Integrand size = 20, antiderivative size = 331 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} c^{5/4}}-\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}}+\frac {\left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} a^{9/4} c^{5/4}} \]
-1/128*ln(a^(1/2)+x*c^(1/2)-a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(3*B*a^(1/2)- 5*A*c^(1/2))/a^(9/4)/c^(5/4)*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))*(3*B*a^(1/2)-5*A*c^(1/2))/a^(9/4)/c^(5/4)*2^(1/2)-1 /64*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(3*B*a^(1/2)+5*A*c^(1/2))/a^ (9/4)/c^(5/4)*2^(1/2)+1/64*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(3*B* a^(1/2)+5*A*c^(1/2))/a^(9/4)/c^(5/4)*2^(1/2)-1/4*(-A*c*x+B*a)*x^(1/2)/a/c/ (c*x^2+a)^2+1/16*(5*A*c*x+B*a)*x^(1/2)/a^2/c/(c*x^2+a)
Time = 0.61 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {x} \left (-3 a^2 B+5 A c^2 x^3+a c x (9 A+B x)\right )}{\left (a+c x^2\right )^2}-\sqrt {2} \left (3 \sqrt {a} B+5 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )+\sqrt {2} \left (3 \sqrt {a} B-5 A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{64 a^{9/4} c^{5/4}} \]
((4*a^(1/4)*c^(1/4)*Sqrt[x]*(-3*a^2*B + 5*A*c^2*x^3 + a*c*x*(9*A + B*x)))/ (a + c*x^2)^2 - Sqrt[2]*(3*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[(Sqrt[a] - Sqrt [c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])] + Sqrt[2]*(3*Sqrt[a]*B - 5*A*Sqr t[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/(6 4*a^(9/4)*c^(5/4))
Time = 0.55 (sec) , antiderivative size = 328, 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 = {550, 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.
| \(\displaystyle \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 550 |
| \(\displaystyle \frac {\int \frac {a B+5 A c x}{2 \sqrt {x} \left (c x^2+a\right )^2}dx}{4 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a B+5 A c x}{\sqrt {x} \left (c x^2+a\right )^2}dx}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}-\frac {\int -\frac {3 a B+5 A c x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{2 a}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 a B+5 A c x}{\sqrt {x} \left (c x^2+a\right )}dx}{4 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\frac {\int \frac {3 a B+5 A c x}{c x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}-\frac {1}{2} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {c} \left (\frac {3 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (5 A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 a}+\frac {\sqrt {x} (a B+5 A c x)}{2 a \left (a+c x^2\right )}}{8 a c}-\frac {\sqrt {x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2}\) |
-1/4*(Sqrt[x]*(a*B - A*c*x))/(a*c*(a + c*x^2)^2) + ((Sqrt[x]*(a*B + 5*A*c* x))/(2*a*(a + c*x^2)) + (((5*A + (3*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 - (( 5*A - (3*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)) + 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)/ (2*a))/(8*a*c)
3.5.29.3.1 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[((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 = 266, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(\frac {\frac {5 A c \,x^{\frac {7}{2}}}{16 a^{2}}+\frac {B \,x^{\frac {5}{2}}}{16 a}+\frac {9 A \,x^{\frac {3}{2}}}{16 a}-\frac {3 B \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {3 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {5 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 a^{2} c}\) | \(266\) |
| default | \(\frac {\frac {5 A c \,x^{\frac {7}{2}}}{16 a^{2}}+\frac {B \,x^{\frac {5}{2}}}{16 a}+\frac {9 A \,x^{\frac {3}{2}}}{16 a}-\frac {3 B \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {3 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {5 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{16 a^{2} c}\) | \(266\) |
2*(5/32*A*c/a^2*x^(7/2)+1/32*B/a*x^(5/2)+9/32*A/a*x^(3/2)-3/32*B*x^(1/2)/c )/(c*x^2+a)^2+1/16/a^2/c*(3/8*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*a rctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)- 1))+5/8*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. 1022 vs. \(2 (235) = 470\).
Time = 0.36 (sec) , antiderivative size = 1022, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=-\frac {{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 27 \, B^{3} a^{4} c - 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}}\right ) - {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 27 \, B^{3} a^{4} c - 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {-\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} + 30 \, A B}{a^{4} c^{2}}}\right ) - {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 27 \, B^{3} a^{4} c + 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}}\right ) + {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}} \log \left (-{\left (81 \, B^{4} a^{2} - 625 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (5 \, A a^{7} c^{4} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 27 \, B^{3} a^{4} c + 75 \, A^{2} B a^{3} c^{2}\right )} \sqrt {\frac {a^{4} c^{2} \sqrt {-\frac {81 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 625 \, A^{4} c^{2}}{a^{9} c^{5}}} - 30 \, A B}{a^{4} c^{2}}}\right ) - 4 \, {\left (5 \, A c^{2} x^{3} + B a c x^{2} + 9 \, A a c x - 3 \, B a^{2}\right )} \sqrt {x}}{64 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]
-1/64*((a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^4*c^2*sqrt(-(81*B^4* a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) + 30*A*B)/(a^4*c^2))*log(- (81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) + (5*A*a^7*c^4*sqrt(-(81*B^4*a^2 - 450* A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) + 27*B^3*a^4*c - 75*A^2*B*a^3*c^2)*s qrt(-(a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5) ) + 30*A*B)/(a^4*c^2))) - (a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^4 *c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) + 30*A* B)/(a^4*c^2))*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (5*A*a^7*c^4*sqrt( -(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) + 27*B^3*a^4*c - 75*A^2*B*a^3*c^2)*sqrt(-(a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625 *A^4*c^2)/(a^9*c^5)) + 30*A*B)/(a^4*c^2))) - (a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/ (a^9*c^5)) - 30*A*B)/(a^4*c^2))*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) + (5*A*a^7*c^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) - 27*B^3*a^4*c + 75*A^2*B*a^3*c^2)*sqrt((a^4*c^2*sqrt(-(81*B^4*a^2 - 450* A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) - 30*A*B)/(a^4*c^2))) + (a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a* c + 625*A^4*c^2)/(a^9*c^5)) - 30*A*B)/(a^4*c^2))*log(-(81*B^4*a^2 - 625*A^ 4*c^2)*sqrt(x) - (5*A*a^7*c^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^ 4*c^2)/(a^9*c^5)) - 27*B^3*a^4*c + 75*A^2*B*a^3*c^2)*sqrt((a^4*c^2*sqrt...
Timed out. \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {5 \, A c^{2} x^{\frac {7}{2}} + B a c x^{\frac {5}{2}} + 9 \, A a c x^{\frac {3}{2}} - 3 \, B a^{2} \sqrt {x}}{16 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B a \sqrt {c} + 5 \, A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, B a \sqrt {c} + 5 \, A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, B a \sqrt {c} - 5 \, A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, B a \sqrt {c} - 5 \, A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{128 \, a^{2} c} \]
1/16*(5*A*c^2*x^(7/2) + B*a*c*x^(5/2) + 9*A*a*c*x^(3/2) - 3*B*a^2*sqrt(x)) /(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c) + 1/128*(2*sqrt(2)*(3*B*a*sqrt(c) + 5*A*sqrt(a)*c)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4) + 2*sqrt(c)*sq rt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2* sqrt(2)*(3*B*a*sqrt(c) + 5*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)) + sqrt(2)*(3*B*a*sqrt(c) - 5*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 )*(3*B*a*sqrt(c) - 5*A*sqrt(a)*c)*log(-sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + s qrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a^2*c)
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{3} c^{3}} + \frac {5 \, A c^{2} x^{\frac {7}{2}} + B a c x^{\frac {5}{2}} + 9 \, A a c x^{\frac {3}{2}} - 3 \, B a^{2} \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a^{2} c} \]
1/64*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c + 5*(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^3) + 1/64*sqrt(2)* (3*(a*c^3)^(1/4)*B*a*c + 5*(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^3) + 1/128*sqrt(2)*(3*(a*c^3)^ (1/4)*B*a*c - 5*(a*c^3)^(3/4)*A)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqr t(a/c))/(a^3*c^3) - 1/128*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c - 5*(a*c^3)^(3/4) *A)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^3*c^3) + 1/16*(5* A*c^2*x^(7/2) + B*a*c*x^(5/2) + 9*A*a*c*x^(3/2) - 3*B*a^2*sqrt(x))/((c*x^2 + a)^2*a^2*c)
Time = 10.04 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {9\,A\,x^{3/2}}{16\,a}+\frac {B\,x^{5/2}}{16\,a}-\frac {3\,B\,\sqrt {x}}{16\,c}+\frac {5\,A\,c\,x^{7/2}}{16\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}-\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a^2}-\frac {125\,A^3\,c}{2048\,a^3}+\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^3}-\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^7\,c^2}\right )}-\frac {25\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}-\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a}-\frac {125\,A^3\,c}{2048\,a^2}+\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^5\,c^3}-\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^2}\right )}\right )\,\sqrt {-\frac {9\,B^2\,a\,\sqrt {-a^9\,c^5}-25\,A^2\,c\,\sqrt {-a^9\,c^5}+30\,A\,B\,a^5\,c^3}{4096\,a^9\,c^5}}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}-\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a^2}-\frac {125\,A^3\,c}{2048\,a^3}-\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^3}+\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^7\,c^2}\right )}-\frac {25\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^9\,c^5}}{4096\,a^8\,c^5}-\frac {25\,A^2\,\sqrt {-a^9\,c^5}}{4096\,a^9\,c^4}-\frac {15\,A\,B}{2048\,a^4\,c^2}}}{32\,\left (\frac {45\,A\,B^2}{2048\,a}-\frac {125\,A^3\,c}{2048\,a^2}-\frac {27\,B^3\,\sqrt {-a^9\,c^5}}{2048\,a^5\,c^3}+\frac {75\,A^2\,B\,\sqrt {-a^9\,c^5}}{2048\,a^6\,c^2}\right )}\right )\,\sqrt {-\frac {25\,A^2\,c\,\sqrt {-a^9\,c^5}-9\,B^2\,a\,\sqrt {-a^9\,c^5}+30\,A\,B\,a^5\,c^3}{4096\,a^9\,c^5}} \]
((9*A*x^(3/2))/(16*a) + (B*x^(5/2))/(16*a) - (3*B*x^(1/2))/(16*c) + (5*A*c *x^(7/2))/(16*a^2))/(a^2 + c^2*x^4 + 2*a*c*x^2) - 2*atanh((9*B^2*c*x^(1/2) *((25*A^2*(-a^9*c^5)^(1/2))/(4096*a^9*c^4) - (15*A*B)/(2048*a^4*c^2) - (9* B^2*(-a^9*c^5)^(1/2))/(4096*a^8*c^5))^(1/2))/(32*((45*A*B^2)/(2048*a^2) - (125*A^3*c)/(2048*a^3) + (27*B^3*(-a^9*c^5)^(1/2))/(2048*a^6*c^3) - (75*A^ 2*B*(-a^9*c^5)^(1/2))/(2048*a^7*c^2))) - (25*A^2*c^2*x^(1/2)*((25*A^2*(-a^ 9*c^5)^(1/2))/(4096*a^9*c^4) - (15*A*B)/(2048*a^4*c^2) - (9*B^2*(-a^9*c^5) ^(1/2))/(4096*a^8*c^5))^(1/2))/(32*((45*A*B^2)/(2048*a) - (125*A^3*c)/(204 8*a^2) + (27*B^3*(-a^9*c^5)^(1/2))/(2048*a^5*c^3) - (75*A^2*B*(-a^9*c^5)^( 1/2))/(2048*a^6*c^2))))*(-(9*B^2*a*(-a^9*c^5)^(1/2) - 25*A^2*c*(-a^9*c^5)^ (1/2) + 30*A*B*a^5*c^3)/(4096*a^9*c^5))^(1/2) - 2*atanh((9*B^2*c*x^(1/2)*( (9*B^2*(-a^9*c^5)^(1/2))/(4096*a^8*c^5) - (25*A^2*(-a^9*c^5)^(1/2))/(4096* a^9*c^4) - (15*A*B)/(2048*a^4*c^2))^(1/2))/(32*((45*A*B^2)/(2048*a^2) - (1 25*A^3*c)/(2048*a^3) - (27*B^3*(-a^9*c^5)^(1/2))/(2048*a^6*c^3) + (75*A^2* B*(-a^9*c^5)^(1/2))/(2048*a^7*c^2))) - (25*A^2*c^2*x^(1/2)*((9*B^2*(-a^9*c ^5)^(1/2))/(4096*a^8*c^5) - (25*A^2*(-a^9*c^5)^(1/2))/(4096*a^9*c^4) - (15 *A*B)/(2048*a^4*c^2))^(1/2))/(32*((45*A*B^2)/(2048*a) - (125*A^3*c)/(2048* a^2) - (27*B^3*(-a^9*c^5)^(1/2))/(2048*a^5*c^3) + (75*A^2*B*(-a^9*c^5)^(1/ 2))/(2048*a^6*c^2))))*(-(25*A^2*c*(-a^9*c^5)^(1/2) - 9*B^2*a*(-a^9*c^5)^(1 /2) + 30*A*B*a^5*c^3)/(4096*a^9*c^5))^(1/2)