\(\int (a+\frac {b}{x})^3 x^3 \, dx\) [1573]

Optimal result

Integrand size = 13, antiderivative size = 14 \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {(b+a x)^4}{4 a} \]

[Out]

1/4*(a*x+b)^4/a

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 32} \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {(a x+b)^4}{4 a} \]

[In]

Int[(a + b/x)^3*x^3,x]

[Out]

(b + a*x)^4/(4*a)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rubi steps \begin{align*} \text {integral}& = \int (b+a x)^3 \, dx \\ & = \frac {(b+a x)^4}{4 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {(b+a x)^4}{4 a} \]

[In]

Integrate[(a + b/x)^3*x^3,x]

[Out]

(b + a*x)^4/(4*a)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\left (a x +b \right )^{4}}{4 a}\)	\(13\)
parallelrisch	\(\frac {1}{4} a^{3} x^{4}+a^{2} b \,x^{3}+\frac {3}{2} a \,b^{2} x^{2}+b^{3} x\)	\(32\)
gosper	\(\frac {x \left (a^{3} x^{3}+4 a^{2} b \,x^{2}+6 a \,b^{2} x +4 b^{3}\right )}{4}\)	\(33\)
norman	\(\frac {b^{3} x^{3}+a^{2} b \,x^{5}+\frac {1}{4} x^{6} a^{3}+\frac {3}{2} a \,b^{2} x^{4}}{x^{2}}\)	\(38\)
risch	\(\frac {a^{3} x^{4}}{4}+a^{2} b \,x^{3}+\frac {3 a \,b^{2} x^{2}}{2}+b^{3} x +\frac {b^{4}}{4 a}\)	\(40\)

[In]

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

[Out]

1/4*(a*x+b)^4/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {1}{4} \, a^{3} x^{4} + a^{2} b x^{3} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} x \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (8) = 16\).

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.29 \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {a^{3} x^{4}}{4} + a^{2} b x^{3} + \frac {3 a b^{2} x^{2}}{2} + b^{3} x \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {1}{4} \, a^{3} x^{4} + a^{2} b x^{3} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {1}{4} \, a^{3} x^{4} + a^{2} b x^{3} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \left (a+\frac {b}{x}\right )^3 x^3 \, dx=\frac {a^3\,x^4}{4}+a^2\,b\,x^3+\frac {3\,a\,b^2\,x^2}{2}+b^3\,x \]

[In]

int(x^3*(a + b/x)^3,x)

[Out]

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