Integrand size = 30, antiderivative size = 457 \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.21 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 210
Rule 217
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x^2\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a^2 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (2 a^2 d^3 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a^{3/2} d^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a^{3/2} d^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a^{5/4} d^{7/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{2 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a^{5/4} d^{7/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{2 \sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a^{3/2} d^4 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{2 b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a^{3/2} d^4 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{2 b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a^{5/4} d^{7/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a^{5/4} d^{7/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {2 a d^3 \sqrt {d x} \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {2 d (d x)^{5/2} \left (a+b x^2\right )}{5 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{5/4} d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.36 \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {d^3 \sqrt {d x} \left (a+b x^2\right ) \left (4 \sqrt [4]{b} \sqrt {x} \left (-5 a+b x^2\right )-5 \sqrt {2} a^{5/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+5 \sqrt {2} a^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{10 b^{9/4} \sqrt {x} \sqrt {\left (a+b x^2\right )^2}} \]
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Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {2 \left (-b \,x^{2}+5 a \right ) x \,d^{4} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 b^{2} \sqrt {d x}\, \left (b \,x^{2}+a \right )}+\frac {a \,d^{3} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{4 b^{2} \left (b \,x^{2}+a \right )}\) | \(208\) |
default | \(\frac {\left (b \,x^{2}+a \right ) d \left (8 b \left (d x \right )^{\frac {5}{2}}+5 a \,d^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right )+10 a \,d^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+10 a \,d^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-40 a \,d^{2} \sqrt {d x}\right )}{20 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b^{2}}\) | \(242\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.48 \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {5 \, \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {d x} a d^{3} + \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2}\right ) + 5 i \, \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {d x} a d^{3} + i \, \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2}\right ) - 5 i \, \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {d x} a d^{3} - i \, \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2}\right ) - 5 \, \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {d x} a d^{3} - \left (-\frac {a^{5} d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2}\right ) + 4 \, {\left (b d^{3} x^{2} - 5 \, a d^{3}\right )} \sqrt {d x}}{10 \, b^{2}} \]
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Timed out. \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.58 \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\frac {5 \, {\left (\frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a^{2}}{b^{2}} + \frac {8 \, {\left (\left (d x\right )^{\frac {5}{2}} b d^{2} - 5 \, \sqrt {d x} a d^{4}\right )}}{b^{2}}}{20 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.60 \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {1}{20} \, d^{3} {\left (\frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}} + \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{3}} - \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{3}} + \frac {8 \, {\left (\sqrt {d x} b^{4} d^{10} x^{2} - 5 \, \sqrt {d x} a b^{3} d^{10}\right )}}{b^{5} d^{10}}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
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Timed out. \[ \int \frac {(d x)^{7/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {{\left (d\,x\right )}^{7/2}}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]
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