Integrand size = 26, antiderivative size = 247 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {5 a^4 b x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {10 a^3 b^2 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {2 a^2 b^3 x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {5 a b^4 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {b^5 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \]
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Time = 0.04 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1126, 276} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=\frac {b^5 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {5 a b^4 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {2 a^2 b^3 x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {5 a^4 b x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {10 a^3 b^2 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
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Rule 276
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^5}{x^2} \, dx}{b^4 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (5 a^4 b^6+\frac {a^5 b^5}{x^2}+10 a^3 b^7 x^2+10 a^2 b^8 x^4+5 a b^9 x^6+b^{10} x^8\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )} \\ & = -\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac {5 a^4 b x \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {10 a^3 b^2 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {2 a^2 b^3 x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {5 a b^4 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {b^5 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=\frac {\sqrt {\left (a+b x^2\right )^2} \left (-63 a^5+315 a^4 b x^2+210 a^3 b^2 x^4+126 a^2 b^3 x^6+45 a b^4 x^8+7 b^5 x^{10}\right )}{63 x \left (a+b x^2\right )} \]
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Time = 0.70 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(-\frac {\left (-7 x^{10} b^{5}-45 a \,x^{8} b^{4}-126 a^{2} x^{6} b^{3}-210 a^{3} x^{4} b^{2}-315 x^{2} a^{4} b +63 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{63 x \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
default | \(-\frac {\left (-7 x^{10} b^{5}-45 a \,x^{8} b^{4}-126 a^{2} x^{6} b^{3}-210 a^{3} x^{4} b^{2}-315 x^{2} a^{4} b +63 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{63 x \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \left (\frac {1}{9} b^{4} x^{9}+\frac {5}{7} a \,b^{3} x^{7}+2 a^{2} b^{2} x^{5}+\frac {10}{3} a^{3} b \,x^{3}+5 a^{4} x \right )}{b \,x^{2}+a}-\frac {a^{5} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{x \left (b \,x^{2}+a \right )}\) | \(96\) |
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.24 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=\frac {7 \, b^{5} x^{10} + 45 \, a b^{4} x^{8} + 126 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} - 63 \, a^{5}}{63 \, x} \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=\frac {1}{9} \, b^{5} x^{9} + \frac {5}{7} \, a b^{4} x^{7} + 2 \, a^{2} b^{3} x^{5} + \frac {10}{3} \, a^{3} b^{2} x^{3} + 5 \, a^{4} b x - \frac {a^{5}}{x} \]
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Time = 0.39 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=\frac {1}{9} \, b^{5} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{7} \, a b^{4} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, a^{2} b^{3} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{3} \, a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 5 \, a^{4} b x \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{x} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^2} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}}{x^2} \,d x \]
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