Integrand size = 20, antiderivative size = 320 \[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx=\frac {x}{a}+\frac {\left (a+b+2 \sqrt {a} \sqrt {a+b}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{5/4} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {\left (a+b+2 \sqrt {a} \sqrt {a+b}\right ) \arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{5/4} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\left (2 \sqrt {a}-\sqrt {a+b}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt {a+b}+\sqrt {a} x^2}\right )}{2 \sqrt {2} a^{5/4} \sqrt {-\sqrt {a}+\sqrt {a+b}}} \] Output:
x/a+1/4*(a+b+2*a^(1/2)*(a+b)^(1/2))*arctan(((-a^(1/2)+(a+b)^(1/2))^(1/2)-2 ^(1/2)*a^(1/4)*x)/(a^(1/2)+(a+b)^(1/2))^(1/2))*2^(1/2)/a^(5/4)/(a+b)^(1/2) /(a^(1/2)+(a+b)^(1/2))^(1/2)-1/4*(a+b+2*a^(1/2)*(a+b)^(1/2))*arctan(((-a^( 1/2)+(a+b)^(1/2))^(1/2)+2^(1/2)*a^(1/4)*x)/(a^(1/2)+(a+b)^(1/2))^(1/2))*2^ (1/2)/a^(5/4)/(a+b)^(1/2)/(a^(1/2)+(a+b)^(1/2))^(1/2)+1/4*(2*a^(1/2)-(a+b) ^(1/2))*arctanh(2^(1/2)*a^(1/4)*(-a^(1/2)+(a+b)^(1/2))^(1/2)*x/((a+b)^(1/2 )+a^(1/2)*x^2))*2^(1/2)/a^(5/4)/(-a^(1/2)+(a+b)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.51 \[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx=\frac {x}{a}-\frac {i \left (\sqrt {a}-i \sqrt {b}\right )^2 \arctan \left (\frac {\sqrt {a} x}{\sqrt {a-i \sqrt {a} \sqrt {b}}}\right )}{2 a \sqrt {a-i \sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {i \left (\sqrt {a}+i \sqrt {b}\right )^2 \arctan \left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 a \sqrt {a+i \sqrt {a} \sqrt {b}} \sqrt {b}} \] Input:
Integrate[x^4/(a + b + 2*a*x^2 + a*x^4),x]
Output:
x/a - ((I/2)*(Sqrt[a] - I*Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a] *Sqrt[b]]])/(a*Sqrt[a - I*Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((I/2)*(Sqrt[a] + I* Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + I* Sqrt[a]*Sqrt[b]]*Sqrt[b])
Time = 1.58 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.53, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1442, 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.
| \(\displaystyle \int \frac {x^4}{a x^4+2 a x^2+a+b} \, dx\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {x}{a}-\frac {\int \frac {2 a x^2+a+b}{a x^4+2 a x^2+a+b}dx}{a}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\int \frac {\sqrt {2} (a+b) \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} \left (a-2 \sqrt {a+b} \sqrt {a}+b\right ) x}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} (a+b)+\sqrt [4]{a} \left (a-2 \sqrt {a+b} \sqrt {a}+b\right ) x}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\int \frac {\sqrt {2} (a+b) \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} \left (a-2 \sqrt {a+b} \sqrt {a}+b\right ) x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} (a+b)+\sqrt [4]{a} \left (a-2 \sqrt {a+b} \sqrt {a}+b\right ) x}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\frac {1}{2} \sqrt [4]{a} \left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{a} \left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{a} \left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{a} \left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\frac {\left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\frac {\left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{a}-\frac {\frac {\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {1}{2} \sqrt [4]{a} \left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \left (2 \sqrt {a} \sqrt {a+b}+a+b\right ) \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {1}{2} \sqrt [4]{a} \left (-2 \sqrt {a} \sqrt {a+b}+a+b\right ) \log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}}{a}\) |
Input:
Int[x^4/(a + b + 2*a*x^2 + a*x^4),x]
Output:
x/a - (((a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*(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]])])/Sqrt[Sqrt[a] + Sqrt[a + b]] - (a^(1/4)*(a + b - 2*Sqrt[a]*Sqrt[a + b])*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]*Sqr t[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) + ((a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*(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]])] )/Sqrt[Sqrt[a] + Sqrt[a + b]] + (a^(1/4)*(a + b - 2*Sqrt[a]*Sqrt[a + b])*L og[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]]))/a
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^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && 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 = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.18
| method | result | size |
| risch | \(\frac {x}{a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (-2 \textit {\_R}^{2} a -a -b \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a^{2}}\) | \(57\) |
| default | \(\frac {x}{a}+\frac {\frac {\frac {\left (-\sqrt {a +b}\, a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a +b}\, \sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \ln \left (\sqrt {a}\, x^{2}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (-2 \sqrt {a +b}\, a b -\frac {\left (-\sqrt {a +b}\, a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a +b}\, \sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {2 \sqrt {a}\, x +\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 b a}+\frac {-\frac {\left (-\sqrt {a +b}\, a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a +b}\, \sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \ln \left (-\sqrt {a}\, x^{2}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}-\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (2 \sqrt {a +b}\, a b +\frac {\left (-\sqrt {a +b}\, a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-\sqrt {a +b}\, \sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a +2 \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}\right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {-2 \sqrt {a}\, x +\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 b a}}{a}\) | \(777\) |
Input:
int(x^4/(a*x^4+2*a*x^2+a+b),x,method=_RETURNVERBOSE)
Output:
x/a+1/4/a^2*sum((-2*_R^2*a-a-b)/(_R^3+_R)*ln(x-_R),_R=RootOf(_Z^4*a+2*_Z^2 *a+a+b))
Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (222) = 444\).
Time = 0.09 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.92 \[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx=\frac {a \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b - a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}}\right ) - a \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b - a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} + a - 3 \, b}{a^{2} b}}\right ) - a \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b + a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}}\right ) + a \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b + a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {-\frac {9 \, a^{2} - 6 \, a b + b^{2}}{a^{5} b}} - a + 3 \, b}{a^{2} b}}\right ) + 4 \, x}{4 \, a} \] Input:
integrate(x^4/(a*x^4+2*a*x^2+a+b),x, algorithm="fricas")
Output:
1/4*(a*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b) )*log(-(3*a^2 + 2*a*b - b^2)*x + (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b )) + 3*a^2*b - a*b^2)*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b))) - a*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) + a - 3*b)/(a^2*b))*log(-(3*a^2 + 2*a*b - b^2)*x - (a^4*b*sqrt(-(9*a^2 - 6*a* b + b^2)/(a^5*b)) + 3*a^2*b - a*b^2)*sqrt((a^2*b*sqrt(-(9*a^2 - 6*a*b + b^ 2)/(a^5*b)) + a - 3*b)/(a^2*b))) - a*sqrt(-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b ^2)/(a^5*b)) - a + 3*b)/(a^2*b))*log(-(3*a^2 + 2*a*b - b^2)*x + (a^4*b*sqr t(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - 3*a^2*b + a*b^2)*sqrt(-(a^2*b*sqrt(-(9 *a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))) + a*sqrt(-(a^2*b*sqrt(-( 9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))*log(-(3*a^2 + 2*a*b - b^ 2)*x - (a^4*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - 3*a^2*b + a*b^2)*sqrt (-(a^2*b*sqrt(-(9*a^2 - 6*a*b + b^2)/(a^5*b)) - a + 3*b)/(a^2*b))) + 4*x)/ a
Time = 0.75 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.33 \[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{5} b^{2} + t^{2} \left (- 32 a^{4} b + 96 a^{3} b^{2}\right ) + a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{4} b + 4 t a^{3} - 24 t a^{2} b + 4 t a b^{2}}{3 a^{2} + 2 a b - b^{2}} \right )} \right )\right )} + \frac {x}{a} \] Input:
integrate(x**4/(a*x**4+2*a*x**2+a+b),x)
Output:
RootSum(256*_t**4*a**5*b**2 + _t**2*(-32*a**4*b + 96*a**3*b**2) + a**3 + 3 *a**2*b + 3*a*b**2 + b**3, Lambda(_t, _t*log(x + (-64*_t**3*a**4*b + 4*_t* a**3 - 24*_t*a**2*b + 4*_t*a*b**2)/(3*a**2 + 2*a*b - b**2)))) + x/a
\[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx=\int { \frac {x^{4}}{a x^{4} + 2 \, a x^{2} + a + b} \,d x } \] Input:
integrate(x^4/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")
Output:
x/a - integrate((2*a*x^2 + a + b)/(a*x^4 + 2*a*x^2 + a + b), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (222) = 444\).
Time = 0.19 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.67 \[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{4} + \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{3} b - 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{2} b^{2} + 2 \, {\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a b + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} b^{2}\right )} a^{2} - {\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{3} b + 7 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} + \sqrt {a^{4} - {\left (a^{2} + a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b + 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} - \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{4} + \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{3} b - 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{2} b^{2} + 2 \, {\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a b + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} b^{2}\right )} a^{2} + {\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{3} b + 7 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} - \sqrt {a^{4} - {\left (a^{2} + a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b + 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} + \frac {x}{a} \] Input:
integrate(x^4/(a*x^4+2*a*x^2+a+b),x, algorithm="giac")
Output:
1/2*(3*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^4 + sqrt(a^2 + sqrt(-a*b)*a)* sqrt(-a*b)*a^3*b - 4*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^2*b^2 + 2*(3*sq rt(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^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))*arctan(x/sqrt((a^2 + sqrt(a^4 - (a^2 + a*b)*a^2))/a^2))/(3*a^6*b + 7*a^5*b^2 + 4*a^4*b^3) - 1/2*(3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^4 + sqrt(a^2 - sqrt(-a*b)*a)* sqrt(-a*b)*a^3*b - 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a^2*b^2 + 2*(3*sq rt(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^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))*arctan(x/sqrt((a^2 - sqrt(a^4 - (a^2 + a*b)*a^2))/a^2))/(3*a^6*b + 7*a^5*b^2 + 4*a^4*b^3) + x/a
Time = 0.41 (sec) , antiderivative size = 1147, normalized size of antiderivative = 3.58 \[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx =\text {Too large to display} \] Input:
int(x^4/(a + b + 2*a*x^2 + a*x^4),x)
Output:
x/a + 2*atanh((24*x*(-a^5*b^3)^(1/2)*(1/(16*a*b) - 3/(16*a^2) + (3*(-a^5*b ^3)^(1/2))/(16*a^4*b^2) - (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((6*(-a^5*b^ 3)^(1/2))/a + 4*a*b^2 + 6*a^2*b - 2*b^3 - (2*b^2*(-a^5*b^3)^(1/2))/a^3 + ( 4*b*(-a^5*b^3)^(1/2))/a^2) - (8*x*(-a^5*b^3)^(1/2)*(1/(16*a*b) - 3/(16*a^2 ) + (3*(-a^5*b^3)^(1/2))/(16*a^4*b^2) - (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2) )/((4*(-a^5*b^3)^(1/2))/a + (6*(-a^5*b^3)^(1/2))/b - 2*a*b^2 + 4*a^2*b + 6 *a^3 - (2*b*(-a^5*b^3)^(1/2))/a^2) - (8*a*b^2*x*(1/(16*a*b) - 3/(16*a^2) + (3*(-a^5*b^3)^(1/2))/(16*a^4*b^2) - (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/( 4*a*b + (4*(-a^5*b^3)^(1/2))/a^2 + 6*a^2 - 2*b^2 + (6*(-a^5*b^3)^(1/2))/(a *b) - (2*b*(-a^5*b^3)^(1/2))/a^3) + (24*a^2*b*x*(1/(16*a*b) - 3/(16*a^2) + (3*(-a^5*b^3)^(1/2))/(16*a^4*b^2) - (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/( 4*a*b + (4*(-a^5*b^3)^(1/2))/a^2 + 6*a^2 - 2*b^2 + (6*(-a^5*b^3)^(1/2))/(a *b) - (2*b*(-a^5*b^3)^(1/2))/a^3))*((3*a*(-a^5*b^3)^(1/2) - b*(-a^5*b^3)^( 1/2) + a^4*b - 3*a^3*b^2)/(16*a^5*b^2))^(1/2) + 2*atanh((24*x*(-a^5*b^3)^( 1/2)*(1/(16*a*b) - 3/(16*a^2) - (3*(-a^5*b^3)^(1/2))/(16*a^4*b^2) + (-a^5* b^3)^(1/2)/(16*a^5*b))^(1/2))/((6*(-a^5*b^3)^(1/2))/a - 4*a*b^2 - 6*a^2*b + 2*b^3 - (2*b^2*(-a^5*b^3)^(1/2))/a^3 + (4*b*(-a^5*b^3)^(1/2))/a^2) - (8* x*(-a^5*b^3)^(1/2)*(1/(16*a*b) - 3/(16*a^2) - (3*(-a^5*b^3)^(1/2))/(16*a^4 *b^2) + (-a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((4*(-a^5*b^3)^(1/2))/a + (6*( -a^5*b^3)^(1/2))/b + 2*a*b^2 - 4*a^2*b - 6*a^3 - (2*b*(-a^5*b^3)^(1/2))...
Time = 0.20 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.00 \[ \int \frac {x^4}{a+b+2 a x^2+a x^4} \, dx =\text {Too large to display} \] Input:
int(x^4/(a*x^4+2*a*x^2+a+b),x)
Output:
(2*sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*atan((sqrt(sqrt(a)*sq rt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt( 2)))*a - 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*atan((sqrt(sqrt(a )*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*s qrt(2)))*a + 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*atan((sqrt(sq rt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*b - 2*sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*atan( (sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a + 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)*a tan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sq rt(a + b) + a)*sqrt(2)))*a - 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt( 2)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a )*sqrt(a + b) + a)*sqrt(2)))*b - sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) - a) *sqrt(2)*log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sq rt(a)*x**2)*a + sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*log(sqrt (sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a - sqrt (a)*sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a + sqrt(a)*sqrt(sqrt(a)*sqrt (a + b) - a)*sqrt(2)*log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt (a + b) + sqrt(a)*x**2)*b + sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(...