3.1575 \(\int (a+\frac {b}{x})^3 x \, dx\)

Optimal. Leaf size=34 \[ \frac {a^3 x^2}{2}+3 a^2 b x+3 a b^2 \log (x)-\frac {b^3}{x} \]

[Out]

-b^3/x+3*a^2*b*x+1/2*a^3*x^2+3*a*b^2*ln(x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {263, 43} \[ 3 a^2 b x+\frac {a^3 x^2}{2}+3 a b^2 \log (x)-\frac {b^3}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3*x,x]

[Out]

-(b^3/x) + 3*a^2*b*x + (a^3*x^2)/2 + 3*a*b^2*Log[x]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 34, normalized size = 1.00 \[ \frac {a^3 x^2}{2}+3 a^2 b x+3 a b^2 \log (x)-\frac {b^3}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3*x,x]

[Out]

-(b^3/x) + 3*a^2*b*x + (a^3*x^2)/2 + 3*a*b^2*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.89, size = 36, normalized size = 1.06 \[ \frac {a^{3} x^{3} + 6 \, a^{2} b x^{2} + 6 \, a b^{2} x \log \relax (x) - 2 \, b^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^3*x^3 + 6*a^2*b*x^2 + 6*a*b^2*x*log(x) - 2*b^3)/x

________________________________________________________________________________________

giac [A]  time = 0.15, size = 33, normalized size = 0.97 \[ \frac {1}{2} \, a^{3} x^{2} + 3 \, a^{2} b x + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) - \frac {b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*x^2 + 3*a^2*b*x + 3*a*b^2*log(abs(x)) - b^3/x

________________________________________________________________________________________

maple [A]  time = 0.01, size = 33, normalized size = 0.97 \[ \frac {a^{3} x^{2}}{2}+3 a^{2} b x +3 a \,b^{2} \ln \relax (x )-\frac {b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3/x+3*a^2*b*x+1/2*a^3*x^2+3*a*b^2*ln(x)

________________________________________________________________________________________

maxima [A]  time = 1.04, size = 32, normalized size = 0.94 \[ \frac {1}{2} \, a^{3} x^{2} + 3 \, a^{2} b x + 3 \, a b^{2} \log \relax (x) - \frac {b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*x^2 + 3*a^2*b*x + 3*a*b^2*log(x) - b^3/x

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 32, normalized size = 0.94 \[ \frac {a^3\,x^2}{2}-\frac {b^3}{x}+3\,a\,b^2\,\ln \relax (x)+3\,a^2\,b\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b/x)^3,x)

[Out]

(a^3*x^2)/2 - b^3/x + 3*a*b^2*log(x) + 3*a^2*b*x

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 31, normalized size = 0.91 \[ \frac {a^{3} x^{2}}{2} + 3 a^{2} b x + 3 a b^{2} \log {\relax (x )} - \frac {b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**2/2 + 3*a**2*b*x + 3*a*b**2*log(x) - b**3/x

________________________________________________________________________________________