Integrand size = 24, antiderivative size = 333 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}-\frac {\left (b^2 c^2-2 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 c^2 d \left (c+d x^2\right )}-\frac {(b c-a d) (3 b c+5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} d^{7/4}}+\frac {(b c-a d) (3 b c+5 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} d^{7/4}}+\frac {(b c-a d) (3 b c+5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}}-\frac {(b c-a d) (3 b c+5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}} \]
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Time = 0.24 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {473, 468, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=\frac {x^{3/2} \left (-\frac {5 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{2 c \left (c+d x^2\right )}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}-\frac {(b c-a d) (5 a d+3 b c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} d^{7/4}}+\frac {(b c-a d) (5 a d+3 b c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{9/4} d^{7/4}}+\frac {(b c-a d) (5 a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}}-\frac {(b c-a d) (5 a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}} \]
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Rule 210
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 )}+\frac {2 \int \frac {\sqrt {x} \left (\frac {1}{2} a (2 b c-5 a d)+\frac {1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^2} \, dx}{c} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x^{3/2}}{2 c \left (c+d x^2\right )}+\frac {((b c-a d) (3 b c+5 a d)) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{4 c^2 d} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x^{3/2}}{2 c \left (c+d x^2\right )}+\frac {((b c-a d) (3 b c+5 a d)) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^2 d} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x^{3/2}}{2 c \left (c+d x^2\right )}-\frac {((b c-a d) (3 b c+5 a d)) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^2 d^{3/2}}+\frac {((b c-a d) (3 b c+5 a d)) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^2 d^{3/2}} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x^{3/2}}{2 c \left (c+d x^2\right )}+\frac {((b c-a d) (3 b c+5 a d)) \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 )}{8 c^2 d^2}+\frac {((b c-a d) (3 b c+5 a d)) \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 )}{8 c^2 d^2}+\frac {((b c-a d) (3 b c+5 a d)) \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 )}{8 \sqrt {2} c^{9/4} d^{7/4}}+\frac {((b c-a d) (3 b c+5 a d)) \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 )}{8 \sqrt {2} c^{9/4} d^{7/4}} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x^{3/2}}{2 c \left (c+d x^2\right )}+\frac {(b c-a d) (3 b c+5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}}-\frac {(b c-a d) (3 b c+5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}}+\frac {((b c-a d) (3 b c+5 a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} d^{7/4}}-\frac {((b c-a d) (3 b c+5 a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} d^{7/4}} \\ & = -\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x^{3/2}}{2 c \left (c+d x^2\right )}-\frac {(b c-a d) (3 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} d^{7/4}}+\frac {(b c-a d) (3 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{9/4} d^{7/4}}+\frac {(b c-a d) (3 b c+5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}}-\frac {(b c-a d) (3 b c+5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{9/4} d^{7/4}} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.63 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=\frac {-\frac {4 \sqrt [4]{c} d^{3/4} \left (b^2 c^2 x^2-2 a b c d x^2+a^2 d \left (4 c+5 d x^2\right )\right )}{\sqrt {x} \left (c+d x^2\right )}-\sqrt {2} \left (3 b^2 c^2+2 a b c d-5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (3 b^2 c^2+2 a b c d-5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{8 c^{9/4} d^{7/4}} \]
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Time = 2.80 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {2 a^{2}}{c^{2} \sqrt {x}}-\frac {\left (2 a d -2 b c \right ) \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (5 a d +3 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{2}}\) | \(167\) |
derivativedivides | \(-\frac {2 a^{2}}{c^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (5 a^{2} d^{2}-2 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{2}}\) | \(185\) |
default | \(-\frac {2 a^{2}}{c^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (5 a^{2} d^{2}-2 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{2}}\) | \(185\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1449, normalized size of antiderivative = 4.35 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=-\frac {4 \, a^{2} c d + {\left (b^{2} c^{2} - 2 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{2}}{2 \, {\left (c^{2} d^{2} x^{\frac {5}{2}} + c^{3} d \sqrt {x}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d - 5 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, c^{2} d} \]
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Time = 0.28 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=-\frac {b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 5 \, a^{2} d^{2} x^{2} + 4 \, a^{2} c d}{2 \, {\left (d x^{\frac {5}{2}} + c \sqrt {x}\right )} c^{2} d} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{3} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{3} d^{4}} - \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{3} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{3} d^{4}} \]
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Time = 5.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d+3\,b\,c\right )}{4\,{\left (-c\right )}^{9/4}\,d^{7/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d+3\,b\,c\right )}{4\,{\left (-c\right )}^{9/4}\,d^{7/4}}-\frac {\frac {2\,a^2}{c}+\frac {x^2\,\left (5\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^2\,d}}{c\,\sqrt {x}+d\,x^{5/2}} \]
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