Integrand size = 20, antiderivative size = 88 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{3/2}} \] Output:
-1/8*(b*x^2+2*a)*(c*x^4+b*x^2+a)^(1/2)/a/x^4+1/16*(-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^(3/2)
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{8 a x^4}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{8 a^{3/2}} \] Input:
Integrate[Sqrt[a + b*x^2 + c*x^4]/x^5,x]
Output:
-1/8*((2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(a*x^4) - ((b^2 - 4*a*c)*ArcT anh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4])/Sqrt[a]])/(8*a^(3/2))
Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1434, 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.
\(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {c x^4+b x^2+a}}{x^6}dx^2\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {1}{2} \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{8 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}}{4 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{8 a^{3/2}}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )\) |
Input:
Int[Sqrt[a + b*x^2 + c*x^4]/x^5,x]
Output:
(-1/4*((2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(a*x^4) + ((b^2 - 4*a*c)*Arc Tanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(8*a^(3/2)))/2
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[(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 = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {\left (b \,x^{2}+2 a \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a \,x^{4}}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}\) | \(81\) |
pseudoelliptic | \(\frac {-4 c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) a \,x^{4}+b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) x^{4}-2 b \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2} \sqrt {a}-4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{\frac {3}{2}}}{16 a^{\frac {3}{2}} x^{4}}\) | \(132\) |
default | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{2}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{8 a^{2}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a}-\frac {c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 \sqrt {a}}\) | \(193\) |
elliptic | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{2}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{8 a^{2}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a}-\frac {c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 \sqrt {a}}\) | \(193\) |
Input:
int((c*x^4+b*x^2+a)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/8*(b*x^2+2*a)*(c*x^4+b*x^2+a)^(1/2)/a/x^4-1/16*(4*a*c-b^2)/a^(3/2)*ln(( 2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)
Time = 0.09 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {a} x^{4} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b x^{2} + 2 \, a^{2}\right )}}{32 \, a^{2} x^{4}}, -\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b x^{2} + 2 \, a^{2}\right )}}{16 \, a^{2} x^{4}}\right ] \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/x^5,x, algorithm="fricas")
Output:
[-1/32*((b^2 - 4*a*c)*sqrt(a)*x^4*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*sqrt(c*x^4 + b*x^2 + a)*(a*b*x^2 + 2*a^2))/(a^2*x^4), -1/16*((b^2 - 4*a*c)*sqrt(-a)* x^4*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*sqrt(c*x^4 + b*x^2 + a)*(a*b*x^2 + 2*a^2))/(a^2*x^4)]
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{5}}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(1/2)/x**5,x)
Output:
Integral(sqrt(a + b*x**2 + c*x**4)/x**5, x)
Exception generated. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/x^5,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. 241 vs. \(2 (74) = 148\).
Time = 0.13 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.74 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=-\frac {{\left (b^{2} - 4 \, a c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a c + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a b \sqrt {c} + {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} c}{8 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{2} a} \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/x^5,x, algorithm="giac")
Output:
-1/8*(b^2 - 4*a*c)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqrt(-a ))/(sqrt(-a)*a) + 1/8*((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) + (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)/(((sqrt(c)*x^2 - sqr t(c*x^4 + b*x^2 + a))^2 - a)^2*a)
Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^5} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(1/2)/x^5,x)
Output:
int((a + b*x^2 + c*x^4)^(1/2)/x^5, x)
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^5} \, dx=\frac {-2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a -\sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,x^{2}+4 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a c \,x^{4}-\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) b^{2} x^{4}}{8 a \,x^{4}} \] Input:
int((c*x^4+b*x^2+a)^(1/2)/x^5,x)
Output:
( - 2*sqrt(a + b*x**2 + c*x**4)*a - sqrt(a + b*x**2 + c*x**4)*b*x**2 + 4*i nt(sqrt(a + b*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*a*c*x**4 - int(sqr t(a + b*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*b**2*x**4)/(8*a*x**4)