Integrand size = 22, antiderivative size = 68 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\frac {x \left (a+b x^2\right )}{3 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}}+\frac {2 x}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1103, 198, 197} \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\frac {x}{3 a \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac {2 x}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
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Rule 197
Rule 198
Rule 1103
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\frac {b x^2}{a}} \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/2}} \, dx}{a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {x}{3 a \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac {\left (2 \sqrt {1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/2}} \, dx}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {2 x}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac {x}{3 a \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\frac {\left (a+b x^2\right ) \left (3 a x+2 b x^3\right )}{3 a^2 \left (\left (a+b x^2\right )^2\right )^{5/4}} \]
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Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(\frac {\left (b \,x^{2}+a \right ) x \left (2 b \,x^{2}+3 a \right )}{3 a^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{4}}}\) | \(44\) |
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none
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}} {\left (2 \, b x^{3} + 3 \, a x\right )}}{3 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \]
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\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{4}}} \,d x } \]
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Time = 13.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx=\frac {x\,\left (2\,b\,x^2+3\,a\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/4}}{3\,a^2\,{\left (b\,x^2+a\right )}^3} \]
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