3.2.74 \(\int \frac {x^3}{(a+b x^2)^3} \, dx\) [174]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} \frac {x^4}{4 a \left (a+b x^2\right )^2} \end {gather*}

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 270

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*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.26 \begin {gather*} -\frac {a+2 b x^2}{4 b^2 \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^3,x]

[Out]

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

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Maple [A]
time = 0.03, size = 31, normalized size = 1.63

method	result	size
gosper	\(-\frac {2 b \,x^{2}+a}{4 b^{2} \left (b \,x^{2}+a \right )^{2}}\)	\(23\)
norman	\(\frac {-\frac {x^{2}}{2 b}-\frac {a}{4 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}\)	\(26\)
risch	\(\frac {-\frac {x^{2}}{2 b}-\frac {a}{4 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}\)	\(26\)
default	\(-\frac {1}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {a}{4 b^{2} \left (b \,x^{2}+a \right )^{2}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 0.27, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 1.13, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
time = 0.11, size = 36, normalized size = 1.89 \begin {gather*} \frac {- a - 2 b x^{2}}{4 a^{2} b^{2} + 8 a b^{3} x^{2} + 4 b^{4} x^{4}} \end {gather*}

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 = 2.39, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2}} \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 37, normalized size = 1.95 \begin {gather*} -\frac {\frac {a}{4\,b^2}+\frac {x^2}{2\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^2)^3,x)

[Out]

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

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