Integrand size = 22, antiderivative size = 298 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 A b+5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(3 A b+5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}} \]
1/4*(A*b-B*a)*x^(5/2)/a/b/(b*x^2+a)^2-1/64*(3*A*b+5*B*a)*arctan(1-b^(1/4)* 2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(9/4)*2^(1/2)+1/64*(3*A*b+5*B*a)*arctan (1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(9/4)*2^(1/2)-1/128*(3*A*b+5 *B*a)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/b^(9/4 )*2^(1/2)+1/128*(3*A*b+5*B*a)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2) *x^(1/2))/a^(7/4)/b^(9/4)*2^(1/2)-1/16*(3*A*b+5*B*a)*x^(1/2)/a/b^2/(b*x^2+ a)
Time = 0.58 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.58 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (5 a^2 B-A b^2 x^2+3 a b \left (A+3 B x^2\right )\right )}{\left (a+b x^2\right )^2}-\sqrt {2} (3 A b+5 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} (3 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{7/4} b^{9/4}} \]
((-4*a^(3/4)*b^(1/4)*Sqrt[x]*(5*a^2*B - A*b^2*x^2 + 3*a*b*(A + 3*B*x^2)))/ (a + b*x^2)^2 - Sqrt[2]*(3*A*b + 5*a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt [2]*a^(1/4)*b^(1/4)*Sqrt[x])] + Sqrt[2]*(3*A*b + 5*a*B)*ArcTanh[(Sqrt[2]*a ^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(64*a^(7/4)*b^(9/4))
Time = 0.48 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {362, 252, 266, 755, 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.
| \(\displaystyle \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {(5 a B+3 A b) \int \frac {x^{3/2}}{\left (b x^2+a\right )^2}dx}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{4 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\int \frac {1}{b x^2+a}d\sqrt {x}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(5 a B+3 A b) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{2 b}-\frac {\sqrt {x}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\) |
((A*b - a*B)*x^(5/2))/(4*a*b*(a + b*x^2)^2) + ((3*A*b + 5*a*B)*(-1/2*Sqrt[ x]/(b*(a + b*x^2)) + ((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sq rt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(S qrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4) *b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sq rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2 *Sqrt[a]))/(2*b)))/(8*a*b)
3.4.85.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
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]
Time = 2.77 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.56
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (A b -9 B a \right ) x^{\frac {5}{2}}}{16 a b}-\frac {\left (3 A b +5 B a \right ) \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 A b +5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a^{2}}\) | \(167\) |
| default | \(\frac {\frac {\left (A b -9 B a \right ) x^{\frac {5}{2}}}{16 a b}-\frac {\left (3 A b +5 B a \right ) \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 A b +5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a^{2}}\) | \(167\) |
2*(1/32*(A*b-9*B*a)/a/b*x^(5/2)-1/32*(3*A*b+5*B*a)/b^2*x^(1/2))/(b*x^2+a)^ 2+1/128*(3*A*b+5*B*a)/b^2/a^2*(a/b)^(1/4)*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/ 2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arc tan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1) )
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 763, normalized size of antiderivative = 2.56 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (-i \, a b^{4} x^{4} - 2 i \, a^{2} b^{3} x^{2} - i \, a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (i \, a b^{4} x^{4} + 2 i \, a^{2} b^{3} x^{2} + i \, a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (-a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, B a^{2} + 3 \, A a b + {\left (9 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
1/64*((a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(-(625*B^4*a^4 + 1500*A*B^3*a^ 3*b + 1350*A^2*B^2*a^2*b^2 + 540*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9))^(1/4 )*log(a^2*b^2*(-(625*B^4*a^4 + 1500*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 5 40*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9))^(1/4) + (5*B*a + 3*A*b)*sqrt(x)) - (-I*a*b^4*x^4 - 2*I*a^2*b^3*x^2 - I*a^3*b^2)*(-(625*B^4*a^4 + 1500*A*B^3* a^3*b + 1350*A^2*B^2*a^2*b^2 + 540*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9))^(1 /4)*log(I*a^2*b^2*(-(625*B^4*a^4 + 1500*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 540*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9))^(1/4) + (5*B*a + 3*A*b)*sqrt(x )) - (I*a*b^4*x^4 + 2*I*a^2*b^3*x^2 + I*a^3*b^2)*(-(625*B^4*a^4 + 1500*A*B ^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 540*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9)) ^(1/4)*log(-I*a^2*b^2*(-(625*B^4*a^4 + 1500*A*B^3*a^3*b + 1350*A^2*B^2*a^2 *b^2 + 540*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9))^(1/4) + (5*B*a + 3*A*b)*sq rt(x)) - (a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(-(625*B^4*a^4 + 1500*A*B^3 *a^3*b + 1350*A^2*B^2*a^2*b^2 + 540*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9))^( 1/4)*log(-a^2*b^2*(-(625*B^4*a^4 + 1500*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 540*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^9))^(1/4) + (5*B*a + 3*A*b)*sqrt(x )) - 4*(5*B*a^2 + 3*A*a*b + (9*B*a*b - A*b^2)*x^2)*sqrt(x))/(a*b^4*x^4 + 2 *a^2*b^3*x^2 + a^3*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (287) = 574\).
Time = 149.65 (sec) , antiderivative size = 1445, normalized size of antiderivative = 4.85 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0) ), ((2*A*x**(5/2)/5 + 2*B*x**(9/2)/9)/a**3, Eq(b, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(3*x**(3/2)))/b**3, Eq(a, 0)), (-12*A*a**2*b*sqrt(x)/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4*x**4) - 3*A*a**2*b*(-a/b)**(1/4)*log(s qrt(x) - (-a/b)**(1/4))/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4* x**4) + 3*A*a**2*b*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b** 2 + 128*a**3*b**3*x**2 + 64*a**2*b**4*x**4) + 6*A*a**2*b*(-a/b)**(1/4)*ata n(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4 *x**4) + 4*A*a*b**2*x**(5/2)/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2* b**4*x**4) - 6*A*a*b**2*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(6 4*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4*x**4) + 6*A*a*b**2*x**2*(- a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b**2 + 128*a**3*b**3*x** 2 + 64*a**2*b**4*x**4) + 12*A*a*b**2*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b )**(1/4))/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4*x**4) - 3*A*b* *3*x**4*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**4*b**2 + 128*a** 3*b**3*x**2 + 64*a**2*b**4*x**4) + 3*A*b**3*x**4*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4*x**4) + 6*A*b**3*x**4*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4*x**4) - 20*B*a**3*sqrt(x)/(64*a**4*b**2 + 128*a**3*b**3*x**2 + 64*a**2*b**4*x**4) - 5*B*a**3*(-a/b)**(1/4)*log(...
Time = 0.29 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (9 \, B a b - A b^{2}\right )} x^{\frac {5}{2}} + {\left (5 \, B a^{2} + 3 \, A a b\right )} \sqrt {x}}{16 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{128 \, a b^{2}} \]
-1/16*((9*B*a*b - A*b^2)*x^(5/2) + (5*B*a^2 + 3*A*a*b)*sqrt(x))/(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2) + 1/128*(2*sqrt(2)*(5*B*a + 3*A*b)*arctan(1/2* sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b) ))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(5*B*a + 3*A*b)*arctan(-1/2 *sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b )))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(5*B*a + 3*A*b)*log(sqrt(2)* a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2) *(5*B*a + 3*A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a ))/(a^(3/4)*b^(1/4)))/(a*b^2)
Time = 0.33 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac {9 \, B a b x^{\frac {5}{2}} - A b^{2} x^{\frac {5}{2}} + 5 \, B a^{2} \sqrt {x} + 3 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} \]
1/64*sqrt(2)*(5*(a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2 )*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^3) + 1/64*sqrt(2)* (5*(a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*( a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^3) + 1/128*sqrt(2)*(5*(a*b^3)^ (1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqr t(a/b))/(a^2*b^3) - 1/128*sqrt(2)*(5*(a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A *b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^3) - 1/16*(9* B*a*b*x^(5/2) - A*b^2*x^(5/2) + 5*B*a^2*sqrt(x) + 3*A*a*b*sqrt(x))/((b*x^2 + a)^2*a*b^2)
Time = 0.34 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.68 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {\frac {\sqrt {x}\,\left (3\,A\,b+5\,B\,a\right )}{16\,b^2}-\frac {x^{5/2}\,\left (A\,b-9\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}{\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}-\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{32\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}{\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}-\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}\right )\,\left (3\,A\,b+5\,B\,a\right )}{32\,{\left (-a\right )}^{7/4}\,b^{9/4}} \]
(atan((((3*A*b + 5*B*a)*((x^(1/2)*(9*A^2*b^2 + 25*B^2*a^2 + 30*A*B*a*b))/( 64*a^2*b) - ((3*A*b^2 + 5*B*a*b)*(3*A*b + 5*B*a))/(64*(-a)^(7/4)*b^(9/4))) *1i)/(64*(-a)^(7/4)*b^(9/4)) + ((3*A*b + 5*B*a)*((x^(1/2)*(9*A^2*b^2 + 25* B^2*a^2 + 30*A*B*a*b))/(64*a^2*b) + ((3*A*b^2 + 5*B*a*b)*(3*A*b + 5*B*a))/ (64*(-a)^(7/4)*b^(9/4)))*1i)/(64*(-a)^(7/4)*b^(9/4)))/(((3*A*b + 5*B*a)*(( x^(1/2)*(9*A^2*b^2 + 25*B^2*a^2 + 30*A*B*a*b))/(64*a^2*b) - ((3*A*b^2 + 5* B*a*b)*(3*A*b + 5*B*a))/(64*(-a)^(7/4)*b^(9/4))))/(64*(-a)^(7/4)*b^(9/4)) - ((3*A*b + 5*B*a)*((x^(1/2)*(9*A^2*b^2 + 25*B^2*a^2 + 30*A*B*a*b))/(64*a^ 2*b) + ((3*A*b^2 + 5*B*a*b)*(3*A*b + 5*B*a))/(64*(-a)^(7/4)*b^(9/4))))/(64 *(-a)^(7/4)*b^(9/4))))*(3*A*b + 5*B*a)*1i)/(32*(-a)^(7/4)*b^(9/4)) - ((x^( 1/2)*(3*A*b + 5*B*a))/(16*b^2) - (x^(5/2)*(A*b - 9*B*a))/(16*a*b))/(a^2 + b^2*x^4 + 2*a*b*x^2) + (atan((((3*A*b + 5*B*a)*((x^(1/2)*(9*A^2*b^2 + 25*B ^2*a^2 + 30*A*B*a*b))/(64*a^2*b) - ((3*A*b^2 + 5*B*a*b)*(3*A*b + 5*B*a)*1i )/(64*(-a)^(7/4)*b^(9/4))))/(64*(-a)^(7/4)*b^(9/4)) + ((3*A*b + 5*B*a)*((x ^(1/2)*(9*A^2*b^2 + 25*B^2*a^2 + 30*A*B*a*b))/(64*a^2*b) + ((3*A*b^2 + 5*B *a*b)*(3*A*b + 5*B*a)*1i)/(64*(-a)^(7/4)*b^(9/4))))/(64*(-a)^(7/4)*b^(9/4) ))/(((3*A*b + 5*B*a)*((x^(1/2)*(9*A^2*b^2 + 25*B^2*a^2 + 30*A*B*a*b))/(64* a^2*b) - ((3*A*b^2 + 5*B*a*b)*(3*A*b + 5*B*a)*1i)/(64*(-a)^(7/4)*b^(9/4))) *1i)/(64*(-a)^(7/4)*b^(9/4)) - ((3*A*b + 5*B*a)*((x^(1/2)*(9*A^2*b^2 + 25* B^2*a^2 + 30*A*B*a*b))/(64*a^2*b) + ((3*A*b^2 + 5*B*a*b)*(3*A*b + 5*B*a...