3.199 \(\int \frac{x^2}{(a+b x)^4} \, dx\)

Optimal. Leaf size=17 \[ \frac{x^3}{3 a (a+b x)^3} \]

[Out]

x^3/(3*a*(a + b*x)^3)

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Rubi [A]  time = 0.0016848, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {37} \[ \frac{x^3}{3 a (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^4,x]

[Out]

x^3/(3*a*(a + b*x)^3)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{x^2}{(a+b x)^4} \, dx &=\frac{x^3}{3 a (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0171861, size = 31, normalized size = 1.82 \[ -\frac{a^2+3 a b x+3 b^2 x^2}{3 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^4,x]

[Out]

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

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Maple [B]  time = 0.004, size = 41, normalized size = 2.4 \begin{align*} -{\frac{1}{{b}^{3} \left ( bx+a \right ) }}+{\frac{a}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^4,x)

[Out]

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

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Maxima [B]  time = 1.08234, size = 73, normalized size = 4.29 \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.48234, size = 111, normalized size = 6.53 \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.516355, size = 56, normalized size = 3.29 \begin{align*} - \frac{a^{2} + 3 a b x + 3 b^{2} x^{2}}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x+a)**4,x)

[Out]

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

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Giac [A]  time = 1.13832, size = 39, normalized size = 2.29 \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \,{\left (b x + a\right )}^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^4,x, algorithm="giac")

[Out]

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