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

Optimal. Leaf size=32 \[ -\frac{\left (A+B x^2\right )^2}{4 \left (a+b x^2\right )^2 (A b-a B)} \]

[Out]

-(A + B*x^2)^2/(4*(A*b - a*B)*(a + b*x^2)^2)

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Rubi [A]  time = 0.0195665, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 37} \[ -\frac{\left (A+B x^2\right )^2}{4 \left (a+b x^2\right )^2 (A b-a B)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

-(A + B*x^2)^2/(4*(A*b - a*B)*(a + b*x^2)^2)

Rule 444

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

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 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^3} \, dx,x,x^2\right )\\ &=-\frac{\left (A+B x^2\right )^2}{4 (A b-a B) \left (a+b x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0130987, size = 30, normalized size = 0.94 \[ -\frac{B \left (a+2 b x^2\right )+A b}{4 b^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

-(A*b + B*(a + 2*b*x^2))/(4*b^2*(a + b*x^2)^2)

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Maple [A]  time = 0.008, size = 39, normalized size = 1.2 \begin{align*} -{\frac{Ab-Ba}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{B}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

-1/4*(A*b-B*a)/b^2/(b*x^2+a)^2-1/2*B/b^2/(b*x^2+a)

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Maxima [A]  time = 0.997769, size = 57, normalized size = 1.78 \begin{align*} -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*B*b*x^2 + B*a + A*b)/(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)

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Fricas [A]  time = 1.18997, size = 86, normalized size = 2.69 \begin{align*} -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*B*b*x^2 + B*a + A*b)/(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)

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Sympy [A]  time = 0.658565, size = 42, normalized size = 1.31 \begin{align*} - \frac{A b + B a + 2 B b x^{2}}{4 a^{2} b^{2} + 8 a b^{3} x^{2} + 4 b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(A*b + B*a + 2*B*b*x**2)/(4*a**2*b**2 + 8*a*b**3*x**2 + 4*b**4*x**4)

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Giac [A]  time = 1.1122, size = 38, normalized size = 1.19 \begin{align*} -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*B*b*x^2 + B*a + A*b)/((b*x^2 + a)^2*b^2)