Integrand size = 24, antiderivative size = 439 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-10 a b c d+13 a^2 d^2}{20 c^2 d \sqrt {x} \left (c+d x^2\right )^2}-\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{80 c^3 d \sqrt {x} \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}} \]
[Out]
Time = 0.33 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {473, 468, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {-\frac {13 a^2 d}{c}+10 a b-\frac {5 b^2 c}{d}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-9 a d (10 b c-13 a d)\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} d^{3/4}}+\frac {5 b^2 c^2-9 a d (10 b c-13 a d)}{16 c^4 d \sqrt {x}} \]
[In]
[Out]
Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 468
Rule 473
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {2 \int \frac {\frac {1}{2} a (10 b c-13 a d)+\frac {5}{2} b^2 c x^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx}{5 c} \\ & = -\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}+\frac {1}{40} \left (-\frac {5 b^2}{d}+\frac {9 a (10 b c-13 a d)}{c^2}\right ) \int \frac {1}{x^{3/2} \left (c+d x^2\right )^2} \, dx \\ & = -\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}-\frac {\left (\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}\right ) \int \frac {1}{x^{3/2} \left (c+d x^2\right )} \, dx}{32 c} \\ & = \frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{32 c^4} \\ & = \frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^4} \\ & = \frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^4 \sqrt {d}}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^4 \sqrt {d}} \\ & = \frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^4 d}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^4 d}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}} \\ & = \frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}} \\ & = \frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{16 c^2 \sqrt {x}}-\frac {2 a^2}{5 c x^{5/2} \left (c+d x^2\right )^2}+\frac {10 a b-\frac {5 b^2 c}{d}-\frac {13 a^2 d}{c}}{20 c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\frac {5 b^2}{d}-\frac {9 a (10 b c-13 a d)}{c^2}}{80 c \sqrt {x} \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}+\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}}-\frac {\left (5 b^2 c^2-90 a b c d+117 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^{17/4} d^{3/4}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.59 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{c} \left (5 b^2 c^2 x^4 \left (9 c+5 d x^2\right )-10 a b c x^2 \left (32 c^2+81 c d x^2+45 d^2 x^4\right )+a^2 \left (-32 c^3+416 c^2 d x^2+1053 c d^2 x^4+585 d^3 x^6\right )\right )}{x^{5/2} \left (c+d x^2\right )^2}-\frac {5 \sqrt {2} \left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{d^{3/4}}-\frac {5 \sqrt {2} \left (5 b^2 c^2-90 a b c d+117 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{d^{3/4}}}{320 c^{17/4}} \]
[In]
[Out]
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 | \(\frac {\frac {2 \left (\left (\frac {21}{32} a^{2} d^{3}-\frac {13}{16} a b c \,d^{2}+\frac {5}{32} b^{2} c^{2} d \right ) x^{\frac {7}{2}}+\frac {c \left (25 a^{2} d^{2}-34 a b c d +9 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (\frac {117}{32} a^{2} d^{2}-\frac {45}{16} a b c d +\frac {5}{32} b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{4}}-\frac {2 a^{2}}{5 c^{3} x^{\frac {5}{2}}}+\frac {2 a \left (3 a d -2 b c \right )}{c^{4} \sqrt {x}}\) | \(235\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 1557, normalized size of antiderivative = 3.55 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {5 \, {\left (5 \, b^{2} c^{2} d - 90 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{6} - 32 \, a^{2} c^{3} + 9 \, {\left (5 \, b^{2} c^{3} - 90 \, a b c^{2} d + 117 \, a^{2} c d^{2}\right )} x^{4} - 32 \, {\left (10 \, a b c^{3} - 13 \, a^{2} c^{2} d\right )} x^{2}}{80 \, {\left (c^{4} d^{2} x^{\frac {13}{2}} + 2 \, c^{5} d x^{\frac {9}{2}} + c^{6} x^{\frac {5}{2}}\right )}} + \frac {{\left (5 \, b^{2} c^{2} - 90 \, a b c d + 117 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c^{4}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {5 \, b^{2} c^{2} d x^{\frac {7}{2}} - 26 \, a b c d^{2} x^{\frac {7}{2}} + 21 \, a^{2} d^{3} x^{\frac {7}{2}} + 9 \, b^{2} c^{3} x^{\frac {3}{2}} - 34 \, a b c^{2} d x^{\frac {3}{2}} + 25 \, a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{4}} - \frac {2 \, {\left (10 \, a b c x^{2} - 15 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{4} x^{\frac {5}{2}}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{5} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 90 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{5} d^{3}} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a+b x^2\right )^2}{x^{7/2} \left (c+d x^2\right )^3} \, dx=\frac {\frac {9\,x^4\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{80\,c^3}-\frac {2\,a^2}{5\,c}+\frac {2\,a\,x^2\,\left (13\,a\,d-10\,b\,c\right )}{5\,c^2}+\frac {d\,x^6\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{16\,c^4}}{c^2\,x^{5/2}+d^2\,x^{13/2}+2\,c\,d\,x^{9/2}}+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{17/4}\,d^{3/4}}-\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (117\,a^2\,d^2-90\,a\,b\,c\,d+5\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{17/4}\,d^{3/4}} \]
[In]
[Out]