3.1622 \(\int \frac{x^2}{(a+\frac{b}{x})^2} \, dx\)

Optimal. Leaf size=58 \[ -\frac{b^4}{a^5 (a x+b)}+\frac{3 b^2 x}{a^4}-\frac{4 b^3 \log (a x+b)}{a^5}-\frac{b x^2}{a^3}+\frac{x^3}{3 a^2} \]

[Out]

(3*b^2*x)/a^4 - (b*x^2)/a^3 + x^3/(3*a^2) - b^4/(a^5*(b + a*x)) - (4*b^3*Log[b + a*x])/a^5

________________________________________________________________________________________

Rubi [A]  time = 0.0370357, antiderivative size = 58, 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} \[ -\frac{b^4}{a^5 (a x+b)}+\frac{3 b^2 x}{a^4}-\frac{4 b^3 \log (a x+b)}{a^5}-\frac{b x^2}{a^3}+\frac{x^3}{3 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x)^2,x]

[Out]

(3*b^2*x)/a^4 - (b*x^2)/a^3 + x^3/(3*a^2) - b^4/(a^5*(b + a*x)) - (4*b^3*Log[b + a*x])/a^5

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

Mathematica [A]  time = 0.0188467, size = 54, normalized size = 0.93 \[ \frac{-3 a^2 b x^2+a^3 x^3-\frac{3 b^4}{a x+b}+9 a b^2 x-12 b^3 \log (a x+b)}{3 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x)^2,x]

[Out]

(9*a*b^2*x - 3*a^2*b*x^2 + a^3*x^3 - (3*b^4)/(b + a*x) - 12*b^3*Log[b + a*x])/(3*a^5)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 57, normalized size = 1. \begin{align*} 3\,{\frac{{b}^{2}x}{{a}^{4}}}-{\frac{b{x}^{2}}{{a}^{3}}}+{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{b}^{4}}{{a}^{5} \left ( ax+b \right ) }}-4\,{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b/x)^2,x)

[Out]

3*b^2*x/a^4-b*x^2/a^3+1/3*x^3/a^2-b^4/a^5/(a*x+b)-4*b^3*ln(a*x+b)/a^5

________________________________________________________________________________________

Maxima [A]  time = 0.989066, size = 80, normalized size = 1.38 \begin{align*} -\frac{b^{4}}{a^{6} x + a^{5} b} - \frac{4 \, b^{3} \log \left (a x + b\right )}{a^{5}} + \frac{a^{2} x^{3} - 3 \, a b x^{2} + 9 \, b^{2} x}{3 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x)^2,x, algorithm="maxima")

[Out]

-b^4/(a^6*x + a^5*b) - 4*b^3*log(a*x + b)/a^5 + 1/3*(a^2*x^3 - 3*a*b*x^2 + 9*b^2*x)/a^4

________________________________________________________________________________________

Fricas [A]  time = 1.40926, size = 155, normalized size = 2.67 \begin{align*} \frac{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a b^{3} x - 3 \, b^{4} - 12 \,{\left (a b^{3} x + b^{4}\right )} \log \left (a x + b\right )}{3 \,{\left (a^{6} x + a^{5} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x)^2,x, algorithm="fricas")

[Out]

1/3*(a^4*x^4 - 2*a^3*b*x^3 + 6*a^2*b^2*x^2 + 9*a*b^3*x - 3*b^4 - 12*(a*b^3*x + b^4)*log(a*x + b))/(a^6*x + a^5
*b)

________________________________________________________________________________________

Sympy [A]  time = 0.340239, size = 54, normalized size = 0.93 \begin{align*} - \frac{b^{4}}{a^{6} x + a^{5} b} + \frac{x^{3}}{3 a^{2}} - \frac{b x^{2}}{a^{3}} + \frac{3 b^{2} x}{a^{4}} - \frac{4 b^{3} \log{\left (a x + b \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b/x)**2,x)

[Out]

-b**4/(a**6*x + a**5*b) + x**3/(3*a**2) - b*x**2/a**3 + 3*b**2*x/a**4 - 4*b**3*log(a*x + b)/a**5

________________________________________________________________________________________

Giac [A]  time = 1.12542, size = 84, normalized size = 1.45 \begin{align*} -\frac{4 \, b^{3} \log \left ({\left | a x + b \right |}\right )}{a^{5}} - \frac{b^{4}}{{\left (a x + b\right )} a^{5}} + \frac{a^{4} x^{3} - 3 \, a^{3} b x^{2} + 9 \, a^{2} b^{2} x}{3 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x)^2,x, algorithm="giac")

[Out]

-4*b^3*log(abs(a*x + b))/a^5 - b^4/((a*x + b)*a^5) + 1/3*(a^4*x^3 - 3*a^3*b*x^2 + 9*a^2*b^2*x)/a^6