Integrand size = 20, antiderivative size = 292 \[ \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx=-\frac {2 a B \sqrt {x}}{c^2}+\frac {2 A x^{3/2}}{3 c}+\frac {2 B x^{5/2}}{5 c}-\frac {a^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} c^{9/4}}+\frac {a^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} c^{9/4}}-\frac {a^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {a^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}} \]
2/3*A*x^(3/2)/c+2/5*B*x^(5/2)/c-1/2*a^(3/4)*arctan(1-c^(1/4)*2^(1/2)*x^(1/ 2)/a^(1/4))*(B*a^(1/2)-A*c^(1/2))/c^(9/4)*2^(1/2)+1/2*a^(3/4)*arctan(1+c^( 1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)-A*c^(1/2))/c^(9/4)*2^(1/2)-1/4*a^ (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))/c^(9/4)*2^(1/2)+1/4*a^(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))/c^(9/4)*2^(1/2)-2*a*B*x^(1/2)/c^2
Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.57 \[ \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx=\frac {4 \sqrt [4]{c} \sqrt {x} (-15 a B+c x (5 A+3 B x))-15 \sqrt {2} a^{3/4} \left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )+15 \sqrt {2} a^{3/4} \left (\sqrt {a} B+A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{30 c^{9/4}} \]
(4*c^(1/4)*Sqrt[x]*(-15*a*B + c*x*(5*A + 3*B*x)) - 15*Sqrt[2]*a^(3/4)*(Sqr t[a]*B - A*Sqrt[c])*ArcTan[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)* Sqrt[x])] + 15*Sqrt[2]*a^(3/4)*(Sqrt[a]*B + A*Sqrt[c])*ArcTanh[(Sqrt[2]*a^ (1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/(30*c^(9/4))
Time = 0.55 (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 = {552, 27, 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.
| \(\displaystyle \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {2 \int \frac {5 x^{3/2} (a B-A c x)}{2 \left (c x^2+a\right )}dx}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {\int \frac {x^{3/2} (a B-A c x)}{c x^2+a}dx}{c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {-\frac {2 \int -\frac {3 a c \sqrt {x} (A+B x)}{2 \left (c x^2+a\right )}dx}{3 c}-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \int \frac {\sqrt {x} (A+B x)}{c x^2+a}dx-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \int \frac {a B-A c x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {\int \frac {a B-A c x}{\sqrt {x} \left (c x^2+a\right )}dx}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \int \frac {a B-A c x}{c x^2+a}d\sqrt {x}}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}-\frac {1}{2} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 B x^{5/2}}{5 c}-\frac {a \left (\frac {2 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}\right )-\frac {2}{3} A x^{3/2}}{c}\) |
(2*B*x^(5/2))/(5*c) - ((-2*A*x^(3/2))/3 + a*((2*B*Sqrt[x])/c - (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)*Sqrt[x])/ a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4)))) + ((A + (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))/c))/c
3.5.12.3.1 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 = 240, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(\frac {2 \left (3 B c \,x^{2}+5 A c x -15 B a \right ) \sqrt {x}}{15 c^{2}}-\frac {a \left (-\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{c^{2}}\) | \(240\) |
| derivativedivides | \(\frac {\frac {2 B c \,x^{\frac {5}{2}}}{5}+\frac {2 A c \,x^{\frac {3}{2}}}{3}-2 a B \sqrt {x}}{c^{2}}-\frac {2 a \left (-\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{c^{2}}\) | \(242\) |
| default | \(\frac {\frac {2 B c \,x^{\frac {5}{2}}}{5}+\frac {2 A c \,x^{\frac {3}{2}}}{3}-2 a B \sqrt {x}}{c^{2}}-\frac {2 a \left (-\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{c^{2}}\) | \(242\) |
2/15*(3*B*c*x^2+5*A*c*x-15*B*a)*x^(1/2)/c^2-a/c^2*(-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*arctan(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. 862 vs. \(2 (199) = 398\).
Time = 0.58 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.95 \[ \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx=-\frac {15 \, c^{2} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} + {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + B^{3} a^{3} c^{2} - A^{2} B a^{2} c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}}\right ) - 15 \, c^{2} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} - {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + B^{3} a^{3} c^{2} - A^{2} B a^{2} c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} + 2 \, A B a^{2}}{c^{4}}}\right ) - 15 \, c^{2} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} + {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - B^{3} a^{3} c^{2} + A^{2} B a^{2} c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}}\right ) + 15 \, c^{2} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}} \log \left (-{\left (B^{4} a^{4} - A^{4} a^{2} c^{2}\right )} \sqrt {x} - {\left (A c^{7} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - B^{3} a^{3} c^{2} + A^{2} B a^{2} c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {B^{4} a^{5} - 2 \, A^{2} B^{2} a^{4} c + A^{4} a^{3} c^{2}}{c^{9}}} - 2 \, A B a^{2}}{c^{4}}}\right ) - 4 \, {\left (3 \, B c x^{2} + 5 \, A c x - 15 \, B a\right )} \sqrt {x}}{30 \, c^{2}} \]
-1/30*(15*c^2*sqrt((c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^ 9) + 2*A*B*a^2)/c^4)*log(-(B^4*a^4 - A^4*a^2*c^2)*sqrt(x) + (A*c^7*sqrt(-( B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + B^3*a^3*c^2 - A^2*B*a^2*c^ 3)*sqrt((c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + 2*A*B* a^2)/c^4)) - 15*c^2*sqrt((c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c ^2)/c^9) + 2*A*B*a^2)/c^4)*log(-(B^4*a^4 - A^4*a^2*c^2)*sqrt(x) - (A*c^7*s qrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + B^3*a^3*c^2 - A^2*B* a^2*c^3)*sqrt((c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + 2*A*B*a^2)/c^4)) - 15*c^2*sqrt(-(c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^ 4*a^3*c^2)/c^9) - 2*A*B*a^2)/c^4)*log(-(B^4*a^4 - A^4*a^2*c^2)*sqrt(x) + ( A*c^7*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) - B^3*a^3*c^2 + A^2*B*a^2*c^3)*sqrt(-(c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2) /c^9) - 2*A*B*a^2)/c^4)) + 15*c^2*sqrt(-(c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^ 4*c + A^4*a^3*c^2)/c^9) - 2*A*B*a^2)/c^4)*log(-(B^4*a^4 - A^4*a^2*c^2)*sqr t(x) - (A*c^7*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) - B^3*a ^3*c^2 + A^2*B*a^2*c^3)*sqrt(-(c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4* a^3*c^2)/c^9) - 2*A*B*a^2)/c^4)) - 4*(3*B*c*x^2 + 5*A*c*x - 15*B*a)*sqrt(x ))/c^2
Time = 14.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00 \[ \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}}{a} & \text {for}\: c = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{c} & \text {for}\: a = 0 \\- \frac {A a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 c^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {A a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 c^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{c^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A x^{\frac {3}{2}}}{3 c} - \frac {2 B a \sqrt {x}}{c^{2}} - \frac {B a \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 c^{2}} + \frac {B a \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 c^{2}} + \frac {B a \sqrt [4]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{c^{2}} + \frac {2 B x^{\frac {5}{2}}}{5 c} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(2*A*x**(3/2)/3 + 2*B*x**(5/2)/5), Eq(a, 0) & Eq(c, 0)), (( 2*A*x**(7/2)/7 + 2*B*x**(9/2)/9)/a, Eq(c, 0)), ((2*A*x**(3/2)/3 + 2*B*x**( 5/2)/5)/c, Eq(a, 0)), (-A*a*log(sqrt(x) - (-a/c)**(1/4))/(2*c**2*(-a/c)**( 1/4)) + A*a*log(sqrt(x) + (-a/c)**(1/4))/(2*c**2*(-a/c)**(1/4)) - A*a*atan (sqrt(x)/(-a/c)**(1/4))/(c**2*(-a/c)**(1/4)) + 2*A*x**(3/2)/(3*c) - 2*B*a* sqrt(x)/c**2 - B*a*(-a/c)**(1/4)*log(sqrt(x) - (-a/c)**(1/4))/(2*c**2) + B *a*(-a/c)**(1/4)*log(sqrt(x) + (-a/c)**(1/4))/(2*c**2) + B*a*(-a/c)**(1/4) *atan(sqrt(x)/(-a/c)**(1/4))/c**2 + 2*B*x**(5/2)/(5*c), True))
Time = 0.32 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.91 \[ \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx=\frac {a {\left (\frac {2 \, \sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{4 \, c^{2}} + \frac {2 \, {\left (3 \, B c x^{\frac {5}{2}} + 5 \, A c x^{\frac {3}{2}} - 15 \, B a \sqrt {x}\right )}}{15 \, c^{2}} \]
1/4*a*(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(s qrt(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)*sqrt (c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(B*a*sqrt(c) + A*s qrt(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)))/c^2 + 2/15*(3*B*c* x^(5/2) + 5*A*c*x^(3/2) - 15*B*a*sqrt(x))/c^2
Time = 0.29 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.90 \[ \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx=\frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, c^{4}} + \frac {2 \, {\left (3 \, B c^{4} x^{\frac {5}{2}} + 5 \, A c^{4} x^{\frac {3}{2}} - 15 \, B a c^{3} \sqrt {x}\right )}}{15 \, c^{5}} \]
1/2*sqrt(2)*((a*c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*arctan(1/2*sqrt(2)*(sq rt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/c^4 + 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*s qrt(x))/(a/c)^(1/4))/c^4 + 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))/c^4 - 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))/c^4 + 2/15*(3*B*c^4*x^(5/2) + 5*A*c^4*x^(3/2) - 15*B*a*c^3*s qrt(x))/c^5
Time = 10.13 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.28 \[ \int \frac {x^{5/2} (A+B x)}{a+c x^2} \, dx=\frac {2\,A\,x^{3/2}}{3\,c}+\frac {2\,B\,x^{5/2}}{5\,c}-\frac {2\,B\,a\,\sqrt {x}}{c^2}-\mathrm {atan}\left (\frac {A^2\,a^3\,\sqrt {x}\,\sqrt {\frac {A\,B\,a^2}{2\,c^4}-\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c^2}-\frac {16\,A\,B^2\,a^5}{c^3}-\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^8}+\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^7}}-\frac {B^2\,a^4\,\sqrt {x}\,\sqrt {\frac {A\,B\,a^2}{2\,c^4}-\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c}-\frac {16\,A\,B^2\,a^5}{c^2}-\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^7}+\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^6}}\right )\,\sqrt {\frac {B^2\,a\,\sqrt {-a^3\,c^9}-A^2\,c\,\sqrt {-a^3\,c^9}+2\,A\,B\,a^2\,c^5}{4\,c^9}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {A^2\,a^3\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {A\,B\,a^2}{2\,c^4}-\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c^2}-\frac {16\,A\,B^2\,a^5}{c^3}+\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^8}-\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^7}}-\frac {B^2\,a^4\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^9}}{4\,c^8}+\frac {A\,B\,a^2}{2\,c^4}-\frac {B^2\,a\,\sqrt {-a^3\,c^9}}{4\,c^9}}\,32{}\mathrm {i}}{\frac {16\,A^3\,a^4}{c}-\frac {16\,A\,B^2\,a^5}{c^2}+\frac {16\,B^3\,a^4\,\sqrt {-a^3\,c^9}}{c^7}-\frac {16\,A^2\,B\,a^3\,\sqrt {-a^3\,c^9}}{c^6}}\right )\,\sqrt {\frac {A^2\,c\,\sqrt {-a^3\,c^9}-B^2\,a\,\sqrt {-a^3\,c^9}+2\,A\,B\,a^2\,c^5}{4\,c^9}}\,2{}\mathrm {i} \]
(2*A*x^(3/2))/(3*c) - atan((A^2*a^3*x^(1/2)*((A^2*(-a^3*c^9)^(1/2))/(4*c^8 ) + (A*B*a^2)/(2*c^4) - (B^2*a*(-a^3*c^9)^(1/2))/(4*c^9))^(1/2)*32i)/((16* A^3*a^4)/c^2 - (16*A*B^2*a^5)/c^3 + (16*B^3*a^4*(-a^3*c^9)^(1/2))/c^8 - (1 6*A^2*B*a^3*(-a^3*c^9)^(1/2))/c^7) - (B^2*a^4*x^(1/2)*((A^2*(-a^3*c^9)^(1/ 2))/(4*c^8) + (A*B*a^2)/(2*c^4) - (B^2*a*(-a^3*c^9)^(1/2))/(4*c^9))^(1/2)* 32i)/((16*A^3*a^4)/c - (16*A*B^2*a^5)/c^2 + (16*B^3*a^4*(-a^3*c^9)^(1/2))/ c^7 - (16*A^2*B*a^3*(-a^3*c^9)^(1/2))/c^6))*((A^2*c*(-a^3*c^9)^(1/2) - B^2 *a*(-a^3*c^9)^(1/2) + 2*A*B*a^2*c^5)/(4*c^9))^(1/2)*2i - atan((A^2*a^3*x^( 1/2)*((A*B*a^2)/(2*c^4) - (A^2*(-a^3*c^9)^(1/2))/(4*c^8) + (B^2*a*(-a^3*c^ 9)^(1/2))/(4*c^9))^(1/2)*32i)/((16*A^3*a^4)/c^2 - (16*A*B^2*a^5)/c^3 - (16 *B^3*a^4*(-a^3*c^9)^(1/2))/c^8 + (16*A^2*B*a^3*(-a^3*c^9)^(1/2))/c^7) - (B ^2*a^4*x^(1/2)*((A*B*a^2)/(2*c^4) - (A^2*(-a^3*c^9)^(1/2))/(4*c^8) + (B^2* a*(-a^3*c^9)^(1/2))/(4*c^9))^(1/2)*32i)/((16*A^3*a^4)/c - (16*A*B^2*a^5)/c ^2 - (16*B^3*a^4*(-a^3*c^9)^(1/2))/c^7 + (16*A^2*B*a^3*(-a^3*c^9)^(1/2))/c ^6))*((B^2*a*(-a^3*c^9)^(1/2) - A^2*c*(-a^3*c^9)^(1/2) + 2*A*B*a^2*c^5)/(4 *c^9))^(1/2)*2i + (2*B*x^(5/2))/(5*c) - (2*B*a*x^(1/2))/c^2