Integrand size = 20, antiderivative size = 195 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}-\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{7/2}} \]
-3/16*(-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)+(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/x^4/(c*x^4+b*x^2+a)^(1/2)-1/4 *(-12*a*c+5*b^2)*(c*x^4+b*x^2+a)^(1/2)/a^2/(-4*a*c+b^2)/x^4+1/8*b*(-52*a*c +15*b^2)*(c*x^4+b*x^2+a)^(1/2)/a^3/(-4*a*c+b^2)/x^2
Time = 0.69 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-8 a^3 c-15 b^3 x^4 \left (b+c x^2\right )+2 a^2 \left (b^2+10 b c x^2-12 c^2 x^4\right )+a b x^2 \left (-5 b^2+62 b c x^2+52 c^2 x^4\right )}{8 a^3 \left (-b^2+4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}+\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{8 a^{7/2}} \]
(-8*a^3*c - 15*b^3*x^4*(b + c*x^2) + 2*a^2*(b^2 + 10*b*c*x^2 - 12*c^2*x^4) + a*b*x^2*(-5*b^2 + 62*b*c*x^2 + 52*c^2*x^4))/(8*a^3*(-b^2 + 4*a*c)*x^4*S qrt[a + b*x^2 + c*x^4]) + (3*(5*b^2 - 4*a*c)*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4])/Sqrt[a]])/(8*a^(7/2))
Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1434, 1165, 27, 1237, 27, 1228, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (c x^4+b x^2+a\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {2 \int -\frac {5 b^2+4 c x^2 b-12 a c}{2 x^6 \sqrt {c x^4+b x^2+a}}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {5 b^2+4 c x^2 b-12 a c}{x^6 \sqrt {c x^4+b x^2+a}}dx^2}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\right )\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {\int \frac {2 c \left (5 b^2-12 a c\right ) x^2+b \left (15 b^2-52 a c\right )}{2 x^4 \sqrt {c x^4+b x^2+a}}dx^2}{2 a}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {\int \frac {2 c \left (5 b^2-12 a c\right ) x^2+b \left (15 b^2-52 a c\right )}{x^4 \sqrt {c x^4+b x^2+a}}dx^2}{4 a}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\right )\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {-\frac {3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{2 a}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{a x^2}}{4 a}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {\frac {3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}}{a}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{a x^2}}{4 a}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {\frac {3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2}}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{a x^2}}{4 a}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\right )\) |
((2*(b^2 - 2*a*c + b*c*x^2))/(a*(b^2 - 4*a*c)*x^4*Sqrt[a + b*x^2 + c*x^4]) + (-1/2*((5*b^2 - 12*a*c)*Sqrt[a + b*x^2 + c*x^4])/(a*x^4) - (-((b*(15*b^ 2 - 52*a*c)*Sqrt[a + b*x^2 + c*x^4])/(a*x^2)) + (3*(b^2 - 4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(2*a^(3 /2)))/(4*a))/(a*(b^2 - 4*a*c)))/2
3.10.88.3.1 Defintions of rubi rules used
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[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[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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[((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])
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 = 0.18 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {5 \left (-\frac {52}{5} c^{2} x^{4}-\frac {62}{5} b c \,x^{2}+b^{2}\right ) x^{2} b \,a^{\frac {3}{2}}}{32}+\frac {3 \left (-c^{2} x^{4}+\frac {5}{6} b c \,x^{2}+\frac {1}{12} b^{2}\right ) a^{\frac {5}{2}}}{4}-\frac {a^{\frac {7}{2}} c}{4}+\frac {3 x^{4} \left (-10 b^{3} \left (c \,x^{2}+b \right ) \sqrt {a}+\sqrt {c \,x^{4}+b \,x^{2}+a}\, \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) \left (16 a^{2} c^{2}-24 a \,b^{2} c +5 b^{4}\right )\right )}{64}}{a^{\frac {7}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{4} \left (a c -\frac {b^{2}}{4}\right )}\) | \(184\) |
| default | \(-\frac {1}{4 a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c \,x^{2}+b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{8 a}-\frac {3 c \left (\frac {1}{a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{a^{\frac {3}{2}}}\right )}{4 a}\) | \(299\) |
| elliptic | \(-\frac {1}{4 a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{a^{\frac {3}{2}}}\right )}{2 a}-\frac {4 c \left (2 c \,x^{2}+b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{8 a}-\frac {3 c \left (\frac {1}{a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{a^{\frac {3}{2}}}\right )}{4 a}\) | \(299\) |
| risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-7 b \,x^{2}+2 a \right )}{8 a^{3} x^{4}}-\frac {-\frac {7 b^{3} \left (2 c \,x^{2}+b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {4 a \,c^{2} \left (b \,x^{2}+2 a \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {7 b^{2} c \left (b \,x^{2}+2 a \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}-\frac {4 a b c \left (2 c \,x^{2}+b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+3 a \left (4 a c -5 b^{2}\right ) \left (\frac {1}{2 a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\right )}{8 a^{3}}\) | \(311\) |
3/4/a^(7/2)*(-5/24*(-52/5*c^2*x^4-62/5*b*c*x^2+b^2)*x^2*b*a^(3/2)+(-c^2*x^ 4+5/6*b*c*x^2+1/12*b^2)*a^(5/2)-1/3*a^(7/2)*c+1/16*x^4*(-10*b^3*(c*x^2+b)* a^(1/2)+(c*x^4+b*x^2+a)^(1/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2 ))/x^2)*(16*a^2*c^2-24*a*b^2*c+5*b^4)))/(c*x^4+b*x^2+a)^(1/2)/x^4/(a*c-1/4 *b^2)
Time = 0.36 (sec) , antiderivative size = 615, normalized size of antiderivative = 3.15 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \sqrt {a} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left ({\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{6} - 2 \, a^{3} b^{2} + 8 \, a^{4} c + {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{4} + 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{8} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{6} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{4}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{6} - 2 \, a^{3} b^{2} + 8 \, a^{4} c + {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{4} + 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{8} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{6} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{4}\right )}}\right ] \]
[-1/32*(3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^8 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^6 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^4)*sqrt(a)* 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*c - 52*a^2*b*c^2)*x^6 - 2*a^3*b^2 + 8*a^4*c + (15*a*b^4 - 62*a^2*b^2*c + 24*a^3*c^2)*x^4 + 5*(a^2*b^3 - 4*a ^3*b*c)*x^2)*sqrt(c*x^4 + b*x^2 + a))/((a^4*b^2*c - 4*a^5*c^2)*x^8 + (a^4* b^3 - 4*a^5*b*c)*x^6 + (a^5*b^2 - 4*a^6*c)*x^4), 1/16*(3*((5*b^4*c - 24*a* b^2*c^2 + 16*a^2*c^3)*x^8 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^6 + (5*a *b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^4)*sqrt(-a)*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* c - 52*a^2*b*c^2)*x^6 - 2*a^3*b^2 + 8*a^4*c + (15*a*b^4 - 62*a^2*b^2*c + 2 4*a^3*c^2)*x^4 + 5*(a^2*b^3 - 4*a^3*b*c)*x^2)*sqrt(c*x^4 + b*x^2 + a))/((a ^4*b^2*c - 4*a^5*c^2)*x^8 + (a^4*b^3 - 4*a^5*b*c)*x^6 + (a^5*b^2 - 4*a^6*c )*x^4)]
\[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\frac {{\left (a^{3} b^{3} c - 3 \, a^{4} b c^{2}\right )} x^{2}}{a^{6} b^{2} - 4 \, a^{7} c} + \frac {a^{3} b^{4} - 4 \, a^{4} b^{2} c + 2 \, a^{5} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}{\sqrt {c x^{4} + b x^{2} + a}} + \frac {3 \, {\left (5 \, b^{2} - 4 \, a c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{3}} - \frac {7 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} b^{2} - 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a c + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a b \sqrt {c} - 9 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a b^{2} - 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} c - 16 \, a^{2} b \sqrt {c}}{8 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \]
((a^3*b^3*c - 3*a^4*b*c^2)*x^2/(a^6*b^2 - 4*a^7*c) + (a^3*b^4 - 4*a^4*b^2* c + 2*a^5*c^2)/(a^6*b^2 - 4*a^7*c))/sqrt(c*x^4 + b*x^2 + a) + 3/8*(5*b^2 - 4*a*c)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqrt(-a))/(sqrt(-a )*a^3) - 1/8*(7*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*b^2 - 4*(sqrt(c) *x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a*c + 8*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^ 2 + a))^2*a*b*sqrt(c) - 9*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a*b^2 - 4*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a^2*c - 16*a^2*b*sqrt(c))/(((sqr t(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2 - a)^2*a^3)
Timed out. \[ \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x^5\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]