Integrand size = 24, antiderivative size = 399 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) x^{3/2}}{16 c^3 d \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {473, 468, 296, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 c \left (c+d x^2\right )^2}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {x^{3/2} \left (\frac {5 a (2 b c-9 a d)}{c^2}+\frac {3 b^2}{d}\right )}{16 c \left (c+d x^2\right )}+\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}} \]
[In]
[Out]
Rule 210
Rule 296
Rule 303
Rule 335
Rule 468
Rule 473
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {2 \int \frac {\sqrt {x} \left (\frac {1}{2} a (2 b c-9 a d)+\frac {1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^3} \, dx}{c} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {1}{8} \left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) \int \frac {\sqrt {x}}{\left (c+d x^2\right )^2} \, dx \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{32 c} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^3 d^{3/2}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^3 d^{3/2}} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^3 d^2}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^3 d^2}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{13/4} d^{7/4}} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {9 a^2 d}{c}\right ) x^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {3 b^2}{d}+\frac {5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{c} d^{3/4} \left (b^2 c^2 x^2 \left (c-3 d x^2\right )-2 a b c d x^2 \left (9 c+5 d x^2\right )+a^2 d \left (32 c^2+81 c d x^2+45 d^2 x^4\right )\right )}{\sqrt {x} \left (c+d x^2\right )^2}-\sqrt {2} \left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{64 c^{13/4} d^{7/4}} \]
[In]
[Out]
Time = 2.72 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(-\frac {2 a^{2}}{c^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {13}{32} a^{2} d^{2}-\frac {5}{16} a b c d -\frac {3}{32} b^{2} c^{2}\right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-18 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (45 a^{2} d^{2}-10 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3}}\) | \(216\) |
default | \(-\frac {2 a^{2}}{c^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {13}{32} a^{2} d^{2}-\frac {5}{16} a b c d -\frac {3}{32} b^{2} c^{2}\right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-18 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (45 a^{2} d^{2}-10 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3}}\) | \(216\) |
risch | \(-\frac {2 a^{2}}{c^{3} \sqrt {x}}-\frac {\frac {2 \left (\frac {13}{32} a^{2} d^{2}-\frac {5}{16} a b c d -\frac {3}{32} b^{2} c^{2}\right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-18 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{16 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (45 a^{2} d^{2}-10 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{3}}\) | \(217\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1540, normalized size of antiderivative = 3.86 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {32 \, a^{2} c^{2} d - {\left (3 \, b^{2} c^{2} d + 10 \, a b c d^{2} - 45 \, a^{2} d^{3}\right )} x^{4} + {\left (b^{2} c^{3} - 18 \, a b c^{2} d + 81 \, a^{2} c d^{2}\right )} x^{2}}{16 \, {\left (c^{3} d^{3} x^{\frac {9}{2}} + 2 \, c^{4} d^{2} x^{\frac {5}{2}} + c^{5} d \sqrt {x}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 10 \, a b c d - 45 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c^{3} d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {2 \, a^{2}}{c^{3} \sqrt {x}} + \frac {3 \, b^{2} c^{2} d x^{\frac {7}{2}} + 10 \, a b c d^{2} x^{\frac {7}{2}} - 13 \, a^{2} d^{3} x^{\frac {7}{2}} - b^{2} c^{3} x^{\frac {3}{2}} + 18 \, a b c^{2} d x^{\frac {3}{2}} - 17 \, a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{3} d} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{4}} - \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{4}} \]
[In]
[Out]
Time = 5.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{13/4}\,d^{7/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{13/4}\,d^{7/4}}-\frac {\frac {2\,a^2}{c}-\frac {x^4\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{16\,c^3}+\frac {x^2\,\left (81\,a^2\,d^2-18\,a\,b\,c\,d+b^2\,c^2\right )}{16\,c^2\,d}}{c^2\,\sqrt {x}+d^2\,x^{9/2}+2\,c\,d\,x^{5/2}} \]
[In]
[Out]