3.2.96 \(\int \frac {x (2+3 x^2)}{(3+5 x^2+x^4)^{3/2}} \, dx\) [196]

Optimal. Leaf size=25 \[ \frac {8+11 x^2}{13 \sqrt {3+5 x^2+x^4}} \]

[Out]

1/13*(11*x^2+8)/(x^4+5*x^2+3)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1261, 650} \begin {gather*} \frac {11 x^2+8}{13 \sqrt {x^4+5 x^2+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

\begin {align*} \int \frac {x \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {2+3 x}{\left (3+5 x+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {8+11 x^2}{13 \sqrt {3+5 x^2+x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 25, normalized size = 1.00 \begin {gather*} \frac {8+11 x^2}{13 \sqrt {3+5 x^2+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs. \(2(21)=42\).
time = 0.04, size = 44, normalized size = 1.76

method	result	size
gosper	\(\frac {11 x^{2}+8}{13 \sqrt {x^{4}+5 x^{2}+3}}\)	\(22\)
trager	\(\frac {11 x^{2}+8}{13 \sqrt {x^{4}+5 x^{2}+3}}\)	\(22\)
risch	\(\frac {11 x^{2}+8}{13 \sqrt {x^{4}+5 x^{2}+3}}\)	\(22\)
elliptic	\(\frac {11 x^{2}+8}{13 \sqrt {x^{4}+5 x^{2}+3}}\)	\(22\)
default	\(\frac {\frac {15 x^{2}}{13}+\frac {18}{13}}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \left (2 x^{2}+5\right )}{13 \sqrt {x^{4}+5 x^{2}+3}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(3*x^2+2)/(x^4+5*x^2+3)^(3/2),x,method=_RETURNVERBOSE)

[Out]

3/13/(x^4+5*x^2+3)^(1/2)*(5*x^2+6)-2/13*(2*x^2+5)/(x^4+5*x^2+3)^(1/2)

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Maxima [A]
time = 0.28, size = 32, normalized size = 1.28 \begin {gather*} \frac {11 \, x^{2}}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {8}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(3*x^2+2)/(x^4+5*x^2+3)^(3/2),x, algorithm="maxima")

[Out]

11/13*x^2/sqrt(x^4 + 5*x^2 + 3) + 8/13/sqrt(x^4 + 5*x^2 + 3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
time = 0.35, size = 46, normalized size = 1.84 \begin {gather*} \frac {11 \, x^{4} + 55 \, x^{2} + \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (11 \, x^{2} + 8\right )} + 33}{13 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(3*x^2+2)/(x^4+5*x^2+3)^(3/2),x, algorithm="fricas")

[Out]

1/13*(11*x^4 + 55*x^2 + sqrt(x^4 + 5*x^2 + 3)*(11*x^2 + 8) + 33)/(x^4 + 5*x^2 + 3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(3*x**2+2)/(x**4+5*x**2+3)**(3/2),x)

[Out]

Integral(x*(3*x**2 + 2)/(x**4 + 5*x**2 + 3)**(3/2), x)

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Giac [A]
time = 6.33, size = 21, normalized size = 0.84 \begin {gather*} \frac {11 \, x^{2} + 8}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(3*x^2+2)/(x^4+5*x^2+3)^(3/2),x, algorithm="giac")

[Out]

1/13*(11*x^2 + 8)/sqrt(x^4 + 5*x^2 + 3)

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Mupad [B]
time = 0.24, size = 21, normalized size = 0.84 \begin {gather*} \frac {11\,x^2+8}{13\,\sqrt {x^4+5\,x^2+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(3*x^2 + 2))/(5*x^2 + x^4 + 3)^(3/2),x)

[Out]

(11*x^2 + 8)/(13*(5*x^2 + x^4 + 3)^(1/2))

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