Integrand size = 15, antiderivative size = 104 \[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=\frac {b x^2}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 a x^2}-\frac {\sqrt {b} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt [4]{a+b x^4}} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 331, 235, 233, 202} \[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=-\frac {\sqrt {b} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt [4]{a+b x^4}}+\frac {b x^2}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 a x^2} \]
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Rule 202
Rule 233
Rule 235
Rule 281
Rule 331
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right ) \\ & = -\frac {\left (a+b x^4\right )^{3/4}}{2 a x^2}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{4 a} \\ & = -\frac {\left (a+b x^4\right )^{3/4}}{2 a x^2}+\frac {\left (b \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{4 a \sqrt [4]{a+b x^4}} \\ & = \frac {b x^2}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 a x^2}-\frac {\left (b \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{4 a \sqrt [4]{a+b x^4}} \\ & = \frac {b x^2}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 a x^2}-\frac {\sqrt {b} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt [4]{a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=-\frac {\sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},-\frac {b x^4}{a}\right )}{2 x^2 \sqrt [4]{a+b x^4}} \]
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\[\int \frac {1}{x^{3} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x\]
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\[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} x^{2}} \]
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\[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \sqrt [4]{a+b x^4}} \, dx=\int \frac {1}{x^3\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \]
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