Integrand size = 20, antiderivative size = 278 \[ \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx=-\frac {2 A}{3 a x^{3/2}}-\frac {2 B}{a \sqrt {x}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt [4]{c} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{7/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt [4]{c} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{7/4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt [4]{c} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{7/4}} \]
-2/3*A/a/x^(3/2)-1/4*c^(1/4)*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^(7/4)*2^(1/2)+1/4*c^(1/4)*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^(7/4)*2^(1/ 2)+1/2*c^(1/4)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)+A*c^(1 /2))/a^(7/4)*2^(1/2)-1/2*c^(1/4)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4)) *(B*a^(1/2)+A*c^(1/2))/a^(7/4)*2^(1/2)-2*B/a/x^(1/2)
Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.56 \[ \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx=-\frac {2 (A+3 B x)}{3 a x^{3/2}}+\frac {\left (\sqrt {a} B \sqrt [4]{c}+A c^{3/4}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} a^{7/4}}+\frac {\left (\sqrt {a} B \sqrt [4]{c}-A c^{3/4}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} a^{7/4}} \]
(-2*(A + 3*B*x))/(3*a*x^(3/2)) + ((Sqrt[a]*B*c^(1/4) + A*c^(3/4))*ArcTan[( Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(7/4)) + ((Sqrt[a]*B*c^(1/4) - A*c^(3/4))*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[ x])/(Sqrt[a] + Sqrt[c]*x)])/(Sqrt[2]*a^(7/4))
Time = 0.49 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {553, 27, 553, 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.
| \(\displaystyle \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {2 \int -\frac {3 (a B-A c x)}{2 x^{3/2} \left (c x^2+a\right )}dx}{3 a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a B-A c x}{x^{3/2} \left (c x^2+a\right )}dx}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {2 \int \frac {a c (A+B x)}{2 \sqrt {x} \left (c x^2+a\right )}dx}{a}-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \int \frac {A+B x}{\sqrt {x} \left (c x^2+a\right )}dx-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {-2 c \int \frac {A+B x}{c x^2+a}d\sqrt {x}-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}}{2 c}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}}{2 c}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-2 c \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )-\frac {2 B}{\sqrt {x}}}{a}-\frac {2 A}{3 a x^{3/2}}\) |
(-2*A)/(3*a*x^(3/2)) + ((-2*B)/Sqrt[x] - 2*c*(((B + (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]) - ((B - (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])))/a
3.5.17.3.1 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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[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 = 235, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {2 \left (3 B x +A \right )}{3 a \,x^{\frac {3}{2}}}-\frac {c \left (\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a}+\frac {B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a}\) | \(235\) |
| derivativedivides | \(-\frac {2 c \left (\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a}-\frac {2 A}{3 a \,x^{\frac {3}{2}}}-\frac {2 B}{a \sqrt {x}}\) | \(239\) |
| default | \(-\frac {2 c \left (\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a}-\frac {2 A}{3 a \,x^{\frac {3}{2}}}-\frac {2 B}{a \sqrt {x}}\) | \(239\) |
-2/3*(3*B*x+A)/a/x^(3/2)-1/a*c*(1/4*A*(a/c)^(1/4)/a*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))+1/4*B/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. 802 vs. \(2 (189) = 378\).
Time = 0.36 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.88 \[ \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx=-\frac {3 \, a x^{2} \sqrt {-\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt {x} + {\left (B a^{6} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt {-\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{2} \sqrt {-\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt {x} - {\left (B a^{6} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt {-\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{2} \sqrt {\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt {x} + {\left (B a^{6} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt {\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 3 \, a x^{2} \sqrt {\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt {x} - {\left (B a^{6} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt {\frac {a^{3} \sqrt {-\frac {B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 4 \, {\left (3 \, B x + A\right )} \sqrt {x}}{6 \, a x^{2}} \]
-1/6*(3*a*x^2*sqrt(-(a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7 ) + 2*A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) + (B*a^6*sqrt(-(B^4*a ^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - A*B^2*a^3*c + A^3*a^2*c^2)*sqrt(- (a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)) - 3*a*x^2*sqrt(-(a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2 *A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) - (B*a^6*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - A*B^2*a^3*c + A^3*a^2*c^2)*sqrt(-(a^3* sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)) - 3*a* x^2*sqrt((a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c )/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) + (B*a^6*sqrt(-(B^4*a^2*c - 2*A^ 2*B^2*a*c^2 + A^4*c^3)/a^7) + A*B^2*a^3*c - A^3*a^2*c^2)*sqrt((a^3*sqrt(-( B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)) + 3*a*x^2*sqr t((a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)* log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) - (B*a^6*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a *c^2 + A^4*c^3)/a^7) + A*B^2*a^3*c - A^3*a^2*c^2)*sqrt((a^3*sqrt(-(B^4*a^2 *c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)) + 4*(3*B*x + A)*sqrt (x))/(a*x^2)
Time = 19.67 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{c} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{a} & \text {for}\: c = 0 \\- \frac {2 A}{3 a x^{\frac {3}{2}}} + \frac {A c \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 a^{2}} - \frac {A c \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 a^{2}} - \frac {A c \sqrt [4]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{a^{2}} - \frac {B \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 a \sqrt [4]{- \frac {a}{c}}} + \frac {B \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 a \sqrt [4]{- \frac {a}{c}}} - \frac {B \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{a \sqrt [4]{- \frac {a}{c}}} - \frac {2 B}{a \sqrt {x}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2))), Eq(a, 0) & Eq(c, 0) ), ((-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2)))/c, Eq(a, 0)), ((-2*A/(3*x**(3/2 )) - 2*B/sqrt(x))/a, Eq(c, 0)), (-2*A/(3*a*x**(3/2)) + A*c*(-a/c)**(1/4)*l og(sqrt(x) - (-a/c)**(1/4))/(2*a**2) - A*c*(-a/c)**(1/4)*log(sqrt(x) + (-a /c)**(1/4))/(2*a**2) - A*c*(-a/c)**(1/4)*atan(sqrt(x)/(-a/c)**(1/4))/a**2 - B*log(sqrt(x) - (-a/c)**(1/4))/(2*a*(-a/c)**(1/4)) + B*log(sqrt(x) + (-a /c)**(1/4))/(2*a*(-a/c)**(1/4)) - B*atan(sqrt(x)/(-a/c)**(1/4))/(a*(-a/c)* *(1/4)) - 2*B/(a*sqrt(x)), True))
Time = 0.31 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx=-\frac {c {\left (\frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{4 \, a} - \frac {2 \, {\left (3 \, B x + A\right )}}{3 \, a x^{\frac {3}{2}}} \]
-1/4*c*(2*sqrt(2)*(B*sqrt(a) + 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)*(B*sqrt(a) + 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)) - sqrt(2)*(B*sqrt(a) - A*sqrt(c))*l og(sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4) ) + sqrt(2)*(B*sqrt(a) - 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 - 2/3*(3*B*x + A)/(a*x^(3/2))
Time = 0.31 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx=-\frac {2 \, {\left (3 \, B x + A\right )}}{3 \, a x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, a^{2} c^{2}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, a^{2} c^{2}} \]
-2/3*(3*B*x + A)/(a*x^(3/2)) - 1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (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^2*c^2) - 1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (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^2*c^2) - 1/4* sqrt(2)*((a*c^3)^(1/4)*A*c^2 - (a*c^3)^(3/4)*B)*log(sqrt(2)*sqrt(x)*(a/c)^ (1/4) + x + sqrt(a/c))/(a^2*c^2) + 1/4*sqrt(2)*((a*c^3)^(1/4)*A*c^2 - (a*c ^3)^(3/4)*B)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^2)
Time = 10.16 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.18 \[ \int \frac {A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx=-2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^3\,c^5\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^7\,c}}{4\,a^7}-\frac {B^2\,\sqrt {-a^7\,c}}{4\,a^6}-\frac {A\,B\,c}{2\,a^3}}}{16\,B^3\,a^3\,c^4+\frac {16\,A^3\,c^5\,\sqrt {-a^7\,c}}{a^2}-16\,A^2\,B\,a^2\,c^5-\frac {16\,A\,B^2\,c^4\,\sqrt {-a^7\,c}}{a}}-\frac {32\,B^2\,a^4\,c^4\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^7\,c}}{4\,a^7}-\frac {B^2\,\sqrt {-a^7\,c}}{4\,a^6}-\frac {A\,B\,c}{2\,a^3}}}{16\,B^3\,a^3\,c^4+\frac {16\,A^3\,c^5\,\sqrt {-a^7\,c}}{a^2}-16\,A^2\,B\,a^2\,c^5-\frac {16\,A\,B^2\,c^4\,\sqrt {-a^7\,c}}{a}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a^7\,c}-A^2\,c\,\sqrt {-a^7\,c}+2\,A\,B\,a^4\,c}{4\,a^7}}-2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^3\,c^5\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^7\,c}}{4\,a^6}-\frac {A^2\,c\,\sqrt {-a^7\,c}}{4\,a^7}-\frac {A\,B\,c}{2\,a^3}}}{16\,B^3\,a^3\,c^4-\frac {16\,A^3\,c^5\,\sqrt {-a^7\,c}}{a^2}-16\,A^2\,B\,a^2\,c^5+\frac {16\,A\,B^2\,c^4\,\sqrt {-a^7\,c}}{a}}-\frac {32\,B^2\,a^4\,c^4\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^7\,c}}{4\,a^6}-\frac {A^2\,c\,\sqrt {-a^7\,c}}{4\,a^7}-\frac {A\,B\,c}{2\,a^3}}}{16\,B^3\,a^3\,c^4-\frac {16\,A^3\,c^5\,\sqrt {-a^7\,c}}{a^2}-16\,A^2\,B\,a^2\,c^5+\frac {16\,A\,B^2\,c^4\,\sqrt {-a^7\,c}}{a}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a^7\,c}-B^2\,a\,\sqrt {-a^7\,c}+2\,A\,B\,a^4\,c}{4\,a^7}}-\frac {\frac {2\,A}{3\,a}+\frac {2\,B\,x}{a}}{x^{3/2}} \]
- 2*atanh((32*A^2*a^3*c^5*x^(1/2)*((A^2*c*(-a^7*c)^(1/2))/(4*a^7) - (B^2*( -a^7*c)^(1/2))/(4*a^6) - (A*B*c)/(2*a^3))^(1/2))/(16*B^3*a^3*c^4 + (16*A^3 *c^5*(-a^7*c)^(1/2))/a^2 - 16*A^2*B*a^2*c^5 - (16*A*B^2*c^4*(-a^7*c)^(1/2) )/a) - (32*B^2*a^4*c^4*x^(1/2)*((A^2*c*(-a^7*c)^(1/2))/(4*a^7) - (B^2*(-a^ 7*c)^(1/2))/(4*a^6) - (A*B*c)/(2*a^3))^(1/2))/(16*B^3*a^3*c^4 + (16*A^3*c^ 5*(-a^7*c)^(1/2))/a^2 - 16*A^2*B*a^2*c^5 - (16*A*B^2*c^4*(-a^7*c)^(1/2))/a ))*(-(B^2*a*(-a^7*c)^(1/2) - A^2*c*(-a^7*c)^(1/2) + 2*A*B*a^4*c)/(4*a^7))^ (1/2) - 2*atanh((32*A^2*a^3*c^5*x^(1/2)*((B^2*(-a^7*c)^(1/2))/(4*a^6) - (A ^2*c*(-a^7*c)^(1/2))/(4*a^7) - (A*B*c)/(2*a^3))^(1/2))/(16*B^3*a^3*c^4 - ( 16*A^3*c^5*(-a^7*c)^(1/2))/a^2 - 16*A^2*B*a^2*c^5 + (16*A*B^2*c^4*(-a^7*c) ^(1/2))/a) - (32*B^2*a^4*c^4*x^(1/2)*((B^2*(-a^7*c)^(1/2))/(4*a^6) - (A^2* c*(-a^7*c)^(1/2))/(4*a^7) - (A*B*c)/(2*a^3))^(1/2))/(16*B^3*a^3*c^4 - (16* A^3*c^5*(-a^7*c)^(1/2))/a^2 - 16*A^2*B*a^2*c^5 + (16*A*B^2*c^4*(-a^7*c)^(1 /2))/a))*(-(A^2*c*(-a^7*c)^(1/2) - B^2*a*(-a^7*c)^(1/2) + 2*A*B*a^4*c)/(4* a^7))^(1/2) - ((2*A)/(3*a) + (2*B*x)/a)/x^(3/2)