Optimal. Leaf size=166 \[ \frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c} \]
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Rubi [A] time = 0.0885933, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {407, 409, 1218} \[ \frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c} \]
Antiderivative was successfully verified.
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Rule 407
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx &=\left (\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac{\left (\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c}+\frac{\left (\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c}\\ &=\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c}\\ \end{align*}
Mathematica [C] time = 0.153133, size = 160, normalized size = 0.96 \[ \frac{5 a c x \sqrt [4]{a+b x^4} F_1\left (\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (b c F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-4 a d F_1\left (\frac{5}{4};-\frac{1}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.407, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c}\sqrt [4]{b{x}^{4}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{d x^{4} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [4]{a + b x^{4}}}{c + d x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{d x^{4} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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