3.246 \(\int \frac{x^3}{(a-b x^2)^5} \, dx\)

Optimal. Leaf size=36 \[ \frac{a}{8 b^2 \left (a-b x^2\right )^4}-\frac{1}{6 b^2 \left (a-b x^2\right )^3} \]

[Out]

a/(8*b^2*(a - b*x^2)^4) - 1/(6*b^2*(a - b*x^2)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0295651, antiderivative size = 36, 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} \[ \frac{a}{8 b^2 \left (a-b x^2\right )^4}-\frac{1}{6 b^2 \left (a-b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a - b*x^2)^5,x]

[Out]

a/(8*b^2*(a - b*x^2)^4) - 1/(6*b^2*(a - b*x^2)^3)

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 )^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a-b x)^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a}{b (a-b x)^5}-\frac{1}{b (a-b x)^4}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{8 b^2 \left (a-b x^2\right )^4}-\frac{1}{6 b^2 \left (a-b x^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.0096997, size = 25, normalized size = 0.69 \[ -\frac{a-4 b x^2}{24 b^2 \left (a-b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a - b*x^2)^5,x]

[Out]

-(a - 4*b*x^2)/(24*b^2*(a - b*x^2)^4)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 35, normalized size = 1. \begin{align*}{\frac{a}{8\,{b}^{2} \left ( b{x}^{2}-a \right ) ^{4}}}+{\frac{1}{6\,{b}^{2} \left ( b{x}^{2}-a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-b*x^2+a)^5,x)

[Out]

1/8/b^2*a/(b*x^2-a)^4+1/6/b^2/(b*x^2-a)^3

________________________________________________________________________________________

Maxima [A]  time = 1.9082, size = 81, normalized size = 2.25 \begin{align*} \frac{4 \, b x^{2} - a}{24 \,{\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/24*(4*b*x^2 - a)/(b^6*x^8 - 4*a*b^5*x^6 + 6*a^2*b^4*x^4 - 4*a^3*b^3*x^2 + a^4*b^2)

________________________________________________________________________________________

Fricas [A]  time = 1.21492, size = 116, normalized size = 3.22 \begin{align*} \frac{4 \, b x^{2} - a}{24 \,{\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/24*(4*b*x^2 - a)/(b^6*x^8 - 4*a*b^5*x^6 + 6*a^2*b^4*x^4 - 4*a^3*b^3*x^2 + a^4*b^2)

________________________________________________________________________________________

Sympy [A]  time = 0.732412, size = 58, normalized size = 1.61 \begin{align*} \frac{- a + 4 b x^{2}}{24 a^{4} b^{2} - 96 a^{3} b^{3} x^{2} + 144 a^{2} b^{4} x^{4} - 96 a b^{5} x^{6} + 24 b^{6} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(-b*x**2+a)**5,x)

[Out]

(-a + 4*b*x**2)/(24*a**4*b**2 - 96*a**3*b**3*x**2 + 144*a**2*b**4*x**4 - 96*a*b**5*x**6 + 24*b**6*x**8)

________________________________________________________________________________________

Giac [A]  time = 2.33869, size = 53, normalized size = 1.47 \begin{align*} \frac{\frac{4}{{\left (b x^{2} - a\right )}^{3} b} + \frac{3 \, a}{{\left (b x^{2} - a\right )}^{4} b}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^2+a)^5,x, algorithm="giac")

[Out]

1/24*(4/((b*x^2 - a)^3*b) + 3*a/((b*x^2 - a)^4*b))/b