Integrand size = 20, antiderivative size = 292 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}} \]
-1/16*ln(a^(1/2)+x*c^(1/2)-a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(B*a^(1/2)-A*c ^(1/2))/a^(5/4)/c^(5/4)*2^(1/2)+1/16*ln(a^(1/2)+x*c^(1/2)+a^(1/4)*c^(1/4)* 2^(1/2)*x^(1/2))*(B*a^(1/2)-A*c^(1/2))/a^(5/4)/c^(5/4)*2^(1/2)-1/8*arctan( 1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)+A*c^(1/2))/a^(5/4)/c^(5/4)*2 ^(1/2)+1/8*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)+A*c^(1/2)) /a^(5/4)/c^(5/4)*2^(1/2)-1/2*(-A*c*x+B*a)*x^(1/2)/a/c/(c*x^2+a)
Time = 0.67 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {x} (-a B+A c x)}{a+c x^2}-\sqrt {2} \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )+\sqrt {2} \left (\sqrt {a} B-A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{8 a^{5/4} c^{5/4}} \]
((4*a^(1/4)*c^(1/4)*Sqrt[x]*(-(a*B) + A*c*x))/(a + c*x^2) - Sqrt[2]*(Sqrt[ a]*B + A*Sqrt[c])*ArcTan[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sq rt[x])] + Sqrt[2]*(Sqrt[a]*B - A*Sqrt[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4) *Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/(8*a^(5/4)*c^(5/4))
Time = 0.49 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {550, 27, 554, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 550 |
| \(\displaystyle \frac {\int \frac {a B+A c x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a B+A c x}{\sqrt {x} \left (c x^2+a\right )}dx}{4 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\int \frac {a B+A c x}{c x^2+a}d\sqrt {x}}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}-\frac {1}{2} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 a c}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}\) |
-1/2*(Sqrt[x]*(a*B - A*c*x))/(a*c*(a + c*x^2)) + (((A + (Sqrt[a]*B)/Sqrt[c ])*Sqrt[c]*(-(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/ 4)*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1 /4)*c^(1/4))))/2 - ((A - (Sqrt[a]*B)/Sqrt[c])*Sqrt[c]*(-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(Sqrt[2]*a^(1/4)*c^(1/4)) + L og[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(2*Sqrt[2]*a^(1/ 4)*c^(1/4))))/2)/(2*a*c)
3.5.22.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[(e*x)^m*(a*d - b*c*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x ] - Simp[e/(2*a*b*(p + 1)) Int[(e*x)^(m - 1)*(a*d*m - b*c*(m + 2*p + 3)*x )*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && LtQ[0, m, 1]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Time = 0.07 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {\frac {A \,x^{\frac {3}{2}}}{2 a}-\frac {B \sqrt {x}}{2 c}}{c \,x^{2}+a}+\frac {\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{2 a c}\) | \(247\) |
| default | \(\frac {\frac {A \,x^{\frac {3}{2}}}{2 a}-\frac {B \sqrt {x}}{2 c}}{c \,x^{2}+a}+\frac {\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{2 a c}\) | \(247\) |
2*(1/4*A/a*x^(3/2)-1/4*B*x^(1/2)/c)/(c*x^2+a)+1/2/a/c*(1/8*B*(a/c)^(1/4)*2 ^(1/2)*(ln((x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x-(a/c)^(1/4)*x^(1 /2)*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+2*arctan (2^(1/2)/(a/c)^(1/4)*x^(1/2)-1))+1/8*A/(a/c)^(1/4)*2^(1/2)*(ln((x-(a/c)^(1 /4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2 )))+2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x ^(1/2)-1)))
Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (200) = 400\).
Time = 0.42 (sec) , antiderivative size = 901, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx=-\frac {{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + B^{3} a^{3} c - A^{2} B a^{2} c^{2}\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + B^{3} a^{3} c - A^{2} B a^{2} c^{2}\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - B^{3} a^{3} c + A^{2} B a^{2} c^{2}\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}}\right ) + {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - B^{3} a^{3} c + A^{2} B a^{2} c^{2}\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}}\right ) - 4 \, {\left (A c x - B a\right )} \sqrt {x}}{8 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} \]
-1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + B^3*a^3 *c - A^2*B*a^2*c^2)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^ 2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt(-(a^2*c^2*sq rt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))*log (-(B^4*a^2 - A^4*c^2)*sqrt(x) - (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + B^3*a^3*c - A^2*B*a^2*c^2)*sqrt(-(a^2*c^2*sqrt(-(B ^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))) - (a*c^2 *x^2 + a^2*c)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5 *c^5)) - 2*A*B)/(a^2*c^2))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a^4*c^4*s qrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - B^3*a^3*c + A^2*B*a^ 2*c^2)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))) + (a*c^2*x^2 + a^2*c)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))*log(-(B^4*a^2 - A^ 4*c^2)*sqrt(x) - (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5 *c^5)) - B^3*a^3*c + A^2*B*a^2*c^2)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B ^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))) - 4*(A*c*x - B*a)*sqrt(x ))/(a*c^2*x^2 + a^2*c)
Leaf count of result is larger than twice the leaf count of optimal. 886 vs. \(2 (264) = 528\).
Time = 29.30 (sec) , antiderivative size = 886, normalized size of antiderivative = 3.03 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{a^{2}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{c^{2}} & \text {for}\: a = 0 \\\frac {A a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {4 A c x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{c}}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {A c x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A c x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A c x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {4 B a \sqrt {x} \sqrt [4]{- \frac {a}{c}}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {B a \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {B a \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 B a \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {B c x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {B c x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 B c x^{2} \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c \sqrt [4]{- \frac {a}{c}} + 8 a c^{2} x^{2} \sqrt [4]{- \frac {a}{c}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(c, 0) ), ((2*A*x**(3/2)/3 + 2*B*x**(5/2)/5)/a**2, Eq(c, 0)), ((-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2)))/c**2, Eq(a, 0)), (A*a*log(sqrt(x) - (-a/c)**(1/4))/(8 *a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*(-a/c)**(1/4)) - A*a*log(sqrt(x) + ( -a/c)**(1/4))/(8*a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*(-a/c)**(1/4)) + 2*A *a*atan(sqrt(x)/(-a/c)**(1/4))/(8*a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*(-a /c)**(1/4)) + 4*A*c*x**(3/2)*(-a/c)**(1/4)/(8*a**2*c*(-a/c)**(1/4) + 8*a*c **2*x**2*(-a/c)**(1/4)) + A*c*x**2*log(sqrt(x) - (-a/c)**(1/4))/(8*a**2*c* (-a/c)**(1/4) + 8*a*c**2*x**2*(-a/c)**(1/4)) - A*c*x**2*log(sqrt(x) + (-a/ c)**(1/4))/(8*a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*(-a/c)**(1/4)) + 2*A*c* x**2*atan(sqrt(x)/(-a/c)**(1/4))/(8*a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*( -a/c)**(1/4)) - 4*B*a*sqrt(x)*(-a/c)**(1/4)/(8*a**2*c*(-a/c)**(1/4) + 8*a* c**2*x**2*(-a/c)**(1/4)) - B*a*sqrt(-a/c)*log(sqrt(x) - (-a/c)**(1/4))/(8* a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*(-a/c)**(1/4)) + B*a*sqrt(-a/c)*log(s qrt(x) + (-a/c)**(1/4))/(8*a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*(-a/c)**(1 /4)) + 2*B*a*sqrt(-a/c)*atan(sqrt(x)/(-a/c)**(1/4))/(8*a**2*c*(-a/c)**(1/4 ) + 8*a*c**2*x**2*(-a/c)**(1/4)) - B*c*x**2*sqrt(-a/c)*log(sqrt(x) - (-a/c )**(1/4))/(8*a**2*c*(-a/c)**(1/4) + 8*a*c**2*x**2*(-a/c)**(1/4)) + B*c*x** 2*sqrt(-a/c)*log(sqrt(x) + (-a/c)**(1/4))/(8*a**2*c*(-a/c)**(1/4) + 8*a*c* *2*x**2*(-a/c)**(1/4)) + 2*B*c*x**2*sqrt(-a/c)*atan(sqrt(x)/(-a/c)**(1/...
Time = 0.35 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{16 \, a c} \]
1/2*(A*c*x^(3/2) - B*a*sqrt(x))/(a*c^2*x^2 + a^2*c) + 1/16*(2*sqrt(2)*(B*a *sqrt(c) + A*sqrt(a)*c)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4) + 2*sq rt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt( c)) + 2*sqrt(2)*(B*a*sqrt(c) + A*sqrt(a)*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*a ^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(s qrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(B*a*sqrt(c) - A*sqrt(a)*c)*log(sqrt(2) *a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2 )*(B*a*sqrt(c) - A*sqrt(a)*c)*log(-sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt( c)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a*c)
Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} a c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{2} c^{3}} \]
1/2*(A*c*x^(3/2) - B*a*sqrt(x))/((c*x^2 + a)*a*c) + 1/8*sqrt(2)*((a*c^3)^( 1/4)*B*a*c + (a*c^3)^(3/4)*A)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2* sqrt(x))/(a/c)^(1/4))/(a^2*c^3) + 1/8*sqrt(2)*((a*c^3)^(1/4)*B*a*c + (a*c^ 3)^(3/4)*A)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1 /4))/(a^2*c^3) + 1/16*sqrt(2)*((a*c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*log( sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^3) - 1/16*sqrt(2)*((a* c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^3)
Time = 10.14 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx=2\,\mathrm {atanh}\left (\frac {2\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}-\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}}}{\frac {A\,B^2}{4}-\frac {A^3\,c}{4\,a}-\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^2\,c^3}+\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^2}}-\frac {2\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}-\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}}}{\frac {A\,B^2}{4\,a}-\frac {A^3\,c}{4\,a^2}-\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^3}+\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^4\,c^2}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a^5\,c^5}-B^2\,a\,\sqrt {-a^5\,c^5}+2\,A\,B\,a^3\,c^3}{64\,a^5\,c^5}}+2\,\mathrm {atanh}\left (\frac {2\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}-\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}}}{\frac {A\,B^2}{4}-\frac {A^3\,c}{4\,a}+\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^2\,c^3}-\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^2}}-\frac {2\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}-\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}}}{\frac {A\,B^2}{4\,a}-\frac {A^3\,c}{4\,a^2}+\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^3}-\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^4\,c^2}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a^5\,c^5}-A^2\,c\,\sqrt {-a^5\,c^5}+2\,A\,B\,a^3\,c^3}{64\,a^5\,c^5}}+\frac {\frac {A\,x^{3/2}}{2\,a}-\frac {B\,\sqrt {x}}{2\,c}}{c\,x^2+a} \]
2*atanh((2*A^2*c^2*x^(1/2)*((B^2*(-a^5*c^5)^(1/2))/(64*a^4*c^5) - (A^2*(-a ^5*c^5)^(1/2))/(64*a^5*c^4) - (A*B)/(32*a^2*c^2))^(1/2))/((A*B^2)/4 - (A^3 *c)/(4*a) - (B^3*(-a^5*c^5)^(1/2))/(4*a^2*c^3) + (A^2*B*(-a^5*c^5)^(1/2))/ (4*a^3*c^2)) - (2*B^2*c*x^(1/2)*((B^2*(-a^5*c^5)^(1/2))/(64*a^4*c^5) - (A^ 2*(-a^5*c^5)^(1/2))/(64*a^5*c^4) - (A*B)/(32*a^2*c^2))^(1/2))/((A*B^2)/(4* a) - (A^3*c)/(4*a^2) - (B^3*(-a^5*c^5)^(1/2))/(4*a^3*c^3) + (A^2*B*(-a^5*c ^5)^(1/2))/(4*a^4*c^2)))*(-(A^2*c*(-a^5*c^5)^(1/2) - B^2*a*(-a^5*c^5)^(1/2 ) + 2*A*B*a^3*c^3)/(64*a^5*c^5))^(1/2) + 2*atanh((2*A^2*c^2*x^(1/2)*((A^2* (-a^5*c^5)^(1/2))/(64*a^5*c^4) - (A*B)/(32*a^2*c^2) - (B^2*(-a^5*c^5)^(1/2 ))/(64*a^4*c^5))^(1/2))/((A*B^2)/4 - (A^3*c)/(4*a) + (B^3*(-a^5*c^5)^(1/2) )/(4*a^2*c^3) - (A^2*B*(-a^5*c^5)^(1/2))/(4*a^3*c^2)) - (2*B^2*c*x^(1/2)*( (A^2*(-a^5*c^5)^(1/2))/(64*a^5*c^4) - (A*B)/(32*a^2*c^2) - (B^2*(-a^5*c^5) ^(1/2))/(64*a^4*c^5))^(1/2))/((A*B^2)/(4*a) - (A^3*c)/(4*a^2) + (B^3*(-a^5 *c^5)^(1/2))/(4*a^3*c^3) - (A^2*B*(-a^5*c^5)^(1/2))/(4*a^4*c^2)))*(-(B^2*a *(-a^5*c^5)^(1/2) - A^2*c*(-a^5*c^5)^(1/2) + 2*A*B*a^3*c^3)/(64*a^5*c^5))^ (1/2) + ((A*x^(3/2))/(2*a) - (B*x^(1/2))/(2*c))/(a + c*x^2)