Integrand size = 20, antiderivative size = 89 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2}} \] Output:
(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)-1/2*arctanh(1/2*( b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a)^(1/2))/a^(3/2)
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-b^2+2 a c-b c x^2}{a \left (-b^2+4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{a^{3/2}} \] Input:
Integrate[1/(x*(a + b*x^2 + c*x^4)^(3/2)),x]
Output:
(-b^2 + 2*a*c - b*c*x^2)/(a*(-b^2 + 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) + ArcT anh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4])/Sqrt[a]]/a^(3/2)
Time = 0.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1434, 1165, 27, 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.
\(\displaystyle \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (c x^4+b x^2+a\right )^{3/2}}dx^2\) |
\(\Big \downarrow \) 1165 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {2 \int -\frac {b^2-4 a c}{2 x^2 \sqrt {c x^4+b x^2+a}}dx^2}{a \left (b^2-4 a c\right )}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{a}+\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {2 \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}}{a}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{a^{3/2}}\right )\) |
Input:
Int[1/(x*(a + b*x^2 + c*x^4)^(3/2)),x]
Output:
((2*(b^2 - 2*a*c + b*c*x^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - A rcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])]/a^(3/2))/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[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[(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 = 98, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(\frac {1}{2 a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (a c -\frac {b^{2}}{4}\right ) a}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\) | \(98\) |
default | \(\frac {1}{2 a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\) | \(99\) |
elliptic | \(\frac {1}{2 a \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\) | \(99\) |
Input:
int(1/x/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2/a/(c*x^4+b*x^2+a)^(1/2)-1/8*b*(2*c*x^2+b)/(c*x^4+b*x^2+a)^(1/2)/(a*c-1 /4*b^2)/a-1/2/a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (77) = 154\).
Time = 0.11 (sec) , antiderivative size = 389, normalized size of antiderivative = 4.37 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, {\left (a b c x^{2} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {c x^{4} + b x^{2} + a}}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left (a b c x^{2} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {c x^{4} + b x^{2} + a}}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}}\right ] \] Input:
integrate(1/x/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/4*(((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)*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*(a*b*c*x^2 + a*b^2 - 2*a^2*c)*sqrt(c*x^4 + b*x^2 + a))/(a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2), 1/2*(((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)*sqrt(-a)*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*s qrt(-a)/(a*c*x^4 + a*b*x^2 + a^2)) + 2*(a*b*c*x^2 + a*b^2 - 2*a^2*c)*sqrt( c*x^4 + b*x^2 + a))/(a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^ 2*b^3 - 4*a^3*b*c)*x^2)]
\[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x/(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral(1/(x*(a + b*x**2 + c*x**4)**(3/2)), x)
Exception generated. \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x/(c*x^4+b*x^2+a)^(3/2),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
Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\frac {a b c x^{2}}{a^{2} b^{2} - 4 \, a^{3} c} + \frac {a b^{2} - 2 \, a^{2} c}{a^{2} b^{2} - 4 \, a^{3} c}}{\sqrt {c x^{4} + b x^{2} + a}} + \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} \] Input:
integrate(1/x/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
(a*b*c*x^2/(a^2*b^2 - 4*a^3*c) + (a*b^2 - 2*a^2*c)/(a^2*b^2 - 4*a^3*c))/sq rt(c*x^4 + b*x^2 + a) + arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sq rt(-a))/(sqrt(-a)*a)
Timed out. \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(1/(x*(a + b*x^2 + c*x^4)^(3/2)),x)
Output:
int(1/(x*(a + b*x^2 + c*x^4)^(3/2)), x)
Time = 0.20 (sec) , antiderivative size = 887, normalized size of antiderivative = 9.97 \[ \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/x/(c*x^4+b*x^2+a)^(3/2),x)
Output:
( - sqrt(c)*sqrt(a)*sqrt(a + b*x**2 + c*x**4)*sqrt(4*a*c - b**2)*sqrt( - 4 *a*c + b**2)*atan((4*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*a*c - sqrt(c)*sqrt( a + b*x**2 + c*x**4)*b**2 + 4*a*c**2*x**2 - b**2*c*x**2)/(sqrt(c)*sqrt(a)* sqrt(4*a*c - b**2)*sqrt( - 4*a*c + b**2)))*b - 2*sqrt(c)*sqrt(a)*sqrt(a + b*x**2 + c*x**4)*sqrt(4*a*c - b**2)*sqrt( - 4*a*c + b**2)*atan((4*sqrt(c)* sqrt(a + b*x**2 + c*x**4)*a*c - sqrt(c)*sqrt(a + b*x**2 + c*x**4)*b**2 + 4 *a*c**2*x**2 - b**2*c*x**2)/(sqrt(c)*sqrt(a)*sqrt(4*a*c - b**2)*sqrt( - 4* a*c + b**2)))*c*x**2 - 2*sqrt(a)*sqrt(4*a*c - b**2)*sqrt( - 4*a*c + b**2)* atan((4*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*a*c - sqrt(c)*sqrt(a + b*x**2 + c*x**4)*b**2 + 4*a*c**2*x**2 - b**2*c*x**2)/(sqrt(c)*sqrt(a)*sqrt(4*a*c - b**2)*sqrt( - 4*a*c + b**2)))*a*c - 2*sqrt(a)*sqrt(4*a*c - b**2)*sqrt( - 4 *a*c + b**2)*atan((4*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*a*c - sqrt(c)*sqrt( a + b*x**2 + c*x**4)*b**2 + 4*a*c**2*x**2 - b**2*c*x**2)/(sqrt(c)*sqrt(a)* sqrt(4*a*c - b**2)*sqrt( - 4*a*c + b**2)))*b*c*x**2 - 2*sqrt(a)*sqrt(4*a*c - b**2)*sqrt( - 4*a*c + b**2)*atan((4*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*a *c - sqrt(c)*sqrt(a + b*x**2 + c*x**4)*b**2 + 4*a*c**2*x**2 - b**2*c*x**2) /(sqrt(c)*sqrt(a)*sqrt(4*a*c - b**2)*sqrt( - 4*a*c + b**2)))*c**2*x**4 + 4 *sqrt(a + b*x**2 + c*x**4)*a**2*c**2 - 3*sqrt(a + b*x**2 + c*x**4)*a*b**2* c - 4*sqrt(a + b*x**2 + c*x**4)*a*b*c**2*x**2 + 4*sqrt(c)*a**2*c**2*x**2 - sqrt(c)*a*b**3 - 5*sqrt(c)*a*b**2*c*x**2 - 4*sqrt(c)*a*b*c**2*x**4)/(a...