3.150 \(\int \frac{(2+3 x^2) \sqrt{3+5 x^2+x^4}}{x^{11}} \, dx\)

Optimal. Leaf size=132 \[ \frac{173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac{161 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{5184 x^4}+\frac{2093 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{10368 \sqrt{3}} \]

[Out]

(-161*(6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(5184*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(15*x^10) - (3 + 5*x^2 + x^4)^(3
/2)/(36*x^8) + (173*(3 + 5*x^2 + x^4)^(3/2))/(3240*x^6) + (2093*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2
+ x^4])])/(10368*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.108624, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1251, 834, 806, 720, 724, 206} \[ \frac{173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac{161 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{5184 x^4}+\frac{2093 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{10368 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^11,x]

[Out]

(-161*(6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(5184*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(15*x^10) - (3 + 5*x^2 + x^4)^(3
/2)/(36*x^8) + (173*(3 + 5*x^2 + x^4)^(3/2))/(3240*x^6) + (2093*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2
+ x^4])])/(10368*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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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{\left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4}}{x^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \sqrt{3+5 x+x^2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{1}{30} \operatorname{Subst}\left (\int \frac{(-10+4 x) \sqrt{3+5 x+x^2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{1}{360} \operatorname{Subst}\left (\int \frac{(-173-10 x) \sqrt{3+5 x+x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac{161}{432} \operatorname{Subst}\left (\int \frac{\sqrt{3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{161 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{5184 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}-\frac{2093 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )}{10368}\\ &=-\frac{161 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{5184 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac{2093 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{6+5 x^2}{\sqrt{3+5 x^2+x^4}}\right )}{5184}\\ &=-\frac{161 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{5184 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac{2093 \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )}{10368 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0370601, size = 84, normalized size = 0.64 \[ \frac{10465 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )-\frac{6 \sqrt{x^4+5 x^2+3} \left (2641 x^8-1370 x^6+1176 x^4+10800 x^2+5184\right )}{x^{10}}}{155520} \]

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^11,x]

[Out]

((-6*Sqrt[3 + 5*x^2 + x^4]*(5184 + 10800*x^2 + 1176*x^4 - 1370*x^6 + 2641*x^8))/x^10 + 10465*Sqrt[3]*ArcTanh[(
6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/155520

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 152, normalized size = 1.2 \begin{align*} -{\frac{1}{15\,{x}^{10}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{36\,{x}^{8}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{173}{3240\,{x}^{6}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{161}{2592\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{805}{15552\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{2093}{31104}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{2093\,\sqrt{3}}{31104}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{1610\,{x}^{2}+4025}{31104}\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^4+5*x^2+3)^(1/2)/x^11,x)

[Out]

-1/15*(x^4+5*x^2+3)^(3/2)/x^10-1/36*(x^4+5*x^2+3)^(3/2)/x^8+173/3240*(x^4+5*x^2+3)^(3/2)/x^6-161/2592/x^4*(x^4
+5*x^2+3)^(3/2)+805/15552/x^2*(x^4+5*x^2+3)^(3/2)-2093/31104*(x^4+5*x^2+3)^(1/2)+2093/31104*arctanh(1/6*(5*x^2
+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)-805/31104*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.50393, size = 180, normalized size = 1.36 \begin{align*} \frac{2093}{31104} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{161}{2592} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{805 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{5184 \, x^{2}} - \frac{161 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{2592 \, x^{4}} + \frac{173 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{3240 \, x^{6}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{36 \, x^{8}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{15 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x, algorithm="maxima")

[Out]

2093/31104*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 161/2592*sqrt(x^4 + 5*x^2 + 3) + 805
/5184*sqrt(x^4 + 5*x^2 + 3)/x^2 - 161/2592*(x^4 + 5*x^2 + 3)^(3/2)/x^4 + 173/3240*(x^4 + 5*x^2 + 3)^(3/2)/x^6
- 1/36*(x^4 + 5*x^2 + 3)^(3/2)/x^8 - 1/15*(x^4 + 5*x^2 + 3)^(3/2)/x^10

________________________________________________________________________________________

Fricas [A]  time = 1.33106, size = 292, normalized size = 2.21 \begin{align*} \frac{10465 \, \sqrt{3} x^{10} \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 ) - 15846 \, x^{10} - 6 \,{\left (2641 \, x^{8} - 1370 \, x^{6} + 1176 \, x^{4} + 10800 \, x^{2} + 5184\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{155520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x, algorithm="fricas")

[Out]

1/155520*(10465*sqrt(3)*x^10*log((25*x^2 + 2*sqrt(3)*(5*x^2 + 6) + 2*sqrt(x^4 + 5*x^2 + 3)*(5*sqrt(3) + 6) + 3
0)/x^2) - 15846*x^10 - 6*(2641*x^8 - 1370*x^6 + 1176*x^4 + 10800*x^2 + 5184)*sqrt(x^4 + 5*x^2 + 3))/x^10

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{x^{11}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+2)*(x**4+5*x**2+3)**(1/2)/x**11,x)

[Out]

Integral((3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3)/x**11, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )}}{x^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x, algorithm="giac")

[Out]

integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)/x^11, x)