Integrand size = 20, antiderivative size = 278 \[ \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx=\frac {2 A \sqrt {x}}{c}+\frac {2 B x^{3/2}}{3 c}+\frac {\sqrt [4]{a} \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} c^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} c^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}} \]
2/3*B*x^(3/2)/c-1/4*a^(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))/c^(7/4)*2^(1/2)+1/4*a^(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))/c^(7/4)*2^(1/2 )+1/2*a^(1/4)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)+A*c^(1/ 2))/c^(7/4)*2^(1/2)-1/2*a^(1/4)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))* (B*a^(1/2)+A*c^(1/2))/c^(7/4)*2^(1/2)+2*A*x^(1/2)/c
Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.56 \[ \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx=\frac {4 c^{3/4} \sqrt {x} (3 A+B x)+3 \sqrt {2} \sqrt [4]{a} \left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )+3 \sqrt {2} \sqrt [4]{a} \left (\sqrt {a} B-A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{6 c^{7/4}} \]
(4*c^(3/4)*Sqrt[x]*(3*A + B*x) + 3*Sqrt[2]*a^(1/4)*(Sqrt[a]*B + A*Sqrt[c]) *ArcTan[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])] + 3*Sqrt[ 2]*a^(1/4)*(Sqrt[a]*B - A*Sqrt[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x ])/(Sqrt[a] + Sqrt[c]*x)])/(6*c^(7/4))
Time = 0.50 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {552, 27, 552, 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.
| \(\displaystyle \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 \int \frac {3 \sqrt {x} (a B-A c x)}{2 \left (c x^2+a\right )}dx}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {\int \frac {\sqrt {x} (a B-A c x)}{c x^2+a}dx}{c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {-\frac {2 \int -\frac {a c (A+B x)}{2 \sqrt {x} \left (c x^2+a\right )}dx}{c}-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {a \int \frac {A+B x}{\sqrt {x} \left (c x^2+a\right )}dx-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \int \frac {A+B x}{c x^2+a}d\sqrt {x}-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}}{2 c}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}}{2 c}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 B x^{3/2}}{3 c}-\frac {2 a \left (\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\right )-2 A \sqrt {x}}{c}\) |
(2*B*x^(3/2))/(3*c) - (-2*A*Sqrt[x] + 2*a*(((B + (A*Sqrt[c])/Sqrt[a])*(-(A rcTan[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])))/c
3.5.13.3.1 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[d*(e*x)^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[e /(b*(m + 2*p + 2)) Int[(e*x)^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[m, 0] && NeQ[ m + 2*p + 2, 0] && (IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 236, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {2 \left (B x +3 A \right ) \sqrt {x}}{3 c}-\frac {a \left (\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a}+\frac {B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{c}\) | \(236\) |
| derivativedivides | \(\frac {\frac {2 B \,x^{\frac {3}{2}}}{3}+2 A \sqrt {x}}{c}-\frac {2 a \left (\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{c}\) | \(238\) |
| default | \(\frac {\frac {2 B \,x^{\frac {3}{2}}}{3}+2 A \sqrt {x}}{c}-\frac {2 a \left (\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{c}\) | \(238\) |
2/3*(B*x+3*A)*x^(1/2)/c-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. 772 vs. \(2 (189) = 378\).
Time = 0.37 (sec) , antiderivative size = 772, normalized size of antiderivative = 2.78 \[ \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx=-\frac {3 \, c \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - A B^{2} a c^{2} + A^{3} c^{3}\right )} \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}}\right ) - 3 \, c \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - A B^{2} a c^{2} + A^{3} c^{3}\right )} \sqrt {-\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}}\right ) - 3 \, c \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + A B^{2} a c^{2} - A^{3} c^{3}\right )} \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}}\right ) + 3 \, c \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (B c^{5} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + A B^{2} a c^{2} - A^{3} c^{3}\right )} \sqrt {\frac {c^{3} \sqrt {-\frac {B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}}\right ) - 4 \, {\left (B x + 3 \, A\right )} \sqrt {x}}{6 \, c} \]
-1/6*(3*c*sqrt(-(c^3*sqrt(-(B^4*a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7) + 2*A*B*a)/c^3)*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (B*c^5*sqrt(-(B^4*a^3 - 2 *A^2*B^2*a^2*c + A^4*a*c^2)/c^7) - A*B^2*a*c^2 + A^3*c^3)*sqrt(-(c^3*sqrt( -(B^4*a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7) + 2*A*B*a)/c^3)) - 3*c*sqrt( -(c^3*sqrt(-(B^4*a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7) + 2*A*B*a)/c^3)*l og(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (B*c^5*sqrt(-(B^4*a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7) - A*B^2*a*c^2 + A^3*c^3)*sqrt(-(c^3*sqrt(-(B^4*a^3 - 2*A ^2*B^2*a^2*c + A^4*a*c^2)/c^7) + 2*A*B*a)/c^3)) - 3*c*sqrt((c^3*sqrt(-(B^4 *a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7) - 2*A*B*a)/c^3)*log(-(B^4*a^2 - A ^4*c^2)*sqrt(x) + (B*c^5*sqrt(-(B^4*a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7 ) + A*B^2*a*c^2 - A^3*c^3)*sqrt((c^3*sqrt(-(B^4*a^3 - 2*A^2*B^2*a^2*c + A^ 4*a*c^2)/c^7) - 2*A*B*a)/c^3)) + 3*c*sqrt((c^3*sqrt(-(B^4*a^3 - 2*A^2*B^2* a^2*c + A^4*a*c^2)/c^7) - 2*A*B*a)/c^3)*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (B*c^5*sqrt(-(B^4*a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7) + A*B^2*a*c^2 - A^3*c^3)*sqrt((c^3*sqrt(-(B^4*a^3 - 2*A^2*B^2*a^2*c + A^4*a*c^2)/c^7) - 2 *A*B*a)/c^3)) - 4*(B*x + 3*A)*sqrt(x))/c
Time = 3.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx=\begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{a} & \text {for}\: c = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{c} & \text {for}\: a = 0 \\\frac {2 A \sqrt {x}}{c} + \frac {A \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 c} - \frac {A \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 c} - \frac {A \sqrt [4]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{c} - \frac {B a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 c^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {B a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 c^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {B a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{c^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 B x^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(2*A*sqrt(x) + 2*B*x**(3/2)/3), Eq(a, 0) & Eq(c, 0)), ((2*A *x**(5/2)/5 + 2*B*x**(7/2)/7)/a, Eq(c, 0)), ((2*A*sqrt(x) + 2*B*x**(3/2)/3 )/c, Eq(a, 0)), (2*A*sqrt(x)/c + A*(-a/c)**(1/4)*log(sqrt(x) - (-a/c)**(1/ 4))/(2*c) - A*(-a/c)**(1/4)*log(sqrt(x) + (-a/c)**(1/4))/(2*c) - A*(-a/c)* *(1/4)*atan(sqrt(x)/(-a/c)**(1/4))/c - B*a*log(sqrt(x) - (-a/c)**(1/4))/(2 *c**2*(-a/c)**(1/4)) + B*a*log(sqrt(x) + (-a/c)**(1/4))/(2*c**2*(-a/c)**(1 /4)) - B*a*atan(sqrt(x)/(-a/c)**(1/4))/(c**2*(-a/c)**(1/4)) + 2*B*x**(3/2) /(3*c), True))
Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.89 \[ \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx=-\frac {a {\left (\frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{4 \, c} + \frac {2 \, {\left (B x^{\frac {3}{2}} + 3 \, A \sqrt {x}\right )}}{3 \, c} \]
-1/4*a*(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)))/c + 2/3*(B*x^(3/2) + 3*A*sqrt(x)) /c
Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.92 \[ \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx=-\frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, c^{4}} + \frac {2 \, {\left (B c^{2} x^{\frac {3}{2}} + 3 \, A c^{2} \sqrt {x}\right )}}{3 \, c^{3}} \]
-1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (a*c^3)^(3/4)*B)*arctan(1/2*sqrt(2)*(s qrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/c^4 - 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))/c^4 - 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))/c^4 + 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))/c^4 + 2/3*(B*c^2*x^(3/2) + 3*A*c^2*sqrt(x))/c^3
Time = 10.15 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.16 \[ \int \frac {x^{3/2} (A+B x)}{a+c x^2} \, dx=\frac {2\,A\,\sqrt {x}}{c}+\frac {2\,B\,x^{3/2}}{3\,c}-\mathrm {atan}\left (\frac {B^2\,a^3\,\sqrt {x}\,\sqrt {\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}-\frac {A\,B\,a}{2\,c^3}-\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}-\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}+\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}-\frac {A^2\,a^2\,c\,\sqrt {x}\,\sqrt {\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}-\frac {A\,B\,a}{2\,c^3}-\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}-\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}+\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a\,c^7}-B^2\,a\,\sqrt {-a\,c^7}+2\,A\,B\,a\,c^4}{4\,c^7}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {B^2\,a^3\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}-\frac {A\,B\,a}{2\,c^3}-\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}+\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}-\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}-\frac {A^2\,a^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}-\frac {A\,B\,a}{2\,c^3}-\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}+\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}-\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a\,c^7}-A^2\,c\,\sqrt {-a\,c^7}+2\,A\,B\,a\,c^4}{4\,c^7}}\,2{}\mathrm {i} \]
(2*A*x^(1/2))/c - atan((B^2*a^3*x^(1/2)*((A^2*(-a*c^7)^(1/2))/(4*c^6) - (A *B*a)/(2*c^3) - (B^2*a*(-a*c^7)^(1/2))/(4*c^7))^(1/2)*32i)/((16*B^3*a^4)/c ^2 + (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c - (16*A*B^2*a^3*(- a*c^7)^(1/2))/c^5) - (A^2*a^2*c*x^(1/2)*((A^2*(-a*c^7)^(1/2))/(4*c^6) - (A *B*a)/(2*c^3) - (B^2*a*(-a*c^7)^(1/2))/(4*c^7))^(1/2)*32i)/((16*B^3*a^4)/c ^2 + (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c - (16*A*B^2*a^3*(- a*c^7)^(1/2))/c^5))*(-(B^2*a*(-a*c^7)^(1/2) - A^2*c*(-a*c^7)^(1/2) + 2*A*B *a*c^4)/(4*c^7))^(1/2)*2i - atan((B^2*a^3*x^(1/2)*((B^2*a*(-a*c^7)^(1/2))/ (4*c^7) - (A*B*a)/(2*c^3) - (A^2*(-a*c^7)^(1/2))/(4*c^6))^(1/2)*32i)/((16* B^3*a^4)/c^2 - (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c + (16*A* B^2*a^3*(-a*c^7)^(1/2))/c^5) - (A^2*a^2*c*x^(1/2)*((B^2*a*(-a*c^7)^(1/2))/ (4*c^7) - (A*B*a)/(2*c^3) - (A^2*(-a*c^7)^(1/2))/(4*c^6))^(1/2)*32i)/((16* B^3*a^4)/c^2 - (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c + (16*A* B^2*a^3*(-a*c^7)^(1/2))/c^5))*(-(A^2*c*(-a*c^7)^(1/2) - B^2*a*(-a*c^7)^(1/ 2) + 2*A*B*a*c^4)/(4*c^7))^(1/2)*2i + (2*B*x^(3/2))/(3*c)