3.1572 \(\int (a+\frac{b}{x})^3 x^4 \, dx\)

Optimal. Leaf size=30 \[ \frac{(a x+b)^5}{5 a^2}-\frac{b (a x+b)^4}{4 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0099667, antiderivative size = 30, 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} \[ \frac{(a x+b)^5}{5 a^2}-\frac{b (a x+b)^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3*x^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0015151, size = 40, normalized size = 1.33 \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^5}{5}+a b^2 x^3+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3*x^4,x]

[Out]

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

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Maple [A]  time = 0.001, size = 35, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{x}^{3}a{b}^{2}+{\frac{{b}^{3}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^3*x^4,x)

[Out]

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

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Maxima [A]  time = 0.980427, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.40899, size = 74, normalized size = 2.47 \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.061702, size = 36, normalized size = 1.2 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{3 a^{2} b x^{4}}{4} + a b^{2} x^{3} + \frac{b^{3} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**3*x**4,x)

[Out]

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

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Giac [A]  time = 1.13982, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^4,x, algorithm="giac")

[Out]

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