\(\int (a+\frac {b}{x})^3 x^6 \, dx\) [1570]

Optimal result

Integrand size = 13, antiderivative size = 43 \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {b^3 x^4}{4}+\frac {3}{5} a b^2 x^5+\frac {1}{2} a^2 b x^6+\frac {a^3 x^7}{7} \]

[Out]

1/4*b^3*x^4+3/5*a*b^2*x^5+1/2*a^2*b*x^6+1/7*a^3*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {a^3 x^7}{7}+\frac {1}{2} a^2 b x^6+\frac {3}{5} a b^2 x^5+\frac {b^3 x^4}{4} \]

[In]

Int[(a + b/x)^3*x^6,x]

[Out]

(b^3*x^4)/4 + (3*a*b^2*x^5)/5 + (a^2*b*x^6)/2 + (a^3*x^7)/7

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])

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 x^3 (b+a x)^3 \, dx \\ & = \int \left (b^3 x^3+3 a b^2 x^4+3 a^2 b x^5+a^3 x^6\right ) \, dx \\ & = \frac {b^3 x^4}{4}+\frac {3}{5} a b^2 x^5+\frac {1}{2} a^2 b x^6+\frac {a^3 x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {b^3 x^4}{4}+\frac {3}{5} a b^2 x^5+\frac {1}{2} a^2 b x^6+\frac {a^3 x^7}{7} \]

[In]

Integrate[(a + b/x)^3*x^6,x]

[Out]

(b^3*x^4)/4 + (3*a*b^2*x^5)/5 + (a^2*b*x^6)/2 + (a^3*x^7)/7

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {x^{4} \left (20 a^{3} x^{3}+70 a^{2} b \,x^{2}+84 a \,b^{2} x +35 b^{3}\right )}{140}\)	\(36\)
default	\(\frac {1}{4} b^{3} x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{7} x^{7} a^{3}\)	\(36\)
risch	\(\frac {1}{4} b^{3} x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{7} x^{7} a^{3}\)	\(36\)
parallelrisch	\(\frac {1}{4} b^{3} x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{7} x^{7} a^{3}\)	\(36\)
norman	\(\frac {\frac {1}{7} a^{3} x^{9}+\frac {1}{4} b^{3} x^{6}+\frac {3}{5} a \,b^{2} x^{7}+\frac {1}{2} a^{2} b \,x^{8}}{x^{2}}\)	\(40\)

[In]

int((a+b/x)^3*x^6,x,method=_RETURNVERBOSE)

[Out]

1/140*x^4*(20*a^3*x^3+70*a^2*b*x^2+84*a*b^2*x+35*b^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \]

[In]

integrate((a+b/x)^3*x^6,x, algorithm="fricas")

[Out]

1/7*a^3*x^7 + 1/2*a^2*b*x^6 + 3/5*a*b^2*x^5 + 1/4*b^3*x^4

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {a^{3} x^{7}}{7} + \frac {a^{2} b x^{6}}{2} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{4}}{4} \]

[In]

integrate((a+b/x)**3*x**6,x)

[Out]

a**3*x**7/7 + a**2*b*x**6/2 + 3*a*b**2*x**5/5 + b**3*x**4/4

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \]

[In]

integrate((a+b/x)^3*x^6,x, algorithm="maxima")

[Out]

1/7*a^3*x^7 + 1/2*a^2*b*x^6 + 3/5*a*b^2*x^5 + 1/4*b^3*x^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \]

[In]

integrate((a+b/x)^3*x^6,x, algorithm="giac")

[Out]

1/7*a^3*x^7 + 1/2*a^2*b*x^6 + 3/5*a*b^2*x^5 + 1/4*b^3*x^4

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx=\frac {a^3\,x^7}{7}+\frac {a^2\,b\,x^6}{2}+\frac {3\,a\,b^2\,x^5}{5}+\frac {b^3\,x^4}{4} \]

[In]

int(x^6*(a + b/x)^3,x)

[Out]

(a^3*x^7)/7 + (b^3*x^4)/4 + (3*a*b^2*x^5)/5 + (a^2*b*x^6)/2