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

Optimal. Leaf size=33 \[ -\frac{a^2}{b^3 (a+b x)}-\frac{2 a \log (a+b x)}{b^3}+\frac{x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0179673, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{a^2}{b^3 (a+b x)}-\frac{2 a \log (a+b x)}{b^3}+\frac{x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^2,x]

[Out]

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

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}{(a+b x)^2} \, dx &=\int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{x}{b^2}-\frac{a^2}{b^3 (a+b x)}-\frac{2 a \log (a+b x)}{b^3}\\ \end{align*}

Mathematica [A]  time = 0.0141772, size = 29, normalized size = 0.88 \[ \frac{-\frac{a^2}{a+b x}-2 a \log (a+b x)+b x}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 34, normalized size = 1. \begin{align*}{\frac{x}{{b}^{2}}}-{\frac{{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}-2\,{\frac{a\ln \left ( bx+a \right ) }{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07287, size = 49, normalized size = 1.48 \begin{align*} -\frac{a^{2}}{b^{4} x + a b^{3}} + \frac{x}{b^{2}} - \frac{2 \, a \log \left (b x + a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52762, size = 97, normalized size = 2.94 \begin{align*} \frac{b^{2} x^{2} + a b x - a^{2} - 2 \,{\left (a b x + a^{2}\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.453715, size = 31, normalized size = 0.94 \begin{align*} - \frac{a^{2}}{a b^{3} + b^{4} x} - \frac{2 a \log{\left (a + b x \right )}}{b^{3}} + \frac{x}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13514, size = 68, normalized size = 2.06 \begin{align*} \frac{2 \, a \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{b x + a}{b^{3}} - \frac{a^{2}}{{\left (b x + a\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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