Integrand size = 20, antiderivative size = 433 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=-\frac {1}{(a+b) x}+\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} (a+b)^{3/2} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {\sqrt [4]{a} \left (2 \sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} (a+b)^{3/2} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\sqrt [4]{a} \left (2 \sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x+\sqrt {a} x^2\right )}{4 \sqrt {2} (a+b)^{3/2} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\sqrt [4]{a} \left (2 \sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x+\sqrt {a} x^2\right )}{4 \sqrt {2} (a+b)^{3/2} \sqrt {-\sqrt {a}+\sqrt {a+b}}} \]
-1/(a+b)/x+1/8*a^(1/4)*ln(x^2*a^(1/2)+(a+b)^(1/2)-a^(1/4)*x*2^(1/2)*(-a^(1 /2)+(a+b)^(1/2))^(1/2))*(2*a^(1/2)-(a+b)^(1/2))/(a+b)^(3/2)*2^(1/2)/(-a^(1 /2)+(a+b)^(1/2))^(1/2)-1/8*a^(1/4)*ln(x^2*a^(1/2)+(a+b)^(1/2)+a^(1/4)*x*2^ (1/2)*(-a^(1/2)+(a+b)^(1/2))^(1/2))*(2*a^(1/2)-(a+b)^(1/2))/(a+b)^(3/2)*2^ (1/2)/(-a^(1/2)+(a+b)^(1/2))^(1/2)+1/4*a^(1/4)*arctan((-a^(1/4)*x*2^(1/2)+ (-a^(1/2)+(a+b)^(1/2))^(1/2))/(a^(1/2)+(a+b)^(1/2))^(1/2))*(2*a^(1/2)+(a+b )^(1/2))/(a+b)^(3/2)*2^(1/2)/(a^(1/2)+(a+b)^(1/2))^(1/2)-1/4*a^(1/4)*arcta n((a^(1/4)*x*2^(1/2)+(-a^(1/2)+(a+b)^(1/2))^(1/2))/(a^(1/2)+(a+b)^(1/2))^( 1/2))*(2*a^(1/2)+(a+b)^(1/2))/(a+b)^(3/2)*2^(1/2)/(a^(1/2)+(a+b)^(1/2))^(1 /2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\frac {1}{(-a-b) x}+\frac {\left (i a-\sqrt {a} \sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a-i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a-i \sqrt {a} \sqrt {b}} \sqrt {b} (a+b)}+\frac {\left (-i a-\sqrt {a} \sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a+i \sqrt {a} \sqrt {b}} \sqrt {b} (a+b)} \]
1/((-a - b)*x) + ((I*a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sq rt[a]*Sqrt[b]]])/(2*Sqrt[a - I*Sqrt[a]*Sqrt[b]]*Sqrt[b]*(a + b)) + (((-I)* a - Sqrt[a]*Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(2*S qrt[a + I*Sqrt[a]*Sqrt[b]]*Sqrt[b]*(a + b))
Time = 0.77 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {1443, 25, 27, 1483, 27, 1142, 25, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a x^4+2 a x^2+a+b\right )} \, dx\) |
| \(\Big \downarrow \) 1443 |
| \(\displaystyle \frac {\int -\frac {a \left (x^2+2\right )}{a x^4+2 a x^2+a+b}dx}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {a \left (x^2+2\right )}{a x^4+2 a x^2+a+b}dx}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int \frac {x^2+2}{a x^4+2 a x^2+a+b}dx}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x+2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) x+2 \sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}-\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt {a}}+\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {a}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {a}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (\sqrt {a+b}+2 \sqrt {a}\right ) \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {1}{2} \sqrt [4]{a} \left (2-\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\right )}{a+b}-\frac {1}{x (a+b)}\) |
-(1/((a + b)*x)) - (a*(((Sqrt[-Sqrt[a] + Sqrt[a + b]]*(2*Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*(-((Sqrt[2]*Sqrt[-Sqrt[a] + Sqrt[a + b]])/a^(1/4)) + 2*x))/(Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[a + b]])])/(a^(1/4)*Sqrt[Sqrt[a] + Sqr t[a + b]]) - (a^(1/4)*(2 - Sqrt[a + b]/Sqrt[a])*Log[Sqrt[a + b] - Sqrt[2]* a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2])/2)/(2*Sqrt[2]*Sqrt[ a]*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) + ((Sqrt[-Sqrt[a] + Sqrt[a + b]]*(2*Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*((Sqrt[2]*Sqrt[-Sqrt[a] + Sq rt[a + b]])/a^(1/4) + 2*x))/(Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[a + b]])])/(a^(1/ 4)*Sqrt[Sqrt[a] + Sqrt[a + b]]) + (a^(1/4)*(2 - Sqrt[a + b]/Sqrt[a])*Log[S qrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2] )/2)/(2*Sqrt[2]*Sqrt[a]*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]])))/(a + b )
3.10.14.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim p[1/(a*d^2*(m + 1)) Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(-\frac {1}{\left (a +b \right ) x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b^{2}+3 a^{2} b^{3}+3 b^{4} a +b^{5}\right ) \textit {\_Z}^{4}+\left (-2 a^{2} b +6 b^{2} a \right ) \textit {\_Z}^{2}+a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-b \,a^{4}+2 a^{3} b^{2}+12 a^{2} b^{3}+14 b^{4} a +5 b^{5}\right ) \textit {\_R}^{4}+\left (a^{3}-6 a^{2} b +25 b^{2} a \right ) \textit {\_R}^{2}+4 a \right ) x +\left (-3 a^{3} b -5 a^{2} b^{2}-a \,b^{3}+b^{4}\right ) \textit {\_R}^{3}\right )\right )}{4}\) | \(162\) |
| default | \(\text {Expression too large to display}\) | \(1262\) |
-1/(a+b)/x+1/4*sum(_R*ln(((-a^4*b+2*a^3*b^2+12*a^2*b^3+14*a*b^4+5*b^5)*_R^ 4+(a^3-6*a^2*b+25*a*b^2)*_R^2+4*a)*x+(-3*a^3*b-5*a^2*b^2-a*b^3+b^4)*_R^3), _R=RootOf((a^3*b^2+3*a^2*b^3+3*a*b^4+b^5)*_Z^4+(-2*a^2*b+6*a*b^2)*_Z^2+a))
Leaf count of result is larger than twice the leaf count of optimal. 1582 vs. \(2 (299) = 598\).
Time = 0.30 (sec) , antiderivative size = 1582, normalized size of antiderivative = 3.65 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\text {Too large to display} \]
1/4*((a + b)*x*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqr t(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))*log(- (3*a^2 - a*b)*x + (6*a^2*b - 2*a*b^2 + (a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5) *sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3* b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15* a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))) - (a + b)*x*sqrt((a^2 - 3*a*b + (a^3*b + 3*a^2*b^2 + 3*a* b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))*log(-(3*a^2 - a*b)*x - (6*a^2*b - 2*a*b^2 + (a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^2 + 15*a^4 *b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))*sqrt((a^2 - 3*a*b + (a^3 *b + 3*a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6 *a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))) + (a + b)*x*sqrt((a^2 - 3*a*b - (a^3*b + 3* a^2*b^2 + 3*a*b^3 + b^4)*sqrt(-(9*a^3 - 6*a^2*b + a*b^2)/(a^6*b + 6*a^5*b^ 2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)))/(a^3*b + 3*a^2 *b^2 + 3*a*b^3 + b^4))*log(-(3*a^2 - a*b)*x + (6*a^2*b - 2*a*b^2 - (a^4...
Time = 1.89 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} b^{2} + 768 a^{2} b^{3} + 768 a b^{4} + 256 b^{5}\right ) + t^{2} \left (- 32 a^{2} b + 96 a b^{2}\right ) + a, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{4} b - 128 t^{3} a^{3} b^{2} + 128 t^{3} a b^{4} + 64 t^{3} b^{5} + 4 t a^{3} - 40 t a^{2} b + 20 t a b^{2}}{3 a^{2} - a b} \right )} \right )\right )} - \frac {1}{x \left (a + b\right )} \]
RootSum(_t**4*(256*a**3*b**2 + 768*a**2*b**3 + 768*a*b**4 + 256*b**5) + _t **2*(-32*a**2*b + 96*a*b**2) + a, Lambda(_t, _t*log(x + (-64*_t**3*a**4*b - 128*_t**3*a**3*b**2 + 128*_t**3*a*b**4 + 64*_t**3*b**5 + 4*_t*a**3 - 40* _t*a**2*b + 20*_t*a*b**2)/(3*a**2 - a*b)))) - 1/(x*(a + b))
\[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + 2 \, a x^{2} + a + b\right )} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (299) = 598\).
Time = 0.37 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\frac {{\left ({\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a b + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} b^{2}\right )} {\left (a + b\right )}^{2} {\left | a \right |} - 2 \, {\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{3} b + 7 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} a b^{3}\right )} {\left | a \right |} {\left | -a - b \right |} - {\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{4} + 10 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{3} b + 11 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{2} b^{2} + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a^{2} + 2 \, a b + \sqrt {-4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a^{2} + a b\right )} + 4 \, {\left (a^{2} + a b\right )}^{2}}}{a^{2} + a b}}}\right )}{2 \, {\left (3 \, a^{6} b + 13 \, a^{5} b^{2} + 21 \, a^{4} b^{3} + 15 \, a^{3} b^{4} + 4 \, a^{2} b^{5}\right )} {\left | -a - b \right |}} - \frac {{\left ({\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a b + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} b^{2}\right )} {\left (a + b\right )}^{2} {\left | a \right |} + 2 \, {\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{3} b + 7 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} a b^{3}\right )} {\left | a \right |} {\left | -a - b \right |} - {\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{4} + 10 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{3} b + 11 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{2} b^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a^{2} + 2 \, a b - \sqrt {-4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a^{2} + a b\right )} + 4 \, {\left (a^{2} + a b\right )}^{2}}}{a^{2} + a b}}}\right )}{2 \, {\left (3 \, a^{6} b + 13 \, a^{5} b^{2} + 21 \, a^{4} b^{3} + 15 \, a^{3} b^{4} + 4 \, a^{2} b^{5}\right )} {\left | -a - b \right |}} - \frac {1}{{\left (a + b\right )} x} \]
1/2*((3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a*b + 4*sqrt(a^2 - sqrt(-a*b)* a)*sqrt(-a*b)*b^2)*(a + b)^2*abs(a) - 2*(3*sqrt(a^2 - sqrt(-a*b)*a)*a^3*b + 7*sqrt(a^2 - sqrt(-a*b)*a)*a^2*b^2 + 4*sqrt(a^2 - sqrt(-a*b)*a)*a*b^3)*a bs(a)*abs(-a - b) - (3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^4 + 10*sqrt(a ^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^3*b + 11*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a* b)*a^2*b^2 + 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a*b^3)*abs(a))*arctan(2 *sqrt(1/2)*x/sqrt((2*a^2 + 2*a*b + sqrt(-4*(a^2 + 2*a*b + b^2)*(a^2 + a*b) + 4*(a^2 + a*b)^2))/(a^2 + a*b)))/((3*a^6*b + 13*a^5*b^2 + 21*a^4*b^3 + 1 5*a^3*b^4 + 4*a^2*b^5)*abs(-a - b)) - 1/2*((3*sqrt(a^2 + sqrt(-a*b)*a)*sqr t(-a*b)*a*b + 4*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*b^2)*(a + b)^2*abs(a) + 2*(3*sqrt(a^2 + sqrt(-a*b)*a)*a^3*b + 7*sqrt(a^2 + sqrt(-a*b)*a)*a^2*b^2 + 4*sqrt(a^2 + sqrt(-a*b)*a)*a*b^3)*abs(a)*abs(-a - b) - (3*sqrt(a^2 + sq rt(-a*b)*a)*sqrt(-a*b)*a^4 + 10*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^3*b + 11*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^2*b^2 + 4*sqrt(a^2 + sqrt(-a*b) *a)*sqrt(-a*b)*a*b^3)*abs(a))*arctan(2*sqrt(1/2)*x/sqrt((2*a^2 + 2*a*b - s qrt(-4*(a^2 + 2*a*b + b^2)*(a^2 + a*b) + 4*(a^2 + a*b)^2))/(a^2 + a*b)))/( (3*a^6*b + 13*a^5*b^2 + 21*a^4*b^3 + 15*a^3*b^4 + 4*a^2*b^5)*abs(-a - b)) - 1/((a + b)*x)
Time = 14.02 (sec) , antiderivative size = 2848, normalized size of antiderivative = 6.58 \[ \int \frac {1}{x^2 \left (a+b+2 a x^2+a x^4\right )} \, dx=\text {Too large to display} \]
- 1/(x*(a + b)) - atan((((-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b ^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 3 2*a^4*b^5 + 128*a^5*b^4 + 192*a^6*b^3 + 128*a^7*b^2 + x*(-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b + 64*a^4*b^6 + 320*a^5*b^5 + 640*a^6*b^4 + 640 *a^7*b^3 + 320*a^8*b^2)) - x*(8*a^7*b + 4*a^8 - 4*a^4*b^4 - 8*a^5*b^3))*(- (3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b ^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i - ((-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^ (1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2 )*(32*a^8*b + 32*a^4*b^5 + 128*a^5*b^4 + 192*a^6*b^3 + 128*a^7*b^2 - x*(-( 3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^ 5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9*b + 64*a^4*b^6 + 320*a^5*b^5 + 64 0*a^6*b^4 + 640*a^7*b^3 + 320*a^8*b^2)) + x*(8*a^7*b + 4*a^8 - 4*a^4*b^4 - 8*a^5*b^3))*(-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/( 16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*1i)/(6*a^6*b + 2*a^7 + (( -(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b*(-a*b^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(32*a^8*b + 32*a^4*b^5 + 128*a^5*b^4 + 192*a^6*b^3 + 128*a^7*b^2 + x*(-(3*a*b^2 - a^2*b - 3*a*(-a*b^3)^(1/2) + b* (-a*b^3)^(1/2))/(16*(3*a*b^4 + b^5 + 3*a^2*b^3 + a^3*b^2)))^(1/2)*(64*a^9* b + 64*a^4*b^6 + 320*a^5*b^5 + 640*a^6*b^4 + 640*a^7*b^3 + 320*a^8*b^2)...