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

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

[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

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ \frac {a^3 x^7}{7}+\frac {1}{2} a^2 b x^6+\frac {3}{5} a b^2 x^5+\frac {b^3 x^4}{4} \]

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

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fricas [A]  time = 0.97, size = 35, normalized size = 0.81 \[ \frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \]

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

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giac [A]  time = 0.16, size = 35, normalized size = 0.81 \[ \frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \]

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

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ \frac {1}{7} a^{3} x^{7}+\frac {1}{2} a^{2} b \,x^{6}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{4} b^{3} x^{4} \]

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

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maxima [A]  time = 1.11, size = 35, normalized size = 0.81 \[ \frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \]

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

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mupad [B]  time = 0.04, size = 35, normalized size = 0.81 \[ \frac {a^3\,x^7}{7}+\frac {a^2\,b\,x^6}{2}+\frac {3\,a\,b^2\,x^5}{5}+\frac {b^3\,x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.08, size = 37, normalized size = 0.86 \[ \frac {a^{3} x^{7}}{7} + \frac {a^{2} b x^{6}}{2} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{4}}{4} \]

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

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