3.2.95 \(\int \frac {x^3 (2+3 x^2)}{(3+5 x^2+x^4)^{3/2}} \, dx\) [195]

Optimal. Leaf size=56 \[ \frac {-33-47 x^2}{13 \sqrt {3+5 x^2+x^4}}+\frac {3}{2} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right ) \]

[Out]

3/2*arctanh(1/2*(2*x^2+5)/(x^4+5*x^2+3)^(1/2))+1/13*(-47*x^2-33)/(x^4+5*x^2+3)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1265, 791, 635, 212} \begin {gather*} \frac {3}{2} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {47 x^2+33}{13 \sqrt {x^4+5 x^2+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/13*(33 + 47*x^2)/Sqrt[3 + 5*x^2 + x^4] + (3*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/2

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 791

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2
)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*
p + 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(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] &&
IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {x^3 \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 {x (2+3 x)}{\left (3+5 x+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {33+47 x^2}{13 \sqrt {3+5 x^2+x^4}}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {33+47 x^2}{13 \sqrt {3+5 x^2+x^4}}+3 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )\\ &=-\frac {33+47 x^2}{13 \sqrt {3+5 x^2+x^4}}+\frac {3}{2} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 54, normalized size = 0.96 \begin {gather*} \frac {-33-47 x^2}{13 \sqrt {3+5 x^2+x^4}}-\frac {3}{2} \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-33 - 47*x^2)/(13*Sqrt[3 + 5*x^2 + x^4]) - (3*Log[-5 - 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(46)=92\).
time = 0.11, size = 95, normalized size = 1.70

method	result	size
risch	\(-\frac {47 x^{2}+33}{13 \sqrt {x^{4}+5 x^{2}+3}}+\frac {3 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{2}\)	\(43\)
trager	\(-\frac {47 x^{2}+33}{13 \sqrt {x^{4}+5 x^{2}+3}}-\frac {3 \ln \left (-2 x^{2}+2 \sqrt {x^{4}+5 x^{2}+3}-5\right )}{2}\)	\(47\)
elliptic	\(-\frac {3 x^{2}}{2 \sqrt {x^{4}+5 x^{2}+3}}+\frac {11}{4 \sqrt {x^{4}+5 x^{2}+3}}-\frac {55 \left (2 x^{2}+5\right )}{52 \sqrt {x^{4}+5 x^{2}+3}}+\frac {3 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{2}\)	\(74\)
default	\(-\frac {3 x^{2}}{2 \sqrt {x^{4}+5 x^{2}+3}}+\frac {15}{4 \sqrt {x^{4}+5 x^{2}+3}}-\frac {75 \left (2 x^{2}+5\right )}{52 \sqrt {x^{4}+5 x^{2}+3}}+\frac {3 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{2}+\frac {\frac {10 x^{2}}{13}+\frac {12}{13}}{\sqrt {x^{4}+5 x^{2}+3}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x^2/(x^4+5*x^2+3)^(1/2)+15/4/(x^4+5*x^2+3)^(1/2)-75/52*(2*x^2+5)/(x^4+5*x^2+3)^(1/2)+3/2*ln(x^2+5/2+(x^4+
5*x^2+3)^(1/2))+2/13/(x^4+5*x^2+3)^(1/2)*(5*x^2+6)

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Maxima [A]
time = 0.29, size = 56, normalized size = 1.00 \begin {gather*} -\frac {47 \, x^{2}}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} - \frac {33}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {3}{2} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-47/13*x^2/sqrt(x^4 + 5*x^2 + 3) - 33/13/sqrt(x^4 + 5*x^2 + 3) + 3/2*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]
time = 0.40, size = 81, normalized size = 1.45 \begin {gather*} -\frac {94 \, x^{4} + 470 \, x^{2} + 39 \, {\left (x^{4} + 5 \, x^{2} + 3\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (47 \, x^{2} + 33\right )} + 282}{26 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/26*(94*x^4 + 470*x^2 + 39*(x^4 + 5*x^2 + 3)*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) + 2*sqrt(x^4 + 5*x^2
+ 3)*(47*x^2 + 33) + 282)/(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^{3} \cdot \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*(3*x**2+2)/(x**4+5*x**2+3)**(3/2),x)

[Out]

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

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Giac [A]
time = 5.42, size = 46, normalized size = 0.82 \begin {gather*} -\frac {47 \, x^{2} + 33}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} - \frac {3}{2} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13*(47*x^2 + 33)/sqrt(x^4 + 5*x^2 + 3) - 3/2*log(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Mupad [B]
time = 0.31, size = 52, normalized size = 0.93 \begin {gather*} \frac {3\,\ln \left (\sqrt {x^4+5\,x^2+3}+x^2+\frac {5}{2}\right )}{2}-\frac {47\,x^2}{13\,\sqrt {x^4+5\,x^2+3}}-\frac {33}{13\,\sqrt {x^4+5\,x^2+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*log((5*x^2 + x^4 + 3)^(1/2) + x^2 + 5/2))/2 - (47*x^2)/(13*(5*x^2 + x^4 + 3)^(1/2)) - 33/(13*(5*x^2 + x^4 +
3)^(1/2))

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