3.2.49 \(\int \frac {(2+3 x^2) \sqrt {3+5 x^2+x^4}}{x^9} \, dx\) [149]

3.2.49.1 Optimal result
3.2.49.2 Mathematica [A] (verified)
3.2.49.3 Rubi [A] (verified)
3.2.49.4 Maple [A] (verified)
3.2.49.5 Fricas [A] (verification not implemented)
3.2.49.6 Sympy [F]
3.2.49.7 Maxima [A] (verification not implemented)
3.2.49.8 Giac [B] (verification not implemented)
3.2.49.9 Mupad [F(-1)]

3.2.49.1 Optimal result

Integrand size = 25, antiderivative size = 111 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^9} \, dx=\frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {871 \text {arctanh}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{3456 \sqrt {3}} \]

output
-1/12*(x^4+5*x^2+3)^(3/2)/x^8-11/216*(x^4+5*x^2+3)^(3/2)/x^6-871/10368*arc 
tanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)+67/1728*(5*x^2+6)* 
(x^4+5*x^2+3)^(1/2)/x^4
 
3.2.49.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.72 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^9} \, dx=\frac {\sqrt {3+5 x^2+x^4} \left (-432-984 x^2-182 x^4+247 x^6\right )}{1728 x^8}+\frac {871 \text {arctanh}\left (\frac {x^2}{\sqrt {3}}-\frac {\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{1728 \sqrt {3}} \]

input
Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^9,x]
 
output
(Sqrt[3 + 5*x^2 + x^4]*(-432 - 984*x^2 - 182*x^4 + 247*x^6))/(1728*x^8) + 
(871*ArcTanh[x^2/Sqrt[3] - Sqrt[3 + 5*x^2 + x^4]/Sqrt[3]])/(1728*Sqrt[3])
 
3.2.49.3 Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1578, 1237, 25, 1228, 1152, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (3 x^2+2\right ) \sqrt {x^4+5 x^2+3}}{x^9} \, dx\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {1}{2} \int \frac {\left (3 x^2+2\right ) \sqrt {x^4+5 x^2+3}}{x^{10}}dx^2\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{2} \left (-\frac {1}{12} \int -\frac {\left (11-2 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^8}dx^2-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{6 x^8}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {1}{12} \int \frac {\left (11-2 x^2\right ) \sqrt {x^4+5 x^2+3}}{x^8}dx^2-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{6 x^8}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (-\frac {67}{6} \int \frac {\sqrt {x^4+5 x^2+3}}{x^6}dx^2-\frac {11 \left (x^4+5 x^2+3\right )^{3/2}}{9 x^6}\right )-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{6 x^8}\right )\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (-\frac {67}{6} \left (-\frac {13}{24} \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2-\frac {\sqrt {x^4+5 x^2+3} \left (5 x^2+6\right )}{12 x^4}\right )-\frac {11 \left (x^4+5 x^2+3\right )^{3/2}}{9 x^6}\right )-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{6 x^8}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (-\frac {67}{6} \left (\frac {13}{12} \int \frac {1}{12-x^4}d\frac {5 x^2+6}{\sqrt {x^4+5 x^2+3}}-\frac {\left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{12 x^4}\right )-\frac {11 \left (x^4+5 x^2+3\right )^{3/2}}{9 x^6}\right )-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{6 x^8}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {1}{12} \left (-\frac {67}{6} \left (\frac {13 \text {arctanh}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{24 \sqrt {3}}-\frac {\left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{12 x^4}\right )-\frac {11 \left (x^4+5 x^2+3\right )^{3/2}}{9 x^6}\right )-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{6 x^8}\right )\)

input
Int[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^9,x]
 
output
(-1/6*(3 + 5*x^2 + x^4)^(3/2)/x^8 + ((-11*(3 + 5*x^2 + x^4)^(3/2))/(9*x^6) 
 - (67*(-1/12*((6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^4 + (13*ArcTanh[(6 + 5 
*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(24*Sqrt[3])))/6)/12)/2
 

3.2.49.3.1 Defintions of rubi rules used

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219
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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1152
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> 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] + Simp[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] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1228
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] + Simp[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] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1578
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[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] && Int 
egerQ[(m - 1)/2]
 
3.2.49.4 Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.64

method result size
pseudoelliptic \(\frac {-871 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}\, x^{8}+6 \sqrt {x^{4}+5 x^{2}+3}\, \left (247 x^{6}-182 x^{4}-984 x^{2}-432\right )}{10368 x^{8}}\) \(71\)
risch \(\frac {247 x^{10}+1053 x^{8}-1153 x^{6}-5898 x^{4}-5112 x^{2}-1296}{1728 x^{8} \sqrt {x^{4}+5 x^{2}+3}}-\frac {871 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{10368}\) \(76\)
trager \(\frac {\left (247 x^{6}-182 x^{4}-984 x^{2}-432\right ) \sqrt {x^{4}+5 x^{2}+3}}{1728 x^{8}}-\frac {871 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{10368}\) \(84\)
elliptic \(\frac {247 \sqrt {x^{4}+5 x^{2}+3}}{1728 x^{2}}-\frac {871 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{10368}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{4 x^{8}}-\frac {41 \sqrt {x^{4}+5 x^{2}+3}}{72 x^{6}}-\frac {91 \sqrt {x^{4}+5 x^{2}+3}}{864 x^{4}}\) \(100\)
default \(-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{12 x^{8}}-\frac {11 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{216 x^{6}}+\frac {67 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{864 x^{4}}-\frac {335 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{5184 x^{2}}+\frac {871 \sqrt {x^{4}+5 x^{2}+3}}{10368}-\frac {871 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{10368}+\frac {335 \left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{10368}\) \(135\)

input
int((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^9,x,method=_RETURNVERBOSE)
 
output
1/10368*(-871*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)*x 
^8+6*(x^4+5*x^2+3)^(1/2)*(247*x^6-182*x^4-984*x^2-432))/x^8
 
3.2.49.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^9} \, dx=\frac {871 \, \sqrt {3} x^{8} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) + 1482 \, x^{8} + 6 \, {\left (247 \, x^{6} - 182 \, x^{4} - 984 \, x^{2} - 432\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{10368 \, x^{8}} \]

input
integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^9,x, algorithm="fricas")
 
output
1/10368*(871*sqrt(3)*x^8*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) + 1482*x^8 + 6*(247*x^6 - 182*x^4 
- 984*x^2 - 432)*sqrt(x^4 + 5*x^2 + 3))/x^8
 
3.2.49.6 Sympy [F]

\[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^9} \, dx=\int \frac {\left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}}{x^{9}}\, dx \]

input
integrate((3*x**2+2)*(x**4+5*x**2+3)**(1/2)/x**9,x)
 
output
Integral((3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3)/x**9, x)
 
3.2.49.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^9} \, dx=-\frac {871}{10368} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {67}{864} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {335 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{1728 \, x^{2}} + \frac {67 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{864 \, x^{4}} - \frac {11 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{216 \, x^{6}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{12 \, x^{8}} \]

input
integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^9,x, algorithm="maxima")
 
output
-871/10368*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) - 
67/864*sqrt(x^4 + 5*x^2 + 3) - 335/1728*sqrt(x^4 + 5*x^2 + 3)/x^2 + 67/864 
*(x^4 + 5*x^2 + 3)^(3/2)/x^4 - 11/216*(x^4 + 5*x^2 + 3)^(3/2)/x^6 - 1/12*( 
x^4 + 5*x^2 + 3)^(3/2)/x^8
 
3.2.49.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (89) = 178\).

Time = 0.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.10 \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^9} \, dx=\frac {871}{10368} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) - \frac {871 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{7} - 5184 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{6} - 57389 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} - 165888 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{4} - 204807 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} - 93312 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 2403 \, x^{2} + 2403 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5184}{1728 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{4}} \]

input
integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^9,x, algorithm="giac")
 
output
871/10368*sqrt(3)*log((x^2 + sqrt(3) - sqrt(x^4 + 5*x^2 + 3))/(x^2 - sqrt( 
3) - sqrt(x^4 + 5*x^2 + 3))) - 1/1728*(871*(x^2 - sqrt(x^4 + 5*x^2 + 3))^7 
 - 5184*(x^2 - sqrt(x^4 + 5*x^2 + 3))^6 - 57389*(x^2 - sqrt(x^4 + 5*x^2 + 
3))^5 - 165888*(x^2 - sqrt(x^4 + 5*x^2 + 3))^4 - 204807*(x^2 - sqrt(x^4 + 
5*x^2 + 3))^3 - 93312*(x^2 - sqrt(x^4 + 5*x^2 + 3))^2 - 2403*x^2 + 2403*sq 
rt(x^4 + 5*x^2 + 3) - 5184)/((x^2 - sqrt(x^4 + 5*x^2 + 3))^2 - 3)^4
 
3.2.49.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^9} \, dx=\int \frac {\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3}}{x^9} \,d x \]

input
int(((3*x^2 + 2)*(5*x^2 + x^4 + 3)^(1/2))/x^9,x)
 
output
int(((3*x^2 + 2)*(5*x^2 + x^4 + 3)^(1/2))/x^9, x)