Integrand size = 24, antiderivative size = 364 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} d^{11/4}}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} d^{11/4}}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} d^{11/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} d^{11/4}} \]
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Time = 0.22 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {474, 468, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} d^{11/4}}+\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{9/4} d^{11/4}}+\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} d^{11/4}}-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} d^{11/4}}-\frac {x^{3/2} (5 a d+11 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{3/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
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Rule 210
Rule 303
Rule 335
Rule 468
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {\sqrt {x} \left (\frac {1}{2} \left (-8 a^2 d^2+3 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2} \\ & = \frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{32 c^2 d^2} \\ & = \frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^2 d^2} \\ & = \frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (21 b^2 c^2+6 a b c d+5 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^2 d^{5/2}}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^2 d^{5/2}} \\ & = \frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^2 d^3}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^2 d^3}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}} \\ & = \frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}} \\ & = \frac {(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}}+\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 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^{9/4} d^{11/4}} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{c} d^{3/4} (b c-a d) x^{3/2} \left (a d \left (9 c+5 d x^2\right )+b c \left (7 c+11 d x^2\right )\right )}{\left (c+d x^2\right )^2}-\sqrt {2} \left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{64 c^{9/4} d^{11/4}} \]
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Time = 2.72 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(\frac {\frac {\left (5 a^{2} d^{2}+6 a b c d -11 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{16 c^{2} d}+\frac {\left (9 a^{2} d^{2}-2 a b c d -7 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{16 c \,d^{2}}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 d^{3} c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(213\) |
default | \(\frac {\frac {\left (5 a^{2} d^{2}+6 a b c d -11 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{16 c^{2} d}+\frac {\left (9 a^{2} d^{2}-2 a b c d -7 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{16 c \,d^{2}}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 d^{3} c^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\) | \(213\) |
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 1532, normalized size of antiderivative = 4.21 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.35 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=-\frac {{\left (11 \, b^{2} c^{2} d - 6 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{\frac {7}{2}} + {\left (7 \, b^{2} c^{3} + 2 \, a b c^{2} d - 9 \, a^{2} c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )}} + \frac {{\left (21 \, b^{2} c^{2} + 6 \, a b c d + 5 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c^{2} d^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=-\frac {11 \, b^{2} c^{2} d x^{\frac {7}{2}} - 6 \, a b c d^{2} x^{\frac {7}{2}} - 5 \, a^{2} d^{3} x^{\frac {7}{2}} + 7 \, b^{2} c^{3} x^{\frac {3}{2}} + 2 \, a b c^{2} d x^{\frac {3}{2}} - 9 \, a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{2} d^{2}} + \frac {\sqrt {2} {\left (21 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{3} d^{5}} + \frac {\sqrt {2} {\left (21 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{3} d^{5}} - \frac {\sqrt {2} {\left (21 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{3} d^{5}} + \frac {\sqrt {2} {\left (21 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{3} d^{5}} \]
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Time = 5.23 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (5\,a^2\,d^2+6\,a\,b\,c\,d+21\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{9/4}\,d^{11/4}}-\frac {\frac {x^{3/2}\,\left (-9\,a^2\,d^2+2\,a\,b\,c\,d+7\,b^2\,c^2\right )}{16\,c\,d^2}-\frac {x^{7/2}\,\left (5\,a^2\,d^2+6\,a\,b\,c\,d-11\,b^2\,c^2\right )}{16\,c^2\,d}}{c^2+2\,c\,d\,x^2+d^2\,x^4}-\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (5\,a^2\,d^2+6\,a\,b\,c\,d+21\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{9/4}\,d^{11/4}} \]
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