b*x**4/4 + 3*a*b**2*x**5/5 + b**3*x**6/6

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Giac [A]
time = 1.46, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/5*a*b^2*x^5 + 3/4*a^2*b*x^4 + 1/3*a^3*x^3

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Mupad [B]
time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \frac {a^3\,x^3}{3}+\frac {3\,a^2\,b\,x^4}{4}+\frac {3\,a\,b^2\,x^5}{5}+\frac {b^3\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^3,x)

[Out]

(a^3*x^3)/3 + (b^3*x^6)/6 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^5)/5

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