3.1573 \(\int (a+\frac {b}{x})^3 x^3 \, dx\)

Optimal. Leaf size=14 \[ \frac {(a x+b)^4}{4 a} \]

[Out]

1/4*(a*x+b)^4/a

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 32} \[ \frac {(a x+b)^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3*x^3,x]

[Out]

(b + a*x)^4/(4*a)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ \frac {(a x+b)^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3*x^3,x]

[Out]

(b + a*x)^4/(4*a)

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fricas [B]  time = 1.13, size = 31, normalized size = 2.21 \[ \frac {1}{4} \, a^{3} x^{4} + a^{2} b x^{3} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 31, normalized size = 2.21 \[ \frac {1}{4} \, a^{3} x^{4} + a^{2} b x^{3} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \[ \frac {\left (a x +b \right )^{4}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a*x+b)^4/a

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maxima [B]  time = 1.10, size = 31, normalized size = 2.21 \[ \frac {1}{4} \, a^{3} x^{4} + a^{2} b x^{3} + \frac {3}{2} \, a b^{2} x^{2} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 31, normalized size = 2.21 \[ \frac {a^3\,x^4}{4}+a^2\,b\,x^3+\frac {3\,a\,b^2\,x^2}{2}+b^3\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b/x)^3,x)

[Out]

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

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sympy [B]  time = 0.07, size = 32, normalized size = 2.29 \[ \frac {a^{3} x^{4}}{4} + a^{2} b x^{3} + \frac {3 a b^{2} x^{2}}{2} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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