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

Optimal. Leaf size=19 \[ \frac{x^4}{4 a \left (a+b x^2\right )^2} \]

[Out]

x^4/(4*a*(a + b*x^2)^2)

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Rubi [A]  time = 0.0038325, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ \frac{x^4}{4 a \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^3,x]

[Out]

x^4/(4*a*(a + b*x^2)^2)

Rule 264

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

Rubi steps

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

Mathematica [A]  time = 0.0076782, size = 24, normalized size = 1.26 \[ -\frac{a+2 b x^2}{4 b^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^3,x]

[Out]

-(a + 2*b*x^2)/(4*b^2*(a + b*x^2)^2)

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Maple [A]  time = 0.007, size = 31, normalized size = 1.6 \begin{align*}{\frac{a}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{1}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.15848, size = 49, normalized size = 2.58 \begin{align*} -\frac{2 \, b x^{2} + a}{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^3/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.24151, size = 73, normalized size = 3.84 \begin{align*} -\frac{2 \, b x^{2} + a}{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^3/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.41778, size = 36, normalized size = 1.89 \begin{align*} - \frac{a + 2 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**3/(b*x**2+a)**3,x)

[Out]

-(a + 2*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 = 1.36925, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, b x^{2} + a}{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^3/(b*x^2+a)^3,x, algorithm="giac")

[Out]

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