3.185 \(\int \frac{x^2}{(a+b x)^3} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a^2}{2 b^3 (a+b x)^2}+\frac{2 a}{b^3 (a+b x)}+\frac{\log (a+b x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.020299, antiderivative size = 41, 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} \[ -\frac{a^2}{2 b^3 (a+b x)^2}+\frac{2 a}{b^3 (a+b x)}+\frac{\log (a+b x)}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0194683, size = 33, normalized size = 0.8 \[ \frac{\frac{a (3 a+4 b x)}{(a+b x)^2}+2 \log (a+b x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^3,x]

[Out]

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

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Maple [A]  time = 0.005, size = 40, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+2\,{\frac{a}{{b}^{3} \left ( bx+a \right ) }}+{\frac{\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)^3,x)

[Out]

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

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Maxima [A]  time = 1.01963, size = 65, normalized size = 1.59 \begin{align*} \frac{4 \, a b x + 3 \, a^{2}}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56722, size = 132, normalized size = 3.22 \begin{align*} \frac{4 \, a b x + 3 \, a^{2} + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.533647, size = 46, normalized size = 1.12 \begin{align*} \frac{3 a^{2} + 4 a b x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{\log{\left (a + b x \right )}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x+a)**3,x)

[Out]

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

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Giac [A]  time = 1.14395, size = 50, normalized size = 1.22 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{4 \, a x + \frac{3 \, a^{2}}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^3,x, algorithm="giac")

[Out]

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