3.3.41 \(\int \frac {x}{(a-b x^2)^3} \, dx\) [241]

Optimal. Leaf size=17 \[ \frac {1}{4 b \left (a-b x^2\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {267} \begin {gather*} \frac {1}{4 b \left (a-b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a - b*x^2)^3,x]

[Out]

1/(4*b*(a - b*x^2)^2)

Rule 267

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{4 b \left (a-b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a - b*x^2)^3,x]

[Out]

1/(4*b*(a - b*x^2)^2)

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Maple [A]
time = 0.02, size = 16, normalized size = 0.94

method	result	size
gosper	\(\frac {1}{4 b \left (-b \,x^{2}+a \right )^{2}}\)	\(16\)
derivativedivides	\(\frac {1}{4 b \left (-b \,x^{2}+a \right )^{2}}\)	\(16\)
default	\(\frac {1}{4 b \left (-b \,x^{2}+a \right )^{2}}\)	\(16\)
norman	\(\frac {1}{4 b \left (-b \,x^{2}+a \right )^{2}}\)	\(16\)
risch	\(\frac {1}{4 b \left (-b \,x^{2}+a \right )^{2}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 16, normalized size = 0.94 \begin {gather*} \frac {1}{4 \, {\left (b x^{2} - a\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/((b*x^2 - a)^2*b)

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Fricas [A]
time = 0.93, size = 26, normalized size = 1.53 \begin {gather*} \frac {1}{4 \, {\left (b^{3} x^{4} - 2 \, a b^{2} x^{2} + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/(b^3*x^4 - 2*a*b^2*x^2 + a^2*b)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.10, size = 26, normalized size = 1.53 \begin {gather*} \frac {1}{4 a^{2} b - 8 a b^{2} x^{2} + 4 b^{3} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.21, size = 16, normalized size = 0.94 \begin {gather*} \frac {1}{4 \, {\left (b x^{2} - a\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/((b*x^2 - a)^2*b)

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Mupad [B]
time = 0.03, size = 26, normalized size = 1.53 \begin {gather*} \frac {1}{4\,a^2\,b-8\,a\,b^2\,x^2+4\,b^3\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a - b*x^2)^3,x)

[Out]

1/(4*a^2*b + 4*b^3*x^4 - 8*a*b^2*x^2)

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