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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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])

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]]

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 = 0.01, 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]

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

________________________________________________________________________________________

fricas [A]  time = 0.99, size = 60, normalized size = 1.67 \[ \frac {4 \, b x^{2} - a}{24 \, {\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.59, size = 39, normalized size = 1.08 \[ \frac {\frac {4}{{\left (b x^{2} - a\right )}^{3} b} + \frac {3 \, a}{{\left (b x^{2} - a\right )}^{4} b}}{24 \, b} \]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 35, normalized size = 0.97 \[ \frac {a}{8 \left (b \,x^{2}-a \right )^{4} b^{2}}+\frac {1}{6 \left (b \,x^{2}-a \right )^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 60, normalized size = 1.67 \[ \frac {4 \, b x^{2} - a}{24 \, {\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 4.60, size = 59, normalized size = 1.64 \[ -\frac {\frac {a}{24\,b^2}-\frac {x^2}{6\,b}}{a^4-4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^4-4\,a\,b^3\,x^6+b^4\,x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a - b*x^2)^5,x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 0.44, size = 60, normalized size = 1.67 \[ - \frac {a - 4 b x^{2}}{24 a^{4} b^{2} - 96 a^{3} b^{3} x^{2} + 144 a^{2} b^{4} x^{4} - 96 a b^{5} x^{6} + 24 b^{6} x^{8}} \]

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)

________________________________________________________________________________________