Integrand size = 24, antiderivative size = 402 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac {\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt {x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \sqrt {x}}{48 c^3 d \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} d^{5/4}}-\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}}+\frac {\left (3 b^2 c^2+7 a d (6 b c-11 a d)\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} d^{5/4}} \]
-2/3*a^2/c/x^(3/2)/(d*x^2+c)^2-1/64*(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*arct an(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/d^(5/4)*2^(1/2)+1/64*(3*b^2 *c^2+7*a*d*(-11*a*d+6*b*c))*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(1 5/4)/d^(5/4)*2^(1/2)-1/128*(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*ln(c^(1/2)+x* d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(15/4)/d^(5/4)*2^(1/2)+1/128*(3 *b^2*c^2+7*a*d*(-11*a*d+6*b*c))*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/ 2)*x^(1/2))/c^(15/4)/d^(5/4)*2^(1/2)-1/12*(11*a^2*d^2-6*a*b*c*d+3*b^2*c^2) *x^(1/2)/c^2/d/(d*x^2+c)^2+1/48*(3*b^2*c^2+7*a*d*(-11*a*d+6*b*c))*x^(1/2)/ c^3/d/(d*x^2+c)
Time = 0.75 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx=\frac {-\frac {4 c^{3/4} \sqrt [4]{d} \left (3 b^2 c^2 x^2 \left (3 c-d x^2\right )-6 a b c d x^2 \left (11 c+7 d x^2\right )+a^2 d \left (32 c^2+121 c d x^2+77 d^2 x^4\right )\right )}{x^{3/2} \left (c+d x^2\right )^2}-3 \sqrt {2} \left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )+3 \sqrt {2} \left (3 b^2 c^2+42 a b c d-77 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{192 c^{15/4} d^{5/4}} \]
((-4*c^(3/4)*d^(1/4)*(3*b^2*c^2*x^2*(3*c - d*x^2) - 6*a*b*c*d*x^2*(11*c + 7*d*x^2) + a^2*d*(32*c^2 + 121*c*d*x^2 + 77*d^2*x^4)))/(x^(3/2)*(c + d*x^2 )^2) - 3*Sqrt[2]*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*ArcTan[(Sqrt[c] - S qrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + 3*Sqrt[2]*(3*b^2*c^2 + 42*a *b*c*d - 77*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(192*c^(15/4)*d^(5/4))
Time = 0.57 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.86, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {365, 27, 362, 253, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {3 b^2 c x^2+a (6 b c-11 a d)}{2 \sqrt {x} \left (d x^2+c\right )^3}dx}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 b^2 c x^2+a (6 b c-11 a d)}{\sqrt {x} \left (d x^2+c\right )^3}dx}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \int \frac {1}{\sqrt {x} \left (d x^2+c\right )^2}dx+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \int \frac {1}{\sqrt {x} \left (d x^2+c\right )}dx}{4 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \int \frac {1}{d x^2+c}d\sqrt {x}}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )+\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {x} \left (-\frac {11 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}+\frac {1}{8} \left (\frac {7 a (6 b c-11 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c}+\frac {\sqrt {x}}{2 c \left (c+d x^2\right )}\right )}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}\) |
(-2*a^2)/(3*c*x^(3/2)*(c + d*x^2)^2) + (((6*a*b - (3*b^2*c)/d - (11*a^2*d) /c)*Sqrt[x])/(4*(c + d*x^2)^2) + (((3*b^2*c)/d + (7*a*(6*b*c - 11*a*d))/c) *(Sqrt[x]/(2*c*(c + d*x^2)) + (3*((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/ c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4))) + ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x]) /c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqr t[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[ Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)* d^(1/4)))/(2*Sqrt[c])))/(2*c)))/8)/(3*c)
3.5.39.3.1 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*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] :> 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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -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 = 2.72 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.55
| method | result | size |
| derivativedivides | \(-\frac {2 a^{2}}{3 c^{3} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (\frac {15}{32} a^{2} d^{2}-\frac {7}{16} a b c d -\frac {1}{32} b^{2} c^{2}\right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-22 a b c d +3 b^{2} c^{2}\right ) \sqrt {x}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-42 a b c d -3 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d c}\right )}{c^{3}}\) | \(220\) |
| default | \(-\frac {2 a^{2}}{3 c^{3} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (\frac {15}{32} a^{2} d^{2}-\frac {7}{16} a b c d -\frac {1}{32} b^{2} c^{2}\right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-22 a b c d +3 b^{2} c^{2}\right ) \sqrt {x}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-42 a b c d -3 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d c}\right )}{c^{3}}\) | \(220\) |
| risch | \(-\frac {2 a^{2}}{3 c^{3} x^{\frac {3}{2}}}-\frac {\frac {2 \left (\frac {15}{32} a^{2} d^{2}-\frac {7}{16} a b c d -\frac {1}{32} b^{2} c^{2}\right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-22 a b c d +3 b^{2} c^{2}\right ) \sqrt {x}}{16 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-42 a b c d -3 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 d c}}{c^{3}}\) | \(221\) |
-2/3*a^2/c^3/x^(3/2)-2/c^3*(((15/32*a^2*d^2-7/16*a*b*c*d-1/32*b^2*c^2)*x^( 5/2)+1/32*c*(19*a^2*d^2-22*a*b*c*d+3*b^2*c^2)/d*x^(1/2))/(d*x^2+c)^2+1/256 *(77*a^2*d^2-42*a*b*c*d-3*b^2*c^2)/d*(c/d)^(1/4)/c*2^(1/2)*(ln((x+(c/d)^(1 /4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2 )))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x ^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1322, normalized size of antiderivative = 3.29 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-1/192*(3*(c^3*d^3*x^6 + 2*c^4*d^2*x^4 + c^5*d*x^2)*(-(81*b^8*c^8 + 4536*a *b^7*c^7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4* b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^6 - 766975 44*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4)*log(c^4*d*(-(81*b^8*c ^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1 457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d ^6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4) - (3*b^2*c ^2 + 42*a*b*c*d - 77*a^2*d^2)*sqrt(x)) + 3*(I*c^3*d^3*x^6 + 2*I*c^4*d^2*x^ 4 + I*c^5*d*x^2)*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3* d^5 + 57274140*a^6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/ (c^15*d^5))^(1/4)*log(I*c^4*d*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2 *b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 - 13854456 *a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153 041*a^8*d^8)/(c^15*d^5))^(1/4) - (3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*sqr t(x)) + 3*(-I*c^3*d^3*x^6 - 2*I*c^4*d^2*x^4 - I*c^5*d*x^2)*(-(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^5*c^5*d^3 - 14579 46*a^4*b^4*c^4*d^4 - 13854456*a^5*b^3*c^3*d^5 + 57274140*a^6*b^2*c^2*d^6 - 76697544*a^7*b*c*d^7 + 35153041*a^8*d^8)/(c^15*d^5))^(1/4)*log(-I*c^4*d*( -(81*b^8*c^8 + 4536*a*b^7*c^7*d + 86940*a^2*b^6*c^6*d^2 + 539784*a^3*b^...
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx=-\frac {32 \, a^{2} c^{2} d - {\left (3 \, b^{2} c^{2} d + 42 \, a b c d^{2} - 77 \, a^{2} d^{3}\right )} x^{4} + {\left (9 \, b^{2} c^{3} - 66 \, a b c^{2} d + 121 \, a^{2} c d^{2}\right )} x^{2}}{48 \, {\left (c^{3} d^{3} x^{\frac {11}{2}} + 2 \, c^{4} d^{2} x^{\frac {7}{2}} + c^{5} d x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, b^{2} c^{2} + 42 \, a b c d - 77 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, c^{3} d} \]
-1/48*(32*a^2*c^2*d - (3*b^2*c^2*d + 42*a*b*c*d^2 - 77*a^2*d^3)*x^4 + (9*b ^2*c^3 - 66*a*b*c^2*d + 121*a^2*c*d^2)*x^2)/(c^3*d^3*x^(11/2) + 2*c^4*d^2* x^(7/2) + c^5*d*x^(3/2)) + 1/128*(2*sqrt(2)*(3*b^2*c^2 + 42*a*b*c*d - 77*a ^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/s qrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(3*b^2*c ^2 + 42*a*b*c*d - 77*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d) )) + sqrt(2)*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^( 1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(3*b^2*c^2 + 42*a*b*c*d - 77*a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d) *x + sqrt(c))/(c^(3/4)*d^(1/4)))/(c^3*d)
Time = 0.30 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx=-\frac {2 \, a^{2}}{3 \, c^{3} x^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{2}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{2}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{2}} - \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{2}} + \frac {b^{2} c^{2} d x^{\frac {5}{2}} + 14 \, a b c d^{2} x^{\frac {5}{2}} - 15 \, a^{2} d^{3} x^{\frac {5}{2}} - 3 \, b^{2} c^{3} \sqrt {x} + 22 \, a b c^{2} d \sqrt {x} - 19 \, a^{2} c d^{2} \sqrt {x}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{3} d} \]
-2/3*a^2/(c^3*x^(3/2)) + 1/64*sqrt(2)*(3*(c*d^3)^(1/4)*b^2*c^2 + 42*(c*d^3 )^(1/4)*a*b*c*d - 77*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c /d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^2) + 1/64*sqrt(2)*(3*(c*d^3)^(1 /4)*b^2*c^2 + 42*(c*d^3)^(1/4)*a*b*c*d - 77*(c*d^3)^(1/4)*a^2*d^2)*arctan( -1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^2) + 1/ 128*sqrt(2)*(3*(c*d^3)^(1/4)*b^2*c^2 + 42*(c*d^3)^(1/4)*a*b*c*d - 77*(c*d^ 3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^4*d^ 2) - 1/128*sqrt(2)*(3*(c*d^3)^(1/4)*b^2*c^2 + 42*(c*d^3)^(1/4)*a*b*c*d - 7 7*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d)) /(c^4*d^2) + 1/16*(b^2*c^2*d*x^(5/2) + 14*a*b*c*d^2*x^(5/2) - 15*a^2*d^3*x ^(5/2) - 3*b^2*c^3*sqrt(x) + 22*a*b*c^2*d*sqrt(x) - 19*a^2*c*d^2*sqrt(x))/ ((d*x^2 + c)^2*c^3*d)
Time = 5.49 (sec) , antiderivative size = 1508, normalized size of antiderivative = 3.75 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
(atan((((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a* b^3*c^12*d^7 - 105971712*a^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*d^8) - ((3 *b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d^7 - 80740352*a^2*c ^13*d^9 + 44040192*a*b*c^14*d^8))/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 7 7*a^2*d^2 + 42*a*b*c*d)*1i)/(64*(-c)^(15/4)*d^(5/4)) + ((x^(1/2)*(97140736 *a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 105971712*a ^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*d^8) + ((3*b^2*c^2 - 77*a^2*d^2 + 42 *a*b*c*d)*(3145728*b^2*c^15*d^7 - 80740352*a^2*c^13*d^9 + 44040192*a*b*c^1 4*d^8))/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*1i )/(64*(-c)^(15/4)*d^(5/4)))/(((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4 *c^13*d^6 + 4128768*a*b^3*c^12*d^7 - 105971712*a^3*b*c^10*d^9 + 21331968*a ^2*b^2*c^11*d^8) - ((3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^1 5*d^7 - 80740352*a^2*c^13*d^9 + 44040192*a*b*c^14*d^8))/(64*(-c)^(15/4)*d^ (5/4)))*(3*b^2*c^2 - 77*a^2*d^2 + 42*a*b*c*d))/(64*(-c)^(15/4)*d^(5/4)) - ((x^(1/2)*(97140736*a^4*c^9*d^10 + 147456*b^4*c^13*d^6 + 4128768*a*b^3*c^1 2*d^7 - 105971712*a^3*b*c^10*d^9 + 21331968*a^2*b^2*c^11*d^8) + ((3*b^2*c^ 2 - 77*a^2*d^2 + 42*a*b*c*d)*(3145728*b^2*c^15*d^7 - 80740352*a^2*c^13*d^9 + 44040192*a*b*c^14*d^8))/(64*(-c)^(15/4)*d^(5/4)))*(3*b^2*c^2 - 77*a^2*d ^2 + 42*a*b*c*d))/(64*(-c)^(15/4)*d^(5/4))))*(3*b^2*c^2 - 77*a^2*d^2 + 42* a*b*c*d)*1i)/(32*(-c)^(15/4)*d^(5/4)) - ((2*a^2)/(3*c) - (x^4*(3*b^2*c^...