3.232 \(\int \frac{x^3}{(a-b x^2)^2} \, dx\)

Optimal. Leaf size=35 \[ \frac{a}{2 b^2 \left (a-b x^2\right )}+\frac{\log \left (a-b x^2\right )}{2 b^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0269475, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 43} \[ \frac{a}{2 b^2 \left (a-b x^2\right )}+\frac{\log \left (a-b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a - b*x^2)^2,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A]  time = 0.012539, size = 29, normalized size = 0.83 \[ \frac{\frac{a}{a-b x^2}+\log \left (a-b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a - b*x^2)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 34, normalized size = 1. \begin{align*} -{\frac{a}{2\,{b}^{2} \left ( b{x}^{2}-a \right ) }}+{\frac{\ln \left ( b{x}^{2}-a \right ) }{2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.09328, size = 47, normalized size = 1.34 \begin{align*} -\frac{a}{2 \,{\left (b^{3} x^{2} - a b^{2}\right )}} + \frac{\log \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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.24491, size = 76, normalized size = 2.17 \begin{align*} \frac{{\left (b x^{2} - a\right )} \log \left (b x^{2} - a\right ) - a}{2 \,{\left (b^{3} x^{2} - a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.352681, size = 29, normalized size = 0.83 \begin{align*} - \frac{a}{- 2 a b^{2} + 2 b^{3} x^{2}} + \frac{\log{\left (- a + b x^{2} \right )}}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 2.0632, size = 72, normalized size = 2.06 \begin{align*} -\frac{\frac{\log \left (\frac{{\left | b x^{2} - a \right |}}{{\left (b x^{2} - a\right )}^{2}{\left | b \right |}}\right )}{b} + \frac{a}{{\left (b x^{2} - a\right )} b}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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