3.1.67 \(\int x (a+b x)^3 \, dx\) [67]

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

[Out]

-1/4*a*(b*x+a)^4/b^2+1/5*(b*x+a)^5/b^2

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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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {(a+b x)^5}{5 b^2}-\frac {a (a+b x)^4}{4 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^3,x]

[Out]

-1/4*(a*(a + b*x)^4)/b^2 + (a + b*x)^5/(5*b^2)

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

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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.33 \begin {gather*} \frac {a^3 x^2}{2}+a^2 b x^3+\frac {3}{4} a b^2 x^4+\frac {b^3 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^3,x]

[Out]

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

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Maple [A]
time = 0.08, size = 35, normalized size = 1.17

method	result	size
gosper	\(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\)	\(35\)
default	\(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\)	\(35\)
norman	\(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\)	\(35\)
risch	\(\frac {1}{5} b^{3} x^{5}+\frac {3}{4} a \,b^{2} x^{4}+a^{2} b \,x^{3}+\frac {1}{2} a^{3} x^{2}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 34, normalized size = 1.13 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac {1}{2} \, a^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.79, size = 34, normalized size = 1.13 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac {1}{2} \, a^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 36, normalized size = 1.20 \begin {gather*} \frac {a^{3} x^{2}}{2} + a^{2} b x^{3} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.41, size = 34, normalized size = 1.13 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} + \frac {3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac {1}{2} \, a^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 34, normalized size = 1.13 \begin {gather*} \frac {a^3\,x^2}{2}+a^2\,b\,x^3+\frac {3\,a\,b^2\,x^4}{4}+\frac {b^3\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^3,x)

[Out]

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

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