Integrand size = 30, antiderivative size = 412 \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{2 \sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.19 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 327, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{2 \sqrt {2} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 210
Rule 303
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)^{5/2}}{a b+b^2 x^2} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 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 {\sqrt {d x}}{a b+b^2 x^2} \, dx}{b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (2 a d \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a d \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^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a d \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^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a^{3/4} d^{5/2} \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^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a^{3/4} d^{5/2} \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^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a d^3 \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^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a d^3 \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^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{3/4} d^{5/2} \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^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{3/4} d^{5/2} \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^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a^{3/4} d^{5/2} \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^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a^{3/4} d^{5/2} \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^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{3/4} d^{5/2} \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^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{3/4} d^{5/2} \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^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a^{3/4} d^{5/2} \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^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^{3/4} d^{5/2} \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^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.37 \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {(d x)^{5/2} \left (a+b x^2\right ) \left (4 b^{3/4} x^{3/2}+3 \sqrt {2} a^{3/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} a^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{6 b^{7/4} x^{5/2} \sqrt {\left (a+b x^2\right )^2}} \]
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Time = 0.14 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.49
method | result | size |
risch | \(\frac {2 x^{2} d^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b \sqrt {d x}\, \left (b \,x^{2}+a \right )}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) d^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{4 b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )}\) | \(200\) |
default | \(\frac {\left (b \,x^{2}+a \right ) d \left (8 \left (d x \right )^{\frac {3}{2}} b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}-3 \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right ) \sqrt {2}\, a \,d^{2}-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,d^{2}-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,d^{2}\right )}{12 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\) | \(221\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.50 \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {4 \, \sqrt {d x} d^{2} x - 3 \, \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} a^{2} d^{7} + \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {3}{4}} b^{5}\right ) + 3 i \, \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} a^{2} d^{7} + i \, \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {3}{4}} b^{5}\right ) - 3 i \, \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} a^{2} d^{7} - i \, \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {3}{4}} b^{5}\right ) + 3 \, \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (\sqrt {d x} a^{2} d^{7} - \left (-\frac {a^{3} d^{10}}{b^{7}}\right )^{\frac {3}{4}} b^{5}\right )}{6 \, b} \]
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Timed out. \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.58 \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {\frac {3 \, a d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b} - \frac {8 \, \left (d x\right )^{\frac {3}{2}} d^{2}}{b}}{12 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.62 \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {1}{12} \, d^{2} {\left (\frac {8 \, \sqrt {d x} x}{b} - \frac {6 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} d} - \frac {6 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4} d} + \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{4} d} - \frac {3 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{4} d}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
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Timed out. \[ \int \frac {(d x)^{5/2}}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {{\left (d\,x\right )}^{5/2}}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]
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