Integrand size = 20, antiderivative size = 161 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{128 a^3 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{256 a^{7/2}} \] Output:
-1/128*(-4*a*c+5*b^2)*(b*x^2+2*a)*(c*x^4+b*x^2+a)^(1/2)/a^3/x^4-1/8*(c*x^4 +b*x^2+a)^(3/2)/a/x^8+5/48*b*(c*x^4+b*x^2+a)^(3/2)/a^2/x^6+1/256*(-4*a*c+b ^2)*(-4*a*c+5*b^2)*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a)^(1/2))/ a^(7/2)
Time = 0.52 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (-48 a^3-8 a^2 b x^2+10 a b^2 x^4-24 a^2 c x^4-15 b^3 x^6+52 a b c x^6\right )}{384 a^3 x^8}+\frac {\left (-5 b^4+24 a b^2 c-16 a^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{128 a^{7/2}} \] Input:
Integrate[Sqrt[a + b*x^2 + c*x^4]/x^9,x]
Output:
(Sqrt[a + b*x^2 + c*x^4]*(-48*a^3 - 8*a^2*b*x^2 + 10*a*b^2*x^4 - 24*a^2*c* x^4 - 15*b^3*x^6 + 52*a*b*c*x^6))/(384*a^3*x^8) + ((-5*b^4 + 24*a*b^2*c - 16*a^2*c^2)*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4])/Sqrt[a]])/(128 *a^(7/2))
Time = 0.57 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1434, 1167, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {c x^4+b x^2+a}}{x^{10}}dx^2\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (2 c x^2+5 b\right ) \sqrt {c x^4+b x^2+a}}{2 x^8}dx^2}{4 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (2 c x^2+5 b\right ) \sqrt {c x^4+b x^2+a}}{x^8}dx^2}{8 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (5 b^2-4 a c\right ) \int \frac {\sqrt {c x^4+b x^2+a}}{x^6}dx^2}{2 a}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (5 b^2-4 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{8 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (5 b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}}{4 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {\left (5 b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{8 a^{3/2}}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}}{8 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 a x^8}\right )\) |
Input:
Int[Sqrt[a + b*x^2 + c*x^4]/x^9,x]
Output:
(-1/4*(a + b*x^2 + c*x^4)^(3/2)/(a*x^8) - ((-5*b*(a + b*x^2 + c*x^4)^(3/2) )/(3*a*x^6) - ((5*b^2 - 4*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))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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])
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]
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]
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.81
| method | result | size |
| pseudoelliptic | \(\frac {\left (a c -\frac {b^{2}}{4}\right ) \left (a c -\frac {5 b^{2}}{4}\right ) x^{8} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )-\frac {5 \left (-\frac {2 b \left (\frac {26 c \,x^{2}}{5}+b \right ) x^{4} a^{\frac {3}{2}}}{3}+\frac {8 x^{2} \left (3 c \,x^{2}+b \right ) a^{\frac {5}{2}}}{15}+\sqrt {a}\, b^{3} x^{6}+\frac {16 a^{\frac {7}{2}}}{5}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8}}{16 a^{\frac {7}{2}} x^{8}}\) | \(130\) |
| risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-52 a b c \,x^{6}+15 b^{3} x^{6}+24 a^{2} c \,x^{4}-10 b^{2} x^{4} a +8 a^{2} b \,x^{2}+48 a^{3}\right )}{384 x^{8} a^{3}}+\frac {\left (16 a^{2} c^{2}-24 a \,b^{2} c +5 b^{4}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{256 a^{\frac {7}{2}}}\) | \(132\) |
| default | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {5 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{48 a^{2} x^{6}}-\frac {5 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{3} x^{4}}+\frac {5 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{2}}-\frac {5 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{4} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{256 a^{\frac {7}{2}}}-\frac {5 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{128 a^{4}}+\frac {7 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 a^{3}}-\frac {3 b^{2} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}-\frac {c b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{2}}+\frac {c^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{32 a^{3}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{2}}+\frac {c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}\) | \(387\) |
| elliptic | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {5 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{48 a^{2} x^{6}}-\frac {5 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{3} x^{4}}+\frac {5 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{2}}-\frac {5 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{4} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{256 a^{\frac {7}{2}}}-\frac {5 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{128 a^{4}}+\frac {7 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 a^{3}}-\frac {3 b^{2} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}-\frac {c b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{2}}+\frac {c^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{32 a^{3}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{2}}+\frac {c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}\) | \(387\) |
Input:
int((c*x^4+b*x^2+a)^(1/2)/x^9,x,method=_RETURNVERBOSE)
Output:
1/16/a^(7/2)*((a*c-1/4*b^2)*(a*c-5/4*b^2)*x^8*ln((2*a+b*x^2+2*a^(1/2)*(c*x ^4+b*x^2+a)^(1/2))/x^2)-5/8*(-2/3*b*(26/5*c*x^2+b)*x^4*a^(3/2)+8/15*x^2*(3 *c*x^2+b)*a^(5/2)+a^(1/2)*b^3*x^6+16/5*a^(7/2))*(c*x^4+b*x^2+a)^(1/2))/x^8
Time = 0.12 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=\left [\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{8} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left ({\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{6} + 8 \, a^{3} b x^{2} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{4} + 48 \, a^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{1536 \, a^{4} x^{8}}, -\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{8} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{6} + 8 \, a^{3} b x^{2} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{4} + 48 \, a^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{768 \, a^{4} x^{8}}\right ] \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/x^9,x, algorithm="fricas")
Output:
[1/1536*(3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*sqrt(a)*x^8*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*((15*a*b^3 - 52*a^2*b*c)*x^6 + 8*a^3*b*x^2 - 2*(5*a^2*b^2 - 1 2*a^3*c)*x^4 + 48*a^4)*sqrt(c*x^4 + b*x^2 + a))/(a^4*x^8), -1/768*(3*(5*b^ 4 - 24*a*b^2*c + 16*a^2*c^2)*sqrt(-a)*x^8*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*((15*a*b^3 - 52*a ^2*b*c)*x^6 + 8*a^3*b*x^2 - 2*(5*a^2*b^2 - 12*a^3*c)*x^4 + 48*a^4)*sqrt(c* x^4 + b*x^2 + a))/(a^4*x^8)]
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{9}}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(1/2)/x**9,x)
Output:
Integral(sqrt(a + b*x**2 + c*x**4)/x**9, x)
Exception generated. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/x^9,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
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (139) = 278\).
Time = 0.13 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.83 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=-\frac {{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{128 \, \sqrt {-a} a^{3}} + \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} b^{4} - 72 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a b^{2} c + 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{2} c^{2} - 55 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a b^{4} + 264 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{2} b^{2} c + 336 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{3} c^{2} + 1152 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{3} b c^{\frac {3}{2}} + 73 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{2} b^{4} + 648 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{3} b^{2} c + 336 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{4} c^{2} + 384 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{3} b^{3} \sqrt {c} + 256 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{4} b c^{\frac {3}{2}} + 15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{3} b^{4} + 312 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{4} b^{2} c + 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{5} c^{2} + 128 \, a^{5} b c^{\frac {3}{2}}}{384 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{4} a^{3}} \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/x^9,x, algorithm="giac")
Output:
-1/128*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^3) + 1/384*(15*(sqrt(c)*x^2 - sqrt(c* x^4 + b*x^2 + a))^7*b^4 - 72*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a*b ^2*c + 48*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a^2*c^2 - 55*(sqrt(c)* x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a*b^4 + 264*(sqrt(c)*x^2 - sqrt(c*x^4 + b *x^2 + a))^5*a^2*b^2*c + 336*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^3 *c^2 + 1152*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^4*a^3*b*c^(3/2) + 73*( sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^2*b^4 + 648*(sqrt(c)*x^2 - sqrt (c*x^4 + b*x^2 + a))^3*a^3*b^2*c + 336*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^4*c^2 + 384*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*a^3*b^3*sqr t(c) + 256*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*a^4*b*c^(3/2) + 15*(s qrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a^3*b^4 + 312*(sqrt(c)*x^2 - sqrt(c* x^4 + b*x^2 + a))*a^4*b^2*c + 48*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a ^5*c^2 + 128*a^5*b*c^(3/2))/(((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2 - a)^4*a^3)
Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^9} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(1/2)/x^9,x)
Output:
int((a + b*x^2 + c*x^4)^(1/2)/x^9, x)
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx=\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{9}}d x \] Input:
int((c*x^4+b*x^2+a)^(1/2)/x^9,x)
Output:
int((c*x^4+b*x^2+a)^(1/2)/x^9,x)