Integrand size = 26, antiderivative size = 343 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {11 (7 b B-15 A c)}{112 b^3 c x^{7/2}}-\frac {11 (7 b B-15 A c)}{48 b^4 x^{3/2}}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}-\frac {7 b B-15 A c}{16 b^2 c x^{7/2} \left (b+c x^2\right )}+\frac {11 c^{3/4} (7 b B-15 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}-\frac {11 c^{3/4} (7 b B-15 A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}+\frac {11 c^{3/4} (7 b B-15 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}-\frac {11 c^{3/4} (7 b B-15 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}} \]
11/112*(-15*A*c+7*B*b)/b^3/c/x^(7/2)-11/48*(-15*A*c+7*B*b)/b^4/x^(3/2)+1/4 *(A*c-B*b)/b/c/x^(7/2)/(c*x^2+b)^2+1/16*(15*A*c-7*B*b)/b^2/c/x^(7/2)/(c*x^ 2+b)+11/64*c^(3/4)*(-15*A*c+7*B*b)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4 ))/b^(19/4)*2^(1/2)-11/64*c^(3/4)*(-15*A*c+7*B*b)*arctan(1+c^(1/4)*2^(1/2) *x^(1/2)/b^(1/4))/b^(19/4)*2^(1/2)+11/128*c^(3/4)*(-15*A*c+7*B*b)*ln(b^(1/ 2)+x*c^(1/2)-b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/b^(19/4)*2^(1/2)-11/128*c^(3 /4)*(-15*A*c+7*B*b)*ln(b^(1/2)+x*c^(1/2)+b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/ b^(19/4)*2^(1/2)
Time = 0.91 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.61 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 b^{3/4} \left (7 b B x^2 \left (32 b^2+121 b c x^2+77 c^2 x^4\right )+3 A \left (32 b^3-160 b^2 c x^2-605 b c^2 x^4-385 c^3 x^6\right )\right )}{x^{7/2} \left (b+c x^2\right )^2}+231 \sqrt {2} c^{3/4} (7 b B-15 A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+231 \sqrt {2} c^{3/4} (-7 b B+15 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{1344 b^{19/4}} \]
((-4*b^(3/4)*(7*b*B*x^2*(32*b^2 + 121*b*c*x^2 + 77*c^2*x^4) + 3*A*(32*b^3 - 160*b^2*c*x^2 - 605*b*c^2*x^4 - 385*c^3*x^6)))/(x^(7/2)*(b + c*x^2)^2) + 231*Sqrt[2]*c^(3/4)*(7*b*B - 15*A*c)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2 ]*b^(1/4)*c^(1/4)*Sqrt[x])] + 231*Sqrt[2]*c^(3/4)*(-7*b*B + 15*A*c)*ArcTan h[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/(1344*b^(19/4) )
Time = 0.52 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {9, 362, 253, 264, 264, 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.
| \(\displaystyle \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {A+B x^2}{x^{9/2} \left (b+c x^2\right )^3}dx\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle -\frac {(7 b B-15 A c) \int \frac {1}{x^{9/2} \left (c x^2+b\right )^2}dx}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \int \frac {1}{x^{9/2} \left (c x^2+b\right )}dx}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \int \frac {1}{x^{5/2} \left (c x^2+b\right )}dx}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {c \int \frac {1}{\sqrt {x} \left (c x^2+b\right )}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \int \frac {1}{c x^2+b}d\sqrt {x}}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\) |
-1/4*(b*B - A*c)/(b*c*x^(7/2)*(b + c*x^2)^2) - ((7*b*B - 15*A*c)*(1/(2*b*x ^(7/2)*(b + c*x^2)) + (11*(-2/(7*b*x^(7/2)) - (c*(-2/(3*b*x^(3/2)) - (2*c* ((-(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4) )) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4 )))/(2*Sqrt[b]) + (-1/2*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sq rt[c]*x]/(Sqrt[2]*b^(1/4)*c^(1/4)) + Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4) *Sqrt[x] + Sqrt[c]*x]/(2*Sqrt[2]*b^(1/4)*c^(1/4)))/(2*Sqrt[b])))/b))/b))/( 4*b)))/(8*b*c)
3.3.17.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
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*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[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 = 1.84 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.55
| method | result | size |
| derivativedivides | \(-\frac {2 A}{7 b^{3} x^{\frac {7}{2}}}-\frac {2 \left (-3 A c +B b \right )}{3 b^{4} x^{\frac {3}{2}}}+\frac {2 c \left (\frac {\left (\frac {23}{32} A \,c^{2}-\frac {15}{32} B b c \right ) x^{\frac {5}{2}}+\frac {b \left (27 A c -19 B b \right ) \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {11 \left (15 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{b^{4}}\) | \(190\) |
| default | \(-\frac {2 A}{7 b^{3} x^{\frac {7}{2}}}-\frac {2 \left (-3 A c +B b \right )}{3 b^{4} x^{\frac {3}{2}}}+\frac {2 c \left (\frac {\left (\frac {23}{32} A \,c^{2}-\frac {15}{32} B b c \right ) x^{\frac {5}{2}}+\frac {b \left (27 A c -19 B b \right ) \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {11 \left (15 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{b^{4}}\) | \(190\) |
| risch | \(-\frac {2 \left (-21 A c \,x^{2}+7 b B \,x^{2}+3 A b \right )}{21 b^{4} x^{\frac {7}{2}}}+\frac {c \left (\frac {2 \left (\frac {23}{32} A \,c^{2}-\frac {15}{32} B b c \right ) x^{\frac {5}{2}}+\frac {b \left (27 A c -19 B b \right ) \sqrt {x}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {11 \left (15 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b}\right )}{b^{4}}\) | \(192\) |
-2/7*A/b^3/x^(7/2)-2/3*(-3*A*c+B*b)/b^4/x^(3/2)+2/b^4*c*(((23/32*A*c^2-15/ 32*B*b*c)*x^(5/2)+1/32*b*(27*A*c-19*B*b)*x^(1/2))/(c*x^2+b)^2+11/256*(15*A *c-7*B*b)*(1/c*b)^(1/4)/b*2^(1/2)*(ln((x+(1/c*b)^(1/4)*x^(1/2)*2^(1/2)+(1/ c*b)^(1/2))/(x-(1/c*b)^(1/4)*x^(1/2)*2^(1/2)+(1/c*b)^(1/2)))+2*arctan(2^(1 /2)/(1/c*b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(1/c*b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 822, normalized size of antiderivative = 2.40 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {231 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (11 \, b^{5} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} - 11 \, {\left (7 \, B b c - 15 \, A c^{2}\right )} \sqrt {x}\right ) - 231 \, {\left (-i \, b^{4} c^{2} x^{8} - 2 i \, b^{5} c x^{6} - i \, b^{6} x^{4}\right )} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (11 i \, b^{5} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} - 11 \, {\left (7 \, B b c - 15 \, A c^{2}\right )} \sqrt {x}\right ) - 231 \, {\left (i \, b^{4} c^{2} x^{8} + 2 i \, b^{5} c x^{6} + i \, b^{6} x^{4}\right )} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (-11 i \, b^{5} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} - 11 \, {\left (7 \, B b c - 15 \, A c^{2}\right )} \sqrt {x}\right ) - 231 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (-11 \, b^{5} \left (-\frac {2401 \, B^{4} b^{4} c^{3} - 20580 \, A B^{3} b^{3} c^{4} + 66150 \, A^{2} B^{2} b^{2} c^{5} - 94500 \, A^{3} B b c^{6} + 50625 \, A^{4} c^{7}}{b^{19}}\right )^{\frac {1}{4}} - 11 \, {\left (7 \, B b c - 15 \, A c^{2}\right )} \sqrt {x}\right ) - 4 \, {\left (77 \, {\left (7 \, B b c^{2} - 15 \, A c^{3}\right )} x^{6} + 121 \, {\left (7 \, B b^{2} c - 15 \, A b c^{2}\right )} x^{4} + 96 \, A b^{3} + 32 \, {\left (7 \, B b^{3} - 15 \, A b^{2} c\right )} x^{2}\right )} \sqrt {x}}{1344 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )}} \]
1/1344*(231*(b^4*c^2*x^8 + 2*b^5*c*x^6 + b^6*x^4)*(-(2401*B^4*b^4*c^3 - 20 580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4* c^7)/b^19)^(1/4)*log(11*b^5*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66 150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4) - 11* (7*B*b*c - 15*A*c^2)*sqrt(x)) - 231*(-I*b^4*c^2*x^8 - 2*I*b^5*c*x^6 - I*b^ 6*x^4)*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4)*log(11*I*b^5*(-(2401*B^4*b ^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4) - 11*(7*B*b*c - 15*A*c^2)*sqrt(x)) - 231*(I*b^ 4*c^2*x^8 + 2*I*b^5*c*x^6 + I*b^6*x^4)*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b ^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^ (1/4)*log(-11*I*b^5*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2* B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4) - 11*(7*B*b*c - 15*A*c^2)*sqrt(x)) - 231*(b^4*c^2*x^8 + 2*b^5*c*x^6 + b^6*x^4)*(-(2401* B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b* c^6 + 50625*A^4*c^7)/b^19)^(1/4)*log(-11*b^5*(-(2401*B^4*b^4*c^3 - 20580*A *B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/ b^19)^(1/4) - 11*(7*B*b*c - 15*A*c^2)*sqrt(x)) - 4*(77*(7*B*b*c^2 - 15*A*c ^3)*x^6 + 121*(7*B*b^2*c - 15*A*b*c^2)*x^4 + 96*A*b^3 + 32*(7*B*b^3 - 15*A *b^2*c)*x^2)*sqrt(x))/(b^4*c^2*x^8 + 2*b^5*c*x^6 + b^6*x^4)
Timed out. \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {77 \, {\left (7 \, B b c^{2} - 15 \, A c^{3}\right )} x^{6} + 121 \, {\left (7 \, B b^{2} c - 15 \, A b c^{2}\right )} x^{4} + 96 \, A b^{3} + 32 \, {\left (7 \, B b^{3} - 15 \, A b^{2} c\right )} x^{2}}{336 \, {\left (b^{4} c^{2} x^{\frac {15}{2}} + 2 \, b^{5} c x^{\frac {11}{2}} + b^{6} x^{\frac {7}{2}}\right )}} - \frac {11 \, {\left (\frac {2 \, \sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}\right )}}{128 \, b^{4}} \]
-1/336*(77*(7*B*b*c^2 - 15*A*c^3)*x^6 + 121*(7*B*b^2*c - 15*A*b*c^2)*x^4 + 96*A*b^3 + 32*(7*B*b^3 - 15*A*b^2*c)*x^2)/(b^4*c^2*x^(15/2) + 2*b^5*c*x^( 11/2) + b^6*x^(7/2)) - 11/128*(2*sqrt(2)*(7*B*b*c - 15*A*c^2)*arctan(1/2*s qrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)) )/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + 2*sqrt(2)*(7*B*b*c - 15*A*c^2)*arctan( -1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sq rt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + sqrt(2)*(7*B*b*c - 15*A*c^2)*log (sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(3/4)*c^(1/4)) - sqrt(2)*(7*B*b*c - 15*A*c^2)*log(-sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt (c)*x + sqrt(b))/(b^(3/4)*c^(1/4)))/b^4
Time = 0.30 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.92 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} - \frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} - \frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} + \frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} - \frac {15 \, B b c^{2} x^{\frac {5}{2}} - 23 \, A c^{3} x^{\frac {5}{2}} + 19 \, B b^{2} c \sqrt {x} - 27 \, A b c^{2} \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{4}} - \frac {2 \, {\left (7 \, B b x^{2} - 21 \, A c x^{2} + 3 \, A b\right )}}{21 \, b^{4} x^{\frac {7}{2}}} \]
-11/64*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A*c)*arctan(1/2*sqr t(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/b^5 - 11/64*sqrt(2)*(7 *(b*c^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b /c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/b^5 - 11/128*sqrt(2)*(7*(b*c^3)^(1/4)* B*b - 15*(b*c^3)^(1/4)*A*c)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c ))/b^5 + 11/128*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A*c)*log(- sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^5 - 1/16*(15*B*b*c^2*x^(5/2 ) - 23*A*c^3*x^(5/2) + 19*B*b^2*c*sqrt(x) - 27*A*b*c^2*sqrt(x))/((c*x^2 + b)^2*b^4) - 2/21*(7*B*b*x^2 - 21*A*c*x^2 + 3*A*b)/(b^4*x^(7/2))
Time = 9.39 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.83 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {2\,x^2\,\left (15\,A\,c-7\,B\,b\right )}{21\,b^2}-\frac {2\,A}{7\,b}+\frac {11\,c^2\,x^6\,\left (15\,A\,c-7\,B\,b\right )}{48\,b^4}+\frac {121\,c\,x^4\,\left (15\,A\,c-7\,B\,b\right )}{336\,b^3}}{b^2\,x^{7/2}+c^2\,x^{15/2}+2\,b\,c\,x^{11/2}}+\frac {11\,{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {\frac {11\,{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (\sqrt {x}\,\left (446054400\,A^2\,b^{12}\,c^7-416317440\,A\,B\,b^{13}\,c^6+97140736\,B^2\,b^{14}\,c^5\right )-\frac {{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (173015040\,A\,b^{17}\,c^5-80740352\,B\,b^{18}\,c^4\right )\,11{}\mathrm {i}}{64\,b^{19/4}}\right )}{64\,b^{19/4}}+\frac {11\,{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (\sqrt {x}\,\left (446054400\,A^2\,b^{12}\,c^7-416317440\,A\,B\,b^{13}\,c^6+97140736\,B^2\,b^{14}\,c^5\right )+\frac {{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (173015040\,A\,b^{17}\,c^5-80740352\,B\,b^{18}\,c^4\right )\,11{}\mathrm {i}}{64\,b^{19/4}}\right )}{64\,b^{19/4}}}{\frac {{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (\sqrt {x}\,\left (446054400\,A^2\,b^{12}\,c^7-416317440\,A\,B\,b^{13}\,c^6+97140736\,B^2\,b^{14}\,c^5\right )-\frac {{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (173015040\,A\,b^{17}\,c^5-80740352\,B\,b^{18}\,c^4\right )\,11{}\mathrm {i}}{64\,b^{19/4}}\right )\,11{}\mathrm {i}}{64\,b^{19/4}}-\frac {{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (\sqrt {x}\,\left (446054400\,A^2\,b^{12}\,c^7-416317440\,A\,B\,b^{13}\,c^6+97140736\,B^2\,b^{14}\,c^5\right )+\frac {{\left (-c\right )}^{3/4}\,\left (15\,A\,c-7\,B\,b\right )\,\left (173015040\,A\,b^{17}\,c^5-80740352\,B\,b^{18}\,c^4\right )\,11{}\mathrm {i}}{64\,b^{19/4}}\right )\,11{}\mathrm {i}}{64\,b^{19/4}}}\right )\,\left (15\,A\,c-7\,B\,b\right )}{32\,b^{19/4}}-\frac {{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {A^3\,c^8\,\sqrt {x}\,3375{}\mathrm {i}-B^3\,b^3\,c^5\,\sqrt {x}\,343{}\mathrm {i}-A^2\,B\,b\,c^7\,\sqrt {x}\,4725{}\mathrm {i}+A\,B^2\,b^2\,c^6\,\sqrt {x}\,2205{}\mathrm {i}}{b^{1/4}\,{\left (-c\right )}^{19/4}\,\left (c\,\left (c\,\left (3375\,A^3\,c-4725\,A^2\,B\,b\right )+2205\,A\,B^2\,b^2\right )-343\,B^3\,b^3\right )}\right )\,\left (15\,A\,c-7\,B\,b\right )\,11{}\mathrm {i}}{32\,b^{19/4}} \]
((2*x^2*(15*A*c - 7*B*b))/(21*b^2) - (2*A)/(7*b) + (11*c^2*x^6*(15*A*c - 7 *B*b))/(48*b^4) + (121*c*x^4*(15*A*c - 7*B*b))/(336*b^3))/(b^2*x^(7/2) + c ^2*x^(15/2) + 2*b*c*x^(11/2)) + (11*(-c)^(3/4)*atan(((11*(-c)^(3/4)*(15*A* c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2*b^14*c^5 - 4163 17440*A*B*b^13*c^6) - ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015040*A*b^17*c^5 - 80740352*B*b^18*c^4)*11i)/(64*b^(19/4))))/(64*b^(19/4)) + (11*(-c)^(3/4)* (15*A*c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2*b^14*c^5 - 416317440*A*B*b^13*c^6) + ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015040*A*b^17 *c^5 - 80740352*B*b^18*c^4)*11i)/(64*b^(19/4))))/(64*b^(19/4)))/(((-c)^(3/ 4)*(15*A*c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2*b^14*c ^5 - 416317440*A*B*b^13*c^6) - ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015040*A*b ^17*c^5 - 80740352*B*b^18*c^4)*11i)/(64*b^(19/4)))*11i)/(64*b^(19/4)) - (( -c)^(3/4)*(15*A*c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2 *b^14*c^5 - 416317440*A*B*b^13*c^6) + ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015 040*A*b^17*c^5 - 80740352*B*b^18*c^4)*11i)/(64*b^(19/4)))*11i)/(64*b^(19/4 ))))*(15*A*c - 7*B*b))/(32*b^(19/4)) - ((-c)^(3/4)*atan((A^3*c^8*x^(1/2)*3 375i - B^3*b^3*c^5*x^(1/2)*343i - A^2*B*b*c^7*x^(1/2)*4725i + A*B^2*b^2*c^ 6*x^(1/2)*2205i)/(b^(1/4)*(-c)^(19/4)*(c*(c*(3375*A^3*c - 4725*A^2*B*b) + 2205*A*B^2*b^2) - 343*B^3*b^3)))*(15*A*c - 7*B*b)*11i)/(32*b^(19/4))