3.1570 \(\int (a+\frac{b}{x})^3 x^6 \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^7}{7}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^4}{4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0199591, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^7}{7}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[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 263

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]

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 \left (a+\frac{b}{x}\right )^3 x^6 \, dx &=\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]  time = 0.0018376, size = 43, normalized size = 1. \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^7}{7}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[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]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{3\,{x}^{5}a{b}^{2}}{5}}+{\frac{{a}^{2}b{x}^{6}}{2}}+{\frac{{a}^{3}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.992243, size = 47, normalized size = 1.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 1.42561, size = 80, normalized size = 1.86 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.063521, size = 37, normalized size = 0.86 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [A]  time = 1.19027, size = 47, normalized size = 1.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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