Integrand size = 15, antiderivative size = 80 \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=-\frac {5}{4} a b \sqrt {a+\frac {b}{x^4}}-\frac {5}{12} b \left (a+\frac {b}{x^4}\right )^{3/2}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{5/2} x^4+\frac {5}{4} a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 43, 52, 65, 214} \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=\frac {5}{4} a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )+\frac {1}{4} x^4 \left (a+\frac {b}{x^4}\right )^{5/2}-\frac {5}{12} b \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {5}{4} a b \sqrt {a+\frac {b}{x^4}} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^2} \, dx,x,\frac {1}{x^4}\right )\right ) \\ & = \frac {1}{4} \left (a+\frac {b}{x^4}\right )^{5/2} x^4-\frac {1}{8} (5 b) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x^4}\right ) \\ & = -\frac {5}{12} b \left (a+\frac {b}{x^4}\right )^{3/2}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{5/2} x^4-\frac {1}{8} (5 a b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^4}\right ) \\ & = -\frac {5}{4} a b \sqrt {a+\frac {b}{x^4}}-\frac {5}{12} b \left (a+\frac {b}{x^4}\right )^{3/2}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{5/2} x^4-\frac {1}{8} \left (5 a^2 b\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right ) \\ & = -\frac {5}{4} a b \sqrt {a+\frac {b}{x^4}}-\frac {5}{12} b \left (a+\frac {b}{x^4}\right )^{3/2}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{5/2} x^4-\frac {1}{4} \left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right ) \\ & = -\frac {5}{4} a b \sqrt {a+\frac {b}{x^4}}-\frac {5}{12} b \left (a+\frac {b}{x^4}\right )^{3/2}+\frac {1}{4} \left (a+\frac {b}{x^4}\right )^{5/2} x^4+\frac {5}{4} a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.20 \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=\frac {\sqrt {a+\frac {b}{x^4}} \left (\sqrt {b+a x^4} \left (-2 b^2-14 a b x^4+3 a^2 x^8\right )+15 a^{3/2} b x^6 \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )\right )}{12 x^4 \sqrt {b+a x^4}} \]
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Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\left (3 a^{2} x^{8}-14 a b \,x^{4}-2 b^{2}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{12 x^{4}}+\frac {5 a^{\frac {3}{2}} b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{4 \sqrt {a \,x^{4}+b}}\) | \(90\) |
default | \(\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} x^{4} \left (3 a^{2} x^{8} \sqrt {a \,x^{4}+b}+15 a^{\frac {3}{2}} b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) x^{6}-14 a b \sqrt {a \,x^{4}+b}\, x^{4}-2 b^{2} \sqrt {a \,x^{4}+b}\right )}{12 \left (a \,x^{4}+b \right )^{\frac {5}{2}}}\) | \(103\) |
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Time = 0.29 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.12 \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=\left [\frac {15 \, a^{\frac {3}{2}} b x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) + 2 \, {\left (3 \, a^{2} x^{8} - 14 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{24 \, x^{4}}, -\frac {15 \, \sqrt {-a} a b x^{4} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) - {\left (3 \, a^{2} x^{8} - 14 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, x^{4}}\right ] \]
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Time = 2.47 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=\frac {a^{\frac {5}{2}} x^{4} \sqrt {1 + \frac {b}{a x^{4}}}}{4} - \frac {7 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x^{4}}}}{6} - \frac {5 a^{\frac {3}{2}} b \log {\left (\frac {b}{a x^{4}} \right )}}{8} + \frac {5 a^{\frac {3}{2}} b \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{4} - \frac {\sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x^{4}}}}{6 x^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=\frac {1}{4} \, \sqrt {a + \frac {b}{x^{4}}} a^{2} x^{4} - \frac {5}{8} \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{6} \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} b - \sqrt {a + \frac {b}{x^{4}}} a b \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (60) = 120\).
Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.78 \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=\frac {1}{4} \, \sqrt {a x^{4} + b} a^{2} x^{2} - \frac {5}{8} \, a^{\frac {3}{2}} b \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {9 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{4} a^{\frac {3}{2}} b^{2} - 12 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} a^{\frac {3}{2}} b^{3} + 7 \, a^{\frac {3}{2}} b^{4}}{3 \, {\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b\right )}^{3}} \]
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Time = 6.99 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int \left (a+\frac {b}{x^4}\right )^{5/2} x^3 \, dx=\frac {a^2\,x^4\,\sqrt {a+\frac {b}{x^4}}}{4}-\frac {b\,{\left (a+\frac {b}{x^4}\right )}^{3/2}}{6}-a\,b\,\sqrt {a+\frac {b}{x^4}}-\frac {a^{3/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x^4}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{4} \]
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