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

Optimal. Leaf size=40 \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.0225053, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1593, 266, 43} \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a/x + b*x)^3,x]

[Out]

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

Rule 1593

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]

Rule 266

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 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 (\frac{a}{x}+b x\right )^3 \, dx &=\int \frac{\left (a+b x^2\right )^3}{x^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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]  time = 0.0068147, size = 40, normalized size = 1. \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a/x + b*x)^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \begin{align*} -{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.15406, size = 46, normalized size = 1.15 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0.770119, size = 85, normalized size = 2.12 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.419012, size = 37, normalized size = 0.92 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.11189, size = 62, normalized size = 1.55 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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