3.1571 \(\int (a+\frac {b}{x})^3 x^5 \, dx\)

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

[Out]

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

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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} \[ \frac {3}{5} a^2 b x^5+\frac {a^3 x^6}{6}+\frac {3}{4} a b^2 x^4+\frac {b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3*x^5,x]

[Out]

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

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3*x^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 35, normalized size = 0.81 \[ \frac {1}{6} \, a^{3} x^{6} + \frac {3}{5} \, a^{2} b x^{5} + \frac {3}{4} \, a b^{2} x^{4} + \frac {1}{3} \, b^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^5,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^3*x^5,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^5,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b/x)^3,x)

[Out]

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

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sympy [A]  time = 0.07, size = 39, normalized size = 0.91 \[ \frac {a^{3} x^{6}}{6} + \frac {3 a^{2} b x^{5}}{5} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**3*x**5,x)

[Out]

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

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