\(\int (\frac {a}{x}+b x)^3 \, dx\) [345]

Optimal result

Integrand size = 11, antiderivative size = 40 \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=-\frac {a^3}{2 x^2}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^4}{4}+3 a^2 b \log (x) \]

[Out]

-1/2*a^3/x^2+3/2*a*b^2*x^2+1/4*b^3*x^4+3*a^2*b*ln(x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1607, 272, 45} \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=-\frac {a^3}{2 x^2}+3 a^2 b \log (x)+\frac {3}{2} a b^2 x^2+\frac {b^3 x^4}{4} \]

[In]

Int[(a/x + b*x)^3,x]

[Out]

-1/2*a^3/x^2 + (3*a*b^2*x^2)/2 + (b^3*x^4)/4 + 3*a^2*b*Log[x]

Rule 45

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b x^2\right )^3}{x^3} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^3}{x^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (3 a b^2+\frac {a^3}{x^2}+\frac {3 a^2 b}{x}+b^3 x\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^3}{2 x^2}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^4}{4}+3 a^2 b \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=-\frac {a^3}{2 x^2}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^4}{4}+3 a^2 b \log (x) \]

[In]

Integrate[(a/x + b*x)^3,x]

[Out]

-1/2*a^3/x^2 + (3*a*b^2*x^2)/2 + (b^3*x^4)/4 + 3*a^2*b*Log[x]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {a^{3}}{2 x^{2}}+\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{4}}{4}+3 a^{2} b \ln \left (x \right )\)	\(35\)
norman	\(\frac {-\frac {1}{2} a^{3}+\frac {1}{4} b^{3} x^{6}+\frac {3}{2} a \,b^{2} x^{4}}{x^{2}}+3 a^{2} b \ln \left (x \right )\)	\(37\)
parallelrisch	\(\frac {b^{3} x^{6}+6 a \,b^{2} x^{4}+12 a^{2} b \ln \left (x \right ) x^{2}-2 a^{3}}{4 x^{2}}\)	\(39\)
risch	\(\frac {b^{3} x^{4}}{4}+\frac {3 a \,b^{2} x^{2}}{2}+\frac {9 a^{2} b}{4}-\frac {a^{3}}{2 x^{2}}+3 a^{2} b \ln \left (x \right )\)	\(41\)

[In]

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

[Out]

-1/2*a^3/x^2+3/2*a*b^2*x^2+1/4*b^3*x^4+3*a^2*b*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=\frac {b^{3} x^{6} + 6 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} \log \left (x\right ) - 2 \, a^{3}}{4 \, x^{2}} \]

[In]

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

[Out]

1/4*(b^3*x^6 + 6*a*b^2*x^4 + 12*a^2*b*x^2*log(x) - 2*a^3)/x^2

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=- \frac {a^{3}}{2 x^{2}} + 3 a^{2} b \log {\left (x \right )} + \frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{4}}{4} \]

[In]

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

[Out]

-a**3/(2*x**2) + 3*a**2*b*log(x) + 3*a*b**2*x**2/2 + b**3*x**4/4

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=\frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b \log \left (x\right ) - \frac {a^{3}}{2 \, x^{2}} \]

[In]

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

[Out]

1/4*b^3*x^4 + 3/2*a*b^2*x^2 + 3*a^2*b*log(x) - 1/2*a^3/x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=\frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac {3 \, a^{2} b x^{2} + a^{3}}{2 \, x^{2}} \]

[In]

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

[Out]

1/4*b^3*x^4 + 3/2*a*b^2*x^2 + 3/2*a^2*b*log(x^2) - 1/2*(3*a^2*b*x^2 + a^3)/x^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \left (\frac {a}{x}+b x\right )^3 \, dx=\frac {b^3\,x^4}{4}-\frac {a^3}{2\,x^2}+\frac {3\,a\,b^2\,x^2}{2}+3\,a^2\,b\,\ln \left (x\right ) \]

[In]

int((b*x + a/x)^3,x)

[Out]

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