Integrand size = 20, antiderivative size = 320 \[ \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx=\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}-\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}-\frac {\left (5 \sqrt {a} B-21 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{11/4} c^{3/4}} \]
1/128*ln(a^(1/2)+x*c^(1/2)-a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(5*B*a^(1/2)-2 1*A*c^(1/2))/a^(11/4)/c^(3/4)*2^(1/2)-1/128*ln(a^(1/2)+x*c^(1/2)+a^(1/4)*c ^(1/4)*2^(1/2)*x^(1/2))*(5*B*a^(1/2)-21*A*c^(1/2))/a^(11/4)/c^(3/4)*2^(1/2 )-1/64*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(5*B*a^(1/2)+21*A*c^(1/2) )/a^(11/4)/c^(3/4)*2^(1/2)+1/64*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))* (5*B*a^(1/2)+21*A*c^(1/2))/a^(11/4)/c^(3/4)*2^(1/2)+1/4*(B*x+A)*x^(1/2)/a/ (c*x^2+a)^2+1/16*(5*B*x+7*A)*x^(1/2)/a^2/(c*x^2+a)
Time = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.58 \[ \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx=\frac {\frac {4 a^{3/4} \sqrt {x} \left (11 a A+9 a B x+7 A c x^2+5 B c x^3\right )}{\left (a+c x^2\right )^2}-\frac {\sqrt {2} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{c^{3/4}}+\frac {\sqrt {2} \left (-5 \sqrt {a} B+21 A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{c^{3/4}}}{64 a^{11/4}} \]
((4*a^(3/4)*Sqrt[x]*(11*a*A + 9*a*B*x + 7*A*c*x^2 + 5*B*c*x^3))/(a + c*x^2 )^2 - (Sqrt[2]*(5*Sqrt[a]*B + 21*A*Sqrt[c])*ArcTan[(Sqrt[a] - Sqrt[c]*x)/( Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])])/c^(3/4) + (Sqrt[2]*(-5*Sqrt[a]*B + 21*A *Sqrt[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)] )/c^(3/4))/(64*a^(11/4))
Time = 0.55 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {551, 27, 551, 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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}-\frac {\int -\frac {7 A+5 B x}{2 \sqrt {x} \left (c x^2+a\right )^2}dx}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {7 A+5 B x}{\sqrt {x} \left (c x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}-\frac {\int -\frac {21 A+5 B x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{2 a}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {21 A+5 B x}{\sqrt {x} \left (c x^2+a\right )}dx}{4 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\frac {\int \frac {21 A+5 B x}{c x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}}{2 c}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}}{2 c}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {21 A \sqrt {c}}{\sqrt {a}}+5 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (5 B-\frac {21 A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{2 a}+\frac {\sqrt {x} (7 A+5 B x)}{2 a \left (a+c x^2\right )}}{8 a}+\frac {\sqrt {x} (A+B x)}{4 a \left (a+c x^2\right )^2}\) |
(Sqrt[x]*(A + B*x))/(4*a*(a + c*x^2)^2) + ((Sqrt[x]*(7*A + 5*B*x))/(2*a*(a + c*x^2)) + (((5*B + (21*A*Sqrt[c])/Sqrt[a])*(-(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*Sqrt[c]) - ((5*B - (2 1*A*Sqrt[c])/Sqrt[a])*(-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)) + Log[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*Sqrt[c]))/(2*a) )/(8*a)
3.5.30.3.1 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 + 1))*(c + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) )), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p , -1] && LtQ[m, 0]
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 = 0.08 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.95
| method | result | size |
| derivativedivides | \(2 A \left (\frac {\sqrt {x}}{8 a \left (c \,x^{2}+a \right )^{2}}+\frac {\frac {7 \sqrt {x}}{32 a \left (c \,x^{2}+a \right )}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+2 B \left (\frac {x^{\frac {3}{2}}}{8 a \left (c \,x^{2}+a \right )^{2}}+\frac {\frac {5 x^{\frac {3}{2}}}{32 a \left (c \,x^{2}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}\right )\) | \(303\) |
| default | \(2 A \left (\frac {\sqrt {x}}{8 a \left (c \,x^{2}+a \right )^{2}}+\frac {\frac {7 \sqrt {x}}{32 a \left (c \,x^{2}+a \right )}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+2 B \left (\frac {x^{\frac {3}{2}}}{8 a \left (c \,x^{2}+a \right )^{2}}+\frac {\frac {5 x^{\frac {3}{2}}}{32 a \left (c \,x^{2}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}\right )\) | \(303\) |
2*A*(1/8*x^(1/2)/a/(c*x^2+a)^2+7/8/a*(1/4*x^(1/2)/a/(c*x^2+a)+3/32/a^2*(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))))+2*B*(1/8*x^(3/2)/a/(c*x^2+a)^ 2+5/8/a*(1/4*x^(3/2)/a/(c*x^2+a)+1/32/a/c/(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. 981 vs. \(2 (224) = 448\).
Time = 0.36 (sec) , antiderivative size = 981, normalized size of antiderivative = 3.07 \[ \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx=\frac {{\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 525 \, A B^{2} a^{4} c + 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}}\right ) - {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 525 \, A B^{2} a^{4} c + 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {-\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 210 \, A B}{a^{5} c}}\right ) - {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 525 \, A B^{2} a^{4} c - 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}}\right ) + {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}} \log \left (-{\left (625 \, B^{4} a^{2} - 194481 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, B a^{9} c^{2} \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} + 525 \, A B^{2} a^{4} c - 9261 \, A^{3} a^{3} c^{2}\right )} \sqrt {\frac {a^{5} c \sqrt {-\frac {625 \, B^{4} a^{2} - 22050 \, A^{2} B^{2} a c + 194481 \, A^{4} c^{2}}{a^{11} c^{3}}} - 210 \, A B}{a^{5} c}}\right ) + 4 \, {\left (5 \, B c x^{3} + 7 \, A c x^{2} + 9 \, B a x + 11 \, A a\right )} \sqrt {x}}{64 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} \]
1/64*((a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)*sqrt(-(a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 210*A*B)/(a^5*c))*log(-( 625*B^4*a^2 - 194481*A^4*c^2)*sqrt(x) + (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) - 525*A*B^2*a^4*c + 9261*A ^3*a^3*c^2)*sqrt(-(a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A ^4*c^2)/(a^11*c^3)) + 210*A*B)/(a^5*c))) - (a^2*c^2*x^4 + 2*a^3*c*x^2 + a^ 4)*sqrt(-(a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/( a^11*c^3)) + 210*A*B)/(a^5*c))*log(-(625*B^4*a^2 - 194481*A^4*c^2)*sqrt(x) - (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/( a^11*c^3)) - 525*A*B^2*a^4*c + 9261*A^3*a^3*c^2)*sqrt(-(a^5*c*sqrt(-(625*B ^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 210*A*B)/(a^5*c ))) - (a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)*sqrt((a^5*c*sqrt(-(625*B^4*a^2 - 2 2050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) - 210*A*B)/(a^5*c))*log(-(6 25*B^4*a^2 - 194481*A^4*c^2)*sqrt(x) + (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 - 2 2050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 525*A*B^2*a^4*c - 9261*A^ 3*a^3*c^2)*sqrt((a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4 *c^2)/(a^11*c^3)) - 210*A*B)/(a^5*c))) + (a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4) *sqrt((a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^1 1*c^3)) - 210*A*B)/(a^5*c))*log(-(625*B^4*a^2 - 194481*A^4*c^2)*sqrt(x) - (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(...
Timed out. \[ \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx=\frac {5 \, B c x^{\frac {7}{2}} + 7 \, A c x^{\frac {5}{2}} + 9 \, B a x^{\frac {3}{2}} + 11 \, A a \sqrt {x}}{16 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, B \sqrt {a} + 21 \, A \sqrt {c}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (5 \, B \sqrt {a} + 21 \, A \sqrt {c}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (5 \, B \sqrt {a} - 21 \, A \sqrt {c}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (5 \, B \sqrt {a} - 21 \, A \sqrt {c}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{128 \, a^{2}} \]
1/16*(5*B*c*x^(7/2) + 7*A*c*x^(5/2) + 9*B*a*x^(3/2) + 11*A*a*sqrt(x))/(a^2 *c^2*x^4 + 2*a^3*c*x^2 + a^4) + 1/128*(2*sqrt(2)*(5*B*sqrt(a) + 21*A*sqrt( 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(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(5*B *sqrt(a) + 21*A*sqrt(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(sqrt(a)*sqrt(c))*sqr t(c)) - sqrt(2)*(5*B*sqrt(a) - 21*A*sqrt(c))*log(sqrt(2)*a^(1/4)*c^(1/4)*s qrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)) + sqrt(2)*(5*B*sqrt(a) - 2 1*A*sqrt(c))*log(-sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/( a^(3/4)*c^(3/4)))/a^2
Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx=\frac {5 \, B c x^{\frac {7}{2}} + 7 \, A c x^{\frac {5}{2}} + 9 \, B a x^{\frac {3}{2}} + 11 \, A a \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a^{2}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} \]
1/16*(5*B*c*x^(7/2) + 7*A*c*x^(5/2) + 9*B*a*x^(3/2) + 11*A*a*sqrt(x))/((c* x^2 + a)^2*a^2) + 1/64*sqrt(2)*(21*(a*c^3)^(1/4)*A*c^2 + 5*(a*c^3)^(3/4)*B )*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^3*c ^3) + 1/64*sqrt(2)*(21*(a*c^3)^(1/4)*A*c^2 + 5*(a*c^3)^(3/4)*B)*arctan(-1/ 2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^3*c^3) + 1/128 *sqrt(2)*(21*(a*c^3)^(1/4)*A*c^2 - 5*(a*c^3)^(3/4)*B)*log(sqrt(2)*sqrt(x)* (a/c)^(1/4) + x + sqrt(a/c))/(a^3*c^3) - 1/128*sqrt(2)*(21*(a*c^3)^(1/4)*A *c^2 - 5*(a*c^3)^(3/4)*B)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c) )/(a^3*c^3)
Time = 10.17 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.15 \[ \int \frac {A+B x}{\sqrt {x} \left (a+c x^2\right )^3} \, dx=2\,\mathrm {atanh}\left (\frac {441\,A^2\,c^3\,\sqrt {x}\,\sqrt {\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}-\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a}+\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^7}-\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}-\frac {2205\,A^2\,B\,c^2}{2048\,a^2}\right )}-\frac {25\,B^2\,c^2\,\sqrt {x}\,\sqrt {\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}-\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a^2}+\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}-\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^9}-\frac {2205\,A^2\,B\,c^2}{2048\,a^3}\right )}\right )\,\sqrt {-\frac {441\,A^2\,c\,\sqrt {-a^{11}\,c^3}-25\,B^2\,a\,\sqrt {-a^{11}\,c^3}+210\,A\,B\,a^6\,c^2}{4096\,a^{11}\,c^3}}+2\,\mathrm {atanh}\left (\frac {441\,A^2\,c^3\,\sqrt {x}\,\sqrt {\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}-\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a}-\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^7}+\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}-\frac {2205\,A^2\,B\,c^2}{2048\,a^2}\right )}-\frac {25\,B^2\,c^2\,\sqrt {x}\,\sqrt {\frac {441\,A^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{11}\,c^2}-\frac {105\,A\,B}{2048\,a^5\,c}-\frac {25\,B^2\,\sqrt {-a^{11}\,c^3}}{4096\,a^{10}\,c^3}}}{32\,\left (\frac {125\,B^3\,c}{2048\,a^2}-\frac {525\,A\,B^2\,\sqrt {-a^{11}\,c^3}}{2048\,a^8}+\frac {9261\,A^3\,c\,\sqrt {-a^{11}\,c^3}}{2048\,a^9}-\frac {2205\,A^2\,B\,c^2}{2048\,a^3}\right )}\right )\,\sqrt {-\frac {25\,B^2\,a\,\sqrt {-a^{11}\,c^3}-441\,A^2\,c\,\sqrt {-a^{11}\,c^3}+210\,A\,B\,a^6\,c^2}{4096\,a^{11}\,c^3}}+\frac {\frac {11\,A\,\sqrt {x}}{16\,a}+\frac {9\,B\,x^{3/2}}{16\,a}+\frac {7\,A\,c\,x^{5/2}}{16\,a^2}+\frac {5\,B\,c\,x^{7/2}}{16\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \]
2*atanh((441*A^2*c^3*x^(1/2)*((25*B^2*(-a^11*c^3)^(1/2))/(4096*a^10*c^3) - (441*A^2*(-a^11*c^3)^(1/2))/(4096*a^11*c^2) - (105*A*B)/(2048*a^5*c))^(1/ 2))/(32*((125*B^3*c)/(2048*a) + (525*A*B^2*(-a^11*c^3)^(1/2))/(2048*a^7) - (9261*A^3*c*(-a^11*c^3)^(1/2))/(2048*a^8) - (2205*A^2*B*c^2)/(2048*a^2))) - (25*B^2*c^2*x^(1/2)*((25*B^2*(-a^11*c^3)^(1/2))/(4096*a^10*c^3) - (441* A^2*(-a^11*c^3)^(1/2))/(4096*a^11*c^2) - (105*A*B)/(2048*a^5*c))^(1/2))/(3 2*((125*B^3*c)/(2048*a^2) + (525*A*B^2*(-a^11*c^3)^(1/2))/(2048*a^8) - (92 61*A^3*c*(-a^11*c^3)^(1/2))/(2048*a^9) - (2205*A^2*B*c^2)/(2048*a^3))))*(- (441*A^2*c*(-a^11*c^3)^(1/2) - 25*B^2*a*(-a^11*c^3)^(1/2) + 210*A*B*a^6*c^ 2)/(4096*a^11*c^3))^(1/2) + 2*atanh((441*A^2*c^3*x^(1/2)*((441*A^2*(-a^11* c^3)^(1/2))/(4096*a^11*c^2) - (105*A*B)/(2048*a^5*c) - (25*B^2*(-a^11*c^3) ^(1/2))/(4096*a^10*c^3))^(1/2))/(32*((125*B^3*c)/(2048*a) - (525*A*B^2*(-a ^11*c^3)^(1/2))/(2048*a^7) + (9261*A^3*c*(-a^11*c^3)^(1/2))/(2048*a^8) - ( 2205*A^2*B*c^2)/(2048*a^2))) - (25*B^2*c^2*x^(1/2)*((441*A^2*(-a^11*c^3)^( 1/2))/(4096*a^11*c^2) - (105*A*B)/(2048*a^5*c) - (25*B^2*(-a^11*c^3)^(1/2) )/(4096*a^10*c^3))^(1/2))/(32*((125*B^3*c)/(2048*a^2) - (525*A*B^2*(-a^11* c^3)^(1/2))/(2048*a^8) + (9261*A^3*c*(-a^11*c^3)^(1/2))/(2048*a^9) - (2205 *A^2*B*c^2)/(2048*a^3))))*(-(25*B^2*a*(-a^11*c^3)^(1/2) - 441*A^2*c*(-a^11 *c^3)^(1/2) + 210*A*B*a^6*c^2)/(4096*a^11*c^3))^(1/2) + ((11*A*x^(1/2))/(1 6*a) + (9*B*x^(3/2))/(16*a) + (7*A*c*x^(5/2))/(16*a^2) + (5*B*c*x^(7/2)...