3.188 \(\int \frac{2+3 x^2}{x^7 \sqrt{3+5 x^2+x^4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{13 \sqrt{x^4+5 x^2+3}}{108 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{54 x^4}-\frac{\sqrt{x^4+5 x^2+3}}{9 x^6}-\frac{61 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{216 \sqrt{3}} \]

[Out]

-Sqrt[3 + 5*x^2 + x^4]/(9*x^6) - Sqrt[3 + 5*x^2 + x^4]/(54*x^4) + (13*Sqrt[3 + 5*x^2 + x^4])/(108*x^2) - (61*A
rcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(216*Sqrt[3])

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Rubi [A]  time = 0.0881354, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1251, 834, 806, 724, 206} \[ \frac{13 \sqrt{x^4+5 x^2+3}}{108 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{54 x^4}-\frac{\sqrt{x^4+5 x^2+3}}{9 x^6}-\frac{61 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{216 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^2)/(x^7*Sqrt[3 + 5*x^2 + x^4]),x]

[Out]

-Sqrt[3 + 5*x^2 + x^4]/(9*x^6) - Sqrt[3 + 5*x^2 + x^4]/(54*x^4) + (13*Sqrt[3 + 5*x^2 + x^4])/(108*x^2) - (61*A
rcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(216*Sqrt[3])

Rule 1251

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]

Rule 834

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

Rule 806

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

Rule 724

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{2+3 x^2}{x^7 \sqrt{3+5 x^2+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{2+3 x}{x^4 \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{3+5 x^2+x^4}}{9 x^6}-\frac{1}{18} \operatorname{Subst}\left (\int \frac{-2+4 x}{x^3 \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{3+5 x^2+x^4}}{9 x^6}-\frac{\sqrt{3+5 x^2+x^4}}{54 x^4}+\frac{1}{108} \operatorname{Subst}\left (\int \frac{-39-2 x}{x^2 \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{3+5 x^2+x^4}}{9 x^6}-\frac{\sqrt{3+5 x^2+x^4}}{54 x^4}+\frac{13 \sqrt{3+5 x^2+x^4}}{108 x^2}+\frac{61}{216} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{3+5 x^2+x^4}}{9 x^6}-\frac{\sqrt{3+5 x^2+x^4}}{54 x^4}+\frac{13 \sqrt{3+5 x^2+x^4}}{108 x^2}-\frac{61}{108} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{6+5 x^2}{\sqrt{3+5 x^2+x^4}}\right )\\ &=-\frac{\sqrt{3+5 x^2+x^4}}{9 x^6}-\frac{\sqrt{3+5 x^2+x^4}}{54 x^4}+\frac{13 \sqrt{3+5 x^2+x^4}}{108 x^2}-\frac{61 \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )}{216 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.024487, size = 77, normalized size = 0.74 \[ \frac{6 \sqrt{x^4+5 x^2+3} \left (13 x^4-2 x^2-12\right )-61 \sqrt{3} x^6 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{648 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^2)/(x^7*Sqrt[3 + 5*x^2 + x^4]),x]

[Out]

(6*Sqrt[3 + 5*x^2 + x^4]*(-12 - 2*x^2 + 13*x^4) - 61*Sqrt[3]*x^6*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2
+ x^4])])/(648*x^6)

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Maple [A]  time = 0.013, size = 83, normalized size = 0.8 \begin{align*} -{\frac{61\,\sqrt{3}}{648}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{1}{9\,{x}^{6}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{1}{54\,{x}^{4}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{13}{108\,{x}^{2}}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x)

[Out]

-61/648*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)-1/9*(x^4+5*x^2+3)^(1/2)/x^6-1/54*(x^4+5*x^2
+3)^(1/2)/x^4+13/108*(x^4+5*x^2+3)^(1/2)/x^2

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Maxima [A]  time = 1.4628, size = 115, normalized size = 1.11 \begin{align*} -\frac{61}{648} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{108 \, x^{2}} - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{54 \, x^{4}} - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{9 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

-61/648*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 13/108*sqrt(x^4 + 5*x^2 + 3)/x^2 - 1/54
*sqrt(x^4 + 5*x^2 + 3)/x^4 - 1/9*sqrt(x^4 + 5*x^2 + 3)/x^6

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Fricas [A]  time = 1.46903, size = 235, normalized size = 2.26 \begin{align*} \frac{61 \, \sqrt{3} x^{6} \log \left (\frac{25 \, x^{2} - 2 \, \sqrt{3}{\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (5 \, \sqrt{3} - 6\right )} + 30}{x^{2}}\right ) + 78 \, x^{6} + 6 \,{\left (13 \, x^{4} - 2 \, x^{2} - 12\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{648 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

1/648*(61*sqrt(3)*x^6*log((25*x^2 - 2*sqrt(3)*(5*x^2 + 6) - 2*sqrt(x^4 + 5*x^2 + 3)*(5*sqrt(3) - 6) + 30)/x^2)
+ 78*x^6 + 6*(13*x^4 - 2*x^2 - 12)*sqrt(x^4 + 5*x^2 + 3))/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 x^{2} + 2}{x^{7} \sqrt{x^{4} + 5 x^{2} + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+2)/x**7/(x**4+5*x**2+3)**(1/2),x)

[Out]

Integral((3*x**2 + 2)/(x**7*sqrt(x**4 + 5*x**2 + 3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 \, x^{2} + 2}{\sqrt{x^{4} + 5 \, x^{2} + 3} x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate((3*x^2 + 2)/(sqrt(x^4 + 5*x^2 + 3)*x^7), x)