\(\int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx\) [957]

Optimal result

Integrand size = 20, antiderivative size = 199 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx=\frac {b \left (7 b^2-12 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^4 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 a^3 x^6}-\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{9/2}} \] Output:

1/256*b*(-12*a*c+7*b^2)*(b*x^2+2*a)*(c*x^4+b*x^2+a)^(1/2)/a^4/x^4-1/10*(c* 
x^4+b*x^2+a)^(3/2)/a/x^10+7/80*b*(c*x^4+b*x^2+a)^(3/2)/a^2/x^8-1/480*(-32* 
a*c+35*b^2)*(c*x^4+b*x^2+a)^(3/2)/a^3/x^6-1/512*b*(-12*a*c+7*b^2)*(-4*a*c+ 
b^2)*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a)^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (-384 a^4-48 a^3 b x^2+56 a^2 b^2 x^4-128 a^3 c x^4-70 a b^3 x^6+232 a^2 b c x^6+105 b^4 x^8-460 a b^2 c x^8+256 a^2 c^2 x^8\right )}{3840 a^4 x^{10}}+\frac {\left (7 b^5-40 a b^3 c+48 a^2 b c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{256 a^{9/2}} \] Input:

Integrate[Sqrt[a + b*x^2 + c*x^4]/x^11,x]
 

Output:

(Sqrt[a + b*x^2 + c*x^4]*(-384*a^4 - 48*a^3*b*x^2 + 56*a^2*b^2*x^4 - 128*a 
^3*c*x^4 - 70*a*b^3*x^6 + 232*a^2*b*c*x^6 + 105*b^4*x^8 - 460*a*b^2*c*x^8 
+ 256*a^2*c^2*x^8))/(3840*a^4*x^10) + ((7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2) 
*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4])/Sqrt[a]])/(256*a^(9/2))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1434, 1167, 27, 1237, 27, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx\)

\(\Big \downarrow \) 1434

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {c x^4+b x^2+a}}{x^{12}}dx^2\)

\(\Big \downarrow \) 1167

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (4 c x^2+7 b\right ) \sqrt {c x^4+b x^2+a}}{2 x^{10}}dx^2}{5 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (4 c x^2+7 b\right ) \sqrt {c x^4+b x^2+a}}{x^{10}}dx^2}{10 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\int \frac {\left (35 b^2+14 c x^2 b-32 a c\right ) \sqrt {c x^4+b x^2+a}}{2 x^8}dx^2}{4 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}}{10 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\int \frac {\left (35 b^2+14 c x^2 b-32 a c\right ) \sqrt {c x^4+b x^2+a}}{x^8}dx^2}{8 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}}{10 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \int \frac {\sqrt {c x^4+b x^2+a}}{x^6}dx^2}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}}{10 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{8 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}}{10 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}}{4 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}}{10 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (-\frac {-\frac {-\frac {5 b \left (7 b^2-12 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{8 a^{3/2}}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}}{10 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 a x^{10}}\right )\)

Input:

Int[Sqrt[a + b*x^2 + c*x^4]/x^11,x]
 

Output:

(-1/5*(a + b*x^2 + c*x^4)^(3/2)/(a*x^10) - ((-7*b*(a + b*x^2 + c*x^4)^(3/2 
))/(4*a*x^8) - (-1/3*((35*b^2 - 32*a*c)*(a + b*x^2 + c*x^4)^(3/2))/(a*x^6) 
 - (5*b*(7*b^2 - 12*a*c)*(-1/4*((2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(a* 
x^4) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c* 
x^4])])/(8*a^(3/2))))/(2*a))/(8*a))/(10*a))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 1167

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(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)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

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 1434

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp 
[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (a c -\frac {7 b^{2}}{12}\right ) \left (a c -\frac {b^{2}}{4}\right ) b \,x^{10} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )+\frac {16 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (\left (-\frac {2}{3} c^{2} x^{8}-\frac {29}{48} b c \,x^{6}-\frac {7}{48} b^{2} x^{4}\right ) a^{\frac {5}{2}}+\frac {35 \left (\frac {46 c \,x^{2}}{7}+b \right ) b^{2} x^{6} a^{\frac {3}{2}}}{192}+\frac {\left (\frac {8 c \,x^{2}}{3}+b \right ) x^{2} a^{\frac {7}{2}}}{8}-\frac {35 \sqrt {a}\, b^{4} x^{8}}{128}+a^{\frac {9}{2}}\right )}{15}\right )}{32 a^{\frac {9}{2}} x^{10}}\)	\(160\)
risch	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-256 a^{2} c^{2} x^{8}+460 a \,b^{2} c \,x^{8}-105 b^{4} x^{8}-232 a^{2} b c \,x^{6}+70 a \,b^{3} x^{6}+128 a^{3} c \,x^{4}-56 a^{2} b^{2} x^{4}+48 a^{3} b \,x^{2}+384 a^{4}\right )}{3840 x^{10} a^{4}}-\frac {b \left (48 a^{2} c^{2}-40 a \,b^{2} c +7 b^{4}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {9}{2}}}\)	\(167\)
default	\(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{10 a \,x^{10}}+\frac {7 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{80 a^{2} x^{8}}-\frac {7 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{96 a^{3} x^{6}}+\frac {7 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{4}}-\frac {7 b^{4} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{256 a^{5} x^{2}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 a^{5}}-\frac {7 b^{5} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {9}{2}}}+\frac {7 b^{4} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{256 a^{5}}-\frac {13 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{3} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{64 a^{\frac {7}{2}}}-\frac {3 b c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{4}}+\frac {3 b^{2} c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{4} x^{2}}-\frac {3 b^{2} c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{64 a^{4}}+\frac {3 b \,c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a^{3}}-\frac {3 b \,c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{6}}\)	\(442\)
elliptic	\(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{10 a \,x^{10}}+\frac {7 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{80 a^{2} x^{8}}-\frac {7 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{96 a^{3} x^{6}}+\frac {7 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{4}}-\frac {7 b^{4} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{256 a^{5} x^{2}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 a^{5}}-\frac {7 b^{5} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {9}{2}}}+\frac {7 b^{4} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{256 a^{5}}-\frac {13 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{3} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{64 a^{\frac {7}{2}}}-\frac {3 b c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{4}}+\frac {3 b^{2} c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{4} x^{2}}-\frac {3 b^{2} c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{64 a^{4}}+\frac {3 b \,c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a^{3}}-\frac {3 b \,c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{6}}\)	\(442\)

Input:

int((c*x^4+b*x^2+a)^(1/2)/x^11,x,method=_RETURNVERBOSE)
 

Output:

-3/32*((a*c-7/12*b^2)*(a*c-1/4*b^2)*b*x^10*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+ 
b*x^2+a)^(1/2))/x^2)+16/15*(c*x^4+b*x^2+a)^(1/2)*((-2/3*c^2*x^8-29/48*b*c* 
x^6-7/48*b^2*x^4)*a^(5/2)+35/192*(46/7*c*x^2+b)*b^2*x^6*a^(3/2)+1/8*(8/3*c 
*x^2+b)*x^2*a^(7/2)-35/128*a^(1/2)*b^4*x^8+a^(9/2)))/a^(9/2)/x^10
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx=\left [\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {a} x^{10} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, {\left ({\left (105 \, a b^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{8} - 48 \, a^{4} b x^{2} - 2 \, {\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{6} - 384 \, a^{5} + 8 \, {\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{15360 \, a^{5} x^{10}}, \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (105 \, a b^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{8} - 48 \, a^{4} b x^{2} - 2 \, {\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{6} - 384 \, a^{5} + 8 \, {\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{7680 \, a^{5} x^{10}}\right ] \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/x^11,x, algorithm="fricas")
 

Output:

[1/15360*(15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*sqrt(a)*x^10*log(-((b^2 + 
 4*a*c)*x^4 + 8*a*b*x^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) 
+ 8*a^2)/x^4) + 4*((105*a*b^4 - 460*a^2*b^2*c + 256*a^3*c^2)*x^8 - 48*a^4* 
b*x^2 - 2*(35*a^2*b^3 - 116*a^3*b*c)*x^6 - 384*a^5 + 8*(7*a^3*b^2 - 16*a^4 
*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^10), 1/7680*(15*(7*b^5 - 40*a*b^3 
*c + 48*a^2*b*c^2)*sqrt(-a)*x^10*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 
 + 2*a)*sqrt(-a)/(a*c*x^4 + a*b*x^2 + a^2)) + 2*((105*a*b^4 - 460*a^2*b^2* 
c + 256*a^3*c^2)*x^8 - 48*a^4*b*x^2 - 2*(35*a^2*b^3 - 116*a^3*b*c)*x^6 - 3 
84*a^5 + 8*(7*a^3*b^2 - 16*a^4*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^10) 
]
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{11}}\, dx \] Input:

integrate((c*x**4+b*x**2+a)**(1/2)/x**11,x)
 

Output:

Integral(sqrt(a + b*x**2 + c*x**4)/x**11, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/x^11,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (173) = 346\).

Time = 0.16 (sec) , antiderivative size = 842, normalized size of antiderivative = 4.23 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/x^11,x, algorithm="giac")
 

Output:

1/256*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^ 
4 + b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^4) - 1/3840*(105*(sqrt(c)*x^2 - sqrt 
(c*x^4 + b*x^2 + a))^9*b^5 - 600*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9 
*a*b^3*c + 720*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a^2*b*c^2 - 490*( 
sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a*b^5 + 2800*(sqrt(c)*x^2 - sqrt( 
c*x^4 + b*x^2 + a))^7*a^2*b^3*c - 3360*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + 
 a))^7*a^3*b*c^2 - 7680*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^6*a^4*c^(5 
/2) + 896*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^2*b^5 - 5120*(sqrt(c 
)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^3*b^3*c - 15360*(sqrt(c)*x^2 - sqrt(c 
*x^4 + b*x^2 + a))^5*a^4*b*c^2 - 24320*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + 
 a))^4*a^4*b^2*c^(3/2) - 2560*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^4*a^ 
5*c^(5/2) - 790*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^3*b^5 - 9200*( 
sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^4*b^3*c - 12000*(sqrt(c)*x^2 - 
sqrt(c*x^4 + b*x^2 + a))^3*a^5*b*c^2 - 3840*(sqrt(c)*x^2 - sqrt(c*x^4 + b* 
x^2 + a))^2*a^4*b^4*sqrt(c) - 5120*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a)) 
^2*a^5*b^2*c^(3/2) - 2560*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*a^6*c^ 
(5/2) - 105*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a^4*b^5 - 3240*(sqrt(c 
)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a^5*b^3*c - 720*(sqrt(c)*x^2 - sqrt(c*x^4 
 + b*x^2 + a))*a^6*b*c^2 - 1280*a^6*b^2*c^(3/2) + 512*a^7*c^(5/2))/(((sqrt 
(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2 - a)^5*a^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^{11}} \,d x \] Input:

int((a + b*x^2 + c*x^4)^(1/2)/x^11,x)
 

Output:

int((a + b*x^2 + c*x^4)^(1/2)/x^11, x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx=\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{11}}d x \] Input:

int((c*x^4+b*x^2+a)^(1/2)/x^11,x)
 

Output:

int((c*x^4+b*x^2+a)^(1/2)/x^11,x)