Integrand size = 20, antiderivative size = 292 \[ \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx=-\frac {2 A}{5 a x^{5/2}}-\frac {2 B}{3 a x^{3/2}}+\frac {2 A c}{a^2 \sqrt {x}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) c^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) c^{3/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{9/4}} \]
-2/5*A/a/x^(5/2)-2/3*B/a/x^(3/2)+1/2*c^(3/4)*arctan(1-c^(1/4)*2^(1/2)*x^(1 /2)/a^(1/4))*(B*a^(1/2)-A*c^(1/2))/a^(9/4)*2^(1/2)-1/2*c^(3/4)*arctan(1+c^ (1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)-A*c^(1/2))/a^(9/4)*2^(1/2)+1/4*c ^(3/4)*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^(9/4)*2^(1/2)-1/4*c^(3/4)*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^(9/4)*2^(1/2)+2*A*c/a^2/x^(1/2)
Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx=-\frac {2 \left (3 a A+5 a B x-15 A c x^2\right )}{15 a^2 x^{5/2}}+\frac {\left (\sqrt {a} B c^{3/4}-A c^{5/4}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} a^{9/4}}-\frac {\left (\sqrt {a} B c^{3/4}+A c^{5/4}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} a^{9/4}} \]
(-2*(3*a*A + 5*a*B*x - 15*A*c*x^2))/(15*a^2*x^(5/2)) + ((Sqrt[a]*B*c^(3/4) - A*c^(5/4))*ArcTan[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x ])])/(Sqrt[2]*a^(9/4)) - ((Sqrt[a]*B*c^(3/4) + A*c^(5/4))*ArcTanh[(Sqrt[2] *a^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/(Sqrt[2]*a^(9/4))
Time = 0.52 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.99, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {553, 27, 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.
| \(\displaystyle \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {2 \int -\frac {5 (a B-A c x)}{2 x^{5/2} \left (c x^2+a\right )}dx}{5 a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a B-A c x}{x^{5/2} \left (c x^2+a\right )}dx}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {2 \int \frac {3 a c (A+B x)}{2 x^{3/2} \left (c x^2+a\right )}dx}{3 a}-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \int \frac {A+B x}{x^{3/2} \left (c x^2+a\right )}dx-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-c \left (-\frac {2 \int -\frac {a B-A c x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \left (\frac {\int \frac {a B-A c x}{\sqrt {x} \left (c x^2+a\right )}dx}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {-c \left (\frac {2 \int \frac {a B-A c x}{c x^2+a}d\sqrt {x}}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}-\frac {1}{2} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-c \left (\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\right )-\frac {2 B}{3 x^{3/2}}}{a}-\frac {2 A}{5 a x^{5/2}}\) |
(-2*A)/(5*a*x^(5/2)) + ((-2*B)/(3*x^(3/2)) - c*((-2*A)/(a*Sqrt[x]) + (2*(- 1/2*((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)*Sqr t[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4)))) + ((A + (Sqrt[a]*B)/Sqrt[c])*Sq rt[c]*(-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(Sq rt[2]*a^(1/4)*c^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + S qrt[c]*x]/(2*Sqrt[2]*a^(1/4)*c^(1/4))))/2))/a))/a
3.5.18.3.1 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 = 0.08 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(-\frac {2 \left (-15 A c \,x^{2}+5 a B x +3 a A \right )}{15 a^{2} x^{\frac {5}{2}}}+\frac {c \left (-\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a^{2}}\) | \(239\) |
| derivativedivides | \(\frac {2 c \left (-\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a^{2}}-\frac {2 A}{5 a \,x^{\frac {5}{2}}}-\frac {2 B}{3 a \,x^{\frac {3}{2}}}+\frac {2 A c}{a^{2} \sqrt {x}}\) | \(243\) |
| default | \(\frac {2 c \left (-\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a^{2}}-\frac {2 A}{5 a \,x^{\frac {5}{2}}}-\frac {2 B}{3 a \,x^{\frac {3}{2}}}+\frac {2 A c}{a^{2} \sqrt {x}}\) | \(243\) |
-2/15*(-15*A*c*x^2+5*B*a*x+3*A*a)/a^2/x^(5/2)+1/a^2*c*(-1/4*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*arcta n(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1))+1/4*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. 869 vs. \(2 (199) = 398\).
Time = 0.38 (sec) , antiderivative size = 869, normalized size of antiderivative = 2.98 \[ \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx=\frac {15 \, a^{2} x^{3} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} + {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + B^{3} a^{4} c - A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}}\right ) - 15 \, a^{2} x^{3} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} - {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + B^{3} a^{4} c - A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} + 2 \, A B c^{2}}{a^{4}}}\right ) - 15 \, a^{2} x^{3} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} + {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - B^{3} a^{4} c + A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}}\right ) + 15 \, a^{2} x^{3} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}} \log \left (-{\left (B^{4} a^{2} c^{2} - A^{4} c^{4}\right )} \sqrt {x} - {\left (A a^{7} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - B^{3} a^{4} c + A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} \sqrt {-\frac {B^{4} a^{2} c^{3} - 2 \, A^{2} B^{2} a c^{4} + A^{4} c^{5}}{a^{9}}} - 2 \, A B c^{2}}{a^{4}}}\right ) + 4 \, {\left (15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )} \sqrt {x}}{30 \, a^{2} x^{3}} \]
1/30*(15*a^2*x^3*sqrt((a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5) /a^9) + 2*A*B*c^2)/a^4)*log(-(B^4*a^2*c^2 - A^4*c^4)*sqrt(x) + (A*a^7*sqrt (-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + B^3*a^4*c - A^2*B*a^3*c ^2)*sqrt((a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + 2*A*B *c^2)/a^4)) - 15*a^2*x^3*sqrt((a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + 2*A*B*c^2)/a^4)*log(-(B^4*a^2*c^2 - A^4*c^4)*sqrt(x) - (A* a^7*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + B^3*a^4*c - A^2 *B*a^3*c^2)*sqrt((a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + 2*A*B*c^2)/a^4)) - 15*a^2*x^3*sqrt(-(a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2 *a*c^4 + A^4*c^5)/a^9) - 2*A*B*c^2)/a^4)*log(-(B^4*a^2*c^2 - A^4*c^4)*sqrt (x) + (A*a^7*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) - B^3*a^ 4*c + A^2*B*a^3*c^2)*sqrt(-(a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4 *c^5)/a^9) - 2*A*B*c^2)/a^4)) + 15*a^2*x^3*sqrt(-(a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) - 2*A*B*c^2)/a^4)*log(-(B^4*a^2*c^2 - A^4 *c^4)*sqrt(x) - (A*a^7*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9 ) - B^3*a^4*c + A^2*B*a^3*c^2)*sqrt(-(a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a *c^4 + A^4*c^5)/a^9) - 2*A*B*c^2)/a^4)) + 4*(15*A*c*x^2 - 5*B*a*x - 3*A*a) *sqrt(x))/(a^2*x^3)
Time = 66.64 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{c} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{a} & \text {for}\: c = 0 \\- \frac {2 A}{5 a x^{\frac {5}{2}}} + \frac {A c \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 a^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A c \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 a^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {A c \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{a^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A c}{a^{2} \sqrt {x}} - \frac {2 B}{3 a x^{\frac {3}{2}}} + \frac {B c \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 a^{2}} - \frac {B c \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 a^{2}} - \frac {B c \sqrt [4]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{a^{2}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(c, 0) ), ((-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2)))/c, Eq(a, 0)), ((-2*A/(5*x**(5/2 )) - 2*B/(3*x**(3/2)))/a, Eq(c, 0)), (-2*A/(5*a*x**(5/2)) + A*c*log(sqrt(x ) - (-a/c)**(1/4))/(2*a**2*(-a/c)**(1/4)) - A*c*log(sqrt(x) + (-a/c)**(1/4 ))/(2*a**2*(-a/c)**(1/4)) + A*c*atan(sqrt(x)/(-a/c)**(1/4))/(a**2*(-a/c)** (1/4)) + 2*A*c/(a**2*sqrt(x)) - 2*B/(3*a*x**(3/2)) + B*c*(-a/c)**(1/4)*log (sqrt(x) - (-a/c)**(1/4))/(2*a**2) - B*c*(-a/c)**(1/4)*log(sqrt(x) + (-a/c )**(1/4))/(2*a**2) - B*c*(-a/c)**(1/4)*atan(sqrt(x)/(-a/c)**(1/4))/a**2, T rue))
Time = 0.30 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx=-\frac {c {\left (\frac {2 \, \sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{4 \, a^{2}} + \frac {2 \, {\left (15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )}}{15 \, a^{2} x^{\frac {5}{2}}} \]
-1/4*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( 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)*sqr t(c)))/(sqrt(a)*sqrt(sqrt(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^2 + 2/15*(15*A* c*x^2 - 5*B*a*x - 3*A*a)/(a^2*x^(5/2))
Time = 0.30 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx=-\frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} c} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, a^{3} c} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, a^{3} c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, a^{3} c} + \frac {2 \, {\left (15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )}}{15 \, a^{2} x^{\frac {5}{2}}} \]
-1/2*sqrt(2)*((a*c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*arctan(1/2*sqrt(2)*(s qrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^3*c) - 1/2*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^3*c) - 1/4*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^3*c) + 1 /4*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^3*c) + 2/15*(15*A*c*x^2 - 5*B*a*x - 3*A*a)/( a^2*x^(5/2))
Time = 9.99 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.27 \[ \int \frac {A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx=-2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^7\,c^6\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}-\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6+16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}-16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}-\frac {32\,B^2\,a^8\,c^5\,\sqrt {x}\,\sqrt {\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}-\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6+16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}-16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}\right )\,\sqrt {\frac {A^2\,c\,\sqrt {-a^9\,c^3}-B^2\,a\,\sqrt {-a^9\,c^3}+2\,A\,B\,a^5\,c^2}{4\,a^9}}-2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^7\,c^6\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}-\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6-16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}+16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}-\frac {32\,B^2\,a^8\,c^5\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^9\,c^3}}{4\,a^8}-\frac {A^2\,c\,\sqrt {-a^9\,c^3}}{4\,a^9}+\frac {A\,B\,c^2}{2\,a^4}}}{16\,A^3\,a^5\,c^7-16\,A\,B^2\,a^6\,c^6-16\,B^3\,a^2\,c^4\,\sqrt {-a^9\,c^3}+16\,A^2\,B\,a\,c^5\,\sqrt {-a^9\,c^3}}\right )\,\sqrt {\frac {B^2\,a\,\sqrt {-a^9\,c^3}-A^2\,c\,\sqrt {-a^9\,c^3}+2\,A\,B\,a^5\,c^2}{4\,a^9}}-\frac {\frac {2\,A}{5\,a}+\frac {2\,B\,x}{3\,a}-\frac {2\,A\,c\,x^2}{a^2}}{x^{5/2}} \]
- 2*atanh((32*A^2*a^7*c^6*x^(1/2)*((A^2*c*(-a^9*c^3)^(1/2))/(4*a^9) - (B^2 *(-a^9*c^3)^(1/2))/(4*a^8) + (A*B*c^2)/(2*a^4))^(1/2))/(16*A^3*a^5*c^7 - 1 6*A*B^2*a^6*c^6 + 16*B^3*a^2*c^4*(-a^9*c^3)^(1/2) - 16*A^2*B*a*c^5*(-a^9*c ^3)^(1/2)) - (32*B^2*a^8*c^5*x^(1/2)*((A^2*c*(-a^9*c^3)^(1/2))/(4*a^9) - ( B^2*(-a^9*c^3)^(1/2))/(4*a^8) + (A*B*c^2)/(2*a^4))^(1/2))/(16*A^3*a^5*c^7 - 16*A*B^2*a^6*c^6 + 16*B^3*a^2*c^4*(-a^9*c^3)^(1/2) - 16*A^2*B*a*c^5*(-a^ 9*c^3)^(1/2)))*((A^2*c*(-a^9*c^3)^(1/2) - B^2*a*(-a^9*c^3)^(1/2) + 2*A*B*a ^5*c^2)/(4*a^9))^(1/2) - 2*atanh((32*A^2*a^7*c^6*x^(1/2)*((B^2*(-a^9*c^3)^ (1/2))/(4*a^8) - (A^2*c*(-a^9*c^3)^(1/2))/(4*a^9) + (A*B*c^2)/(2*a^4))^(1/ 2))/(16*A^3*a^5*c^7 - 16*A*B^2*a^6*c^6 - 16*B^3*a^2*c^4*(-a^9*c^3)^(1/2) + 16*A^2*B*a*c^5*(-a^9*c^3)^(1/2)) - (32*B^2*a^8*c^5*x^(1/2)*((B^2*(-a^9*c^ 3)^(1/2))/(4*a^8) - (A^2*c*(-a^9*c^3)^(1/2))/(4*a^9) + (A*B*c^2)/(2*a^4))^ (1/2))/(16*A^3*a^5*c^7 - 16*A*B^2*a^6*c^6 - 16*B^3*a^2*c^4*(-a^9*c^3)^(1/2 ) + 16*A^2*B*a*c^5*(-a^9*c^3)^(1/2)))*((B^2*a*(-a^9*c^3)^(1/2) - A^2*c*(-a ^9*c^3)^(1/2) + 2*A*B*a^5*c^2)/(4*a^9))^(1/2) - ((2*A)/(5*a) + (2*B*x)/(3* a) - (2*A*c*x^2)/a^2)/x^(5/2)