Integrand size = 24, antiderivative size = 439 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{80 c^3 d \sqrt {x} \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}} \]
-2/5*a^2/c/x^(5/2)/(d*x^2+c)^2-1/64*(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))*arc tan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(17/4)/d^(3/4)*2^(1/2)+1/64*(5*b^ 2*c^2-9*a*d*(-13*a*d+10*b*c))*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^ (17/4)/d^(3/4)*2^(1/2)+1/128*(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))*ln(c^(1/2) +x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(17/4)/d^(3/4)*2^(1/2)-1/128 *(5*b^2*c^2-9*a*d*(-13*a*d+10*b*c))*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2 ^(1/2)*x^(1/2))/c^(17/4)/d^(3/4)*2^(1/2)+1/16*(5*b^2*c^2-9*a*d*(-13*a*d+10 *b*c))/c^4/d/x^(1/2)+1/20*(-13*a^2*d^2+10*a*b*c*d-5*b^2*c^2)/c^2/d/(d*x^2+ c)^2/x^(1/2)+1/80*(-5*b^2*c^2+9*a*d*(-13*a*d+10*b*c))/c^3/d/(d*x^2+c)/x^(1 /2)
Time = 0.78 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.59 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{c} \left (5 b^2 c^2 x^4 \left (9 c+5 d x^2\right )-10 a b c x^2 \left (32 c^2+81 c d x^2+45 d^2 x^4\right )+a^2 \left (-32 c^3+416 c^2 d x^2+1053 c d^2 x^4+585 d^3 x^6\right )\right )}{x^{5/2} \left (c+d x^2\right )^2}-\frac {5 \sqrt {2} \left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{d^{3/4}}-\frac {5 \sqrt {2} \left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{d^{3/4}}}{320 c^{17/4}} \]
((4*c^(1/4)*(5*b^2*c^2*x^4*(9*c + 5*d*x^2) - 10*a*b*c*x^2*(32*c^2 + 81*c*d *x^2 + 45*d^2*x^4) + a^2*(-32*c^3 + 416*c^2*d*x^2 + 1053*c*d^2*x^4 + 585*d ^3*x^6)))/(x^(5/2)*(c + d*x^2)^2) - (5*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 1 17*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]) ])/d^(3/4) - (5*Sqrt[2]*(5*b^2*c^2 - 90*a*b*c*d + 117*a^2*d^2)*ArcTanh[(Sq rt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/d^(3/4))/(320*c^(17 /4))
Time = 0.60 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.82, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {365, 27, 362, 253, 264, 266, 826, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {5 b^2 c x^2+a (10 b c-13 a d)}{2 x^{3/2} \left (d x^2+c\right )^3}dx}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 b^2 c x^2+a (10 b c-13 a d)}{x^{3/2} \left (d x^2+c\right )^3}dx}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \int \frac {1}{x^{3/2} \left (d x^2+c\right )^2}dx}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \int \frac {1}{x^{3/2} \left (d x^2+c\right )}dx}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {d \int \frac {\sqrt {x}}{d x^2+c}dx}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \int \frac {x}{d x^2+c}d\sqrt {x}}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{4 \sqrt {x} \left (c+d x^2\right )^2}-\frac {1}{8} \left (\frac {5 b^2 c}{d}-\frac {9 a (10 b c-13 a d)}{c}\right ) \left (\frac {5 \left (-\frac {2 d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{c}-\frac {2}{c \sqrt {x}}\right )}{4 c}+\frac {1}{2 c \sqrt {x} \left (c+d x^2\right )}\right )}{5 c}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}\) |
(-2*a^2)/(5*c*x^(5/2)*(c + d*x^2)^2) + ((10*a*b - (5*b^2*c)/d - (13*a^2*d) /c)/(4*Sqrt[x]*(c + d*x^2)^2) - (((5*b^2*c)/d - (9*a*(10*b*c - 13*a*d))/c) *(1/(2*c*Sqrt[x]*(c + d*x^2)) + (5*(-2/(c*Sqrt[x]) - (2*d*((-(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[d]) - (-1/2*Log[Sqrt[c] - Sqrt[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[d])))/c))/(4*c)))/8)/(5*c)
3.5.40.3.1 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] :> 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[((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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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 = 2.71 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(-\frac {2 a \left (-15 a d \,x^{2}+10 c b \,x^{2}+a c \right )}{5 c^{4} x^{\frac {5}{2}}}+\frac {\frac {2 \left (\frac {21}{32} a^{2} d^{3}-\frac {13}{16} a b c \,d^{2}+\frac {5}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}+\frac {c \left (25 a^{2} d^{2}-34 a b c d +9 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{16}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (\frac {117}{32} a^{2} d^{2}-\frac {45}{16} a b c d +\frac {5}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{4}}\) | \(233\) |
| derivativedivides | \(\frac {\frac {2 \left (\left (\frac {21}{32} a^{2} d^{3}-\frac {13}{16} a b c \,d^{2}+\frac {5}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}+\frac {c \left (25 a^{2} d^{2}-34 a b c d +9 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (\frac {117}{32} a^{2} d^{2}-\frac {45}{16} a b c d +\frac {5}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{4}}-\frac {2 a^{2}}{5 c^{3} x^{\frac {5}{2}}}+\frac {2 a \left (3 a d -2 b c \right )}{c^{4} \sqrt {x}}\) | \(235\) |
| default | \(\frac {\frac {2 \left (\left (\frac {21}{32} a^{2} d^{3}-\frac {13}{16} a b c \,d^{2}+\frac {5}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}+\frac {c \left (25 a^{2} d^{2}-34 a b c d +9 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (\frac {117}{32} a^{2} d^{2}-\frac {45}{16} a b c d +\frac {5}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{4}}-\frac {2 a^{2}}{5 c^{3} x^{\frac {5}{2}}}+\frac {2 a \left (3 a d -2 b c \right )}{c^{4} \sqrt {x}}\) | \(235\) |
-2/5*a*(-15*a*d*x^2+10*b*c*x^2+a*c)/c^4/x^(5/2)+1/c^4*(2*((21/32*a^2*d^3-1 3/16*a*b*c*d^2+5/32*b^2*c^2*d)*x^(7/2)+1/32*c*(25*a^2*d^2-34*a*b*c*d+9*b^2 *c^2)*x^(3/2))/(d*x^2+c)^2+1/4*(117/32*a^2*d^2-45/16*a*b*c*d+5/32*b^2*c^2) /d/(c/d)^(1/4)*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 = 0.29 (sec) , antiderivative size = 1557, normalized size of antiderivative = 3.55 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
1/320*(5*(c^4*d^2*x^7 + 2*c^5*d*x^5 + c^6*x^3)*(-(625*b^8*c^8 - 45000*a*b^ 7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a ^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 5 76580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(1/4)*log(c^13*d^2*( -(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3 *b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697 317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17* d^3))^(3/4) + (125*b^6*c^6 - 6750*a*b^5*c^5*d + 130275*a^2*b^4*c^4*d^2 - 1 044900*a^3*b^3*c^3*d^3 + 3048435*a^4*b^2*c^2*d^4 - 3696030*a^5*b*c*d^5 + 1 601613*a^6*d^6)*sqrt(x)) - 5*(I*c^4*d^2*x^7 + 2*I*c^5*d*x^5 + I*c^6*x^3)*( -(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^2*b^6*c^6*d^2 - 17739000*a^3 *b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415092600*a^5*b^3*c^3*d^5 + 697 317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17* d^3))^(1/4)*log(I*c^13*d^2*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 1273500*a^ 2*b^6*c^6*d^2 - 17739000*a^3*b^5*c^5*d^3 + 124525350*a^4*b^4*c^4*d^4 - 415 092600*a^5*b^3*c^3*d^5 + 697317660*a^6*b^2*c^2*d^6 - 576580680*a^7*b*c*d^7 + 187388721*a^8*d^8)/(c^17*d^3))^(3/4) + (125*b^6*c^6 - 6750*a*b^5*c^5*d + 130275*a^2*b^4*c^4*d^2 - 1044900*a^3*b^3*c^3*d^3 + 3048435*a^4*b^2*c^2*d ^4 - 3696030*a^5*b*c*d^5 + 1601613*a^6*d^6)*sqrt(x)) - 5*(-I*c^4*d^2*x^7 - 2*I*c^5*d*x^5 - I*c^6*x^3)*(-(625*b^8*c^8 - 45000*a*b^7*c^7*d + 127350...
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {5 \, {\left (5 \, b^{2} c^{2} d - 90 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{6} - 32 \, a^{2} c^{3} + 9 \, {\left (5 \, b^{2} c^{3} - 90 \, a b c^{2} d + 117 \, a^{2} c d^{2}\right )} x^{4} - 32 \, {\left (10 \, a b c^{3} - 13 \, a^{2} c^{2} d\right )} x^{2}}{80 \, {\left (c^{4} d^{2} x^{\frac {13}{2}} + 2 \, c^{5} d x^{\frac {9}{2}} + c^{6} x^{\frac {5}{2}}\right )}} + \frac {{\left (5 \, b^{2} c^{2} - 90 \, a b c d + 117 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c^{4}} \]
1/80*(5*(5*b^2*c^2*d - 90*a*b*c*d^2 + 117*a^2*d^3)*x^6 - 32*a^2*c^3 + 9*(5 *b^2*c^3 - 90*a*b*c^2*d + 117*a^2*c*d^2)*x^4 - 32*(10*a*b*c^3 - 13*a^2*c^2 *d)*x^2)/(c^4*d^2*x^(13/2) + 2*c^5*d*x^(9/2) + c^6*x^(5/2)) + 1/128*(5*b^2 *c^2 - 90*a*b*c*d + 117*a^2*d^2)*(2*sqrt(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(sqrt(c)*sq rt(d))*sqrt(d)) + 2*sqrt(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(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^( 1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/c^4
Time = 0.32 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {5 \, b^{2} c^{2} d x^{\frac {7}{2}} - 26 \, a b c d^{2} x^{\frac {7}{2}} + 21 \, a^{2} d^{3} x^{\frac {7}{2}} + 9 \, b^{2} c^{3} x^{\frac {3}{2}} - 34 \, a b c^{2} d x^{\frac {3}{2}} + 25 \, a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{4}} - \frac {2 \, {\left (10 \, a b c x^{2} - 15 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{4} x^{\frac {5}{2}}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{5} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{5} d^{3}} \]
1/16*(5*b^2*c^2*d*x^(7/2) - 26*a*b*c*d^2*x^(7/2) + 21*a^2*d^3*x^(7/2) + 9* b^2*c^3*x^(3/2) - 34*a*b*c^2*d*x^(3/2) + 25*a^2*c*d^2*x^(3/2))/((d*x^2 + c )^2*c^4) - 2/5*(10*a*b*c*x^2 - 15*a^2*d*x^2 + a^2*c)/(c^4*x^(5/2)) + 1/64* sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^ (3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^ (1/4))/(c^5*d^3) + 1/64*sqrt(2)*(5*(c*d^3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4 )*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^ (1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^5*d^3) - 1/128*sqrt(2)*(5*(c*d^3)^(3/4) *b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt( 2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^5*d^3) + 1/128*sqrt(2)*(5*(c*d^ 3)^(3/4)*b^2*c^2 - 90*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*l og(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^5*d^3)
Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {\frac {9\,x^4\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{80\,c^3}-\frac {2\,a^2}{5\,c}+\frac {2\,a\,x^2\,\left (13\,a\,d-10\,b\,c\right )}{5\,c^2}+\frac {d\,x^6\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{16\,c^4}}{c^2\,x^{5/2}+d^2\,x^{13/2}+2\,c\,d\,x^{9/2}}+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{17/4}\,d^{3/4}}-\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{17/4}\,d^{3/4}} \]
((9*x^4*(117*a^2*d^2 + 5*b^2*c^2 - 90*a*b*c*d))/(80*c^3) - (2*a^2)/(5*c) + (2*a*x^2*(13*a*d - 10*b*c))/(5*c^2) + (d*x^6*(117*a^2*d^2 + 5*b^2*c^2 - 9 0*a*b*c*d))/(16*c^4))/(c^2*x^(5/2) + d^2*x^(13/2) + 2*c*d*x^(9/2)) + (atan ((d^(1/4)*x^(1/2))/(-c)^(1/4))*(117*a^2*d^2 + 5*b^2*c^2 - 90*a*b*c*d))/(32 *(-c)^(17/4)*d^(3/4)) - (atanh((d^(1/4)*x^(1/2))/(-c)^(1/4))*(117*a^2*d^2 + 5*b^2*c^2 - 90*a*b*c*d))/(32*(-c)^(17/4)*d^(3/4))