Optimal. Leaf size=742 \[ \frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}+\sqrt{b}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+b x^4} (a d+b c)}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} \sqrt{-c} \sqrt{d}+\sqrt{b} c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{4 \sqrt [4]{a} c \sqrt{a+b x^4} (a d+b c)}-\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt{b c-a d}}-\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt{a d-b c}}+\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2 \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} (a d+b c)}+\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{-c}\right )^2 \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} (a d+b c)} \]
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Rubi [A] time = 0.633017, antiderivative size = 742, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {409, 1217, 220, 1707} \[ -\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt{b c-a d}}-\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt{a d-b c}}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}+\sqrt{b}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt{a+b x^4} (a d+b c)}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} \sqrt{-c} \sqrt{d}+\sqrt{b} c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} c \sqrt{a+b x^4} (a d+b c)}+\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2 \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} (a d+b c)}+\frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{-c}\right )^2 \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \sqrt{a+b x^4} (a d+b c)} \]
Antiderivative was successfully verified.
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Rule 409
Rule 1217
Rule 220
Rule 1707
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x^4} \left (c+d x^4\right )} \, dx &=\frac{\int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c}+\frac{\int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c}\\ &=\frac{\left (\sqrt{b} \left (\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 (b c+a d)}+\frac{\left (\sqrt{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 c (b c+a d)}-\frac{\left (\sqrt{a} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) \sqrt{d}\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c (b c+a d)}+\frac{\left (\sqrt{a} \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) \sqrt{d}\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c (b c+a d)}\\ &=-\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt{b c-a d}}-\frac{\sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{a+b x^4}}\right )}{4 (-c)^{3/4} \sqrt{-b c+a d}}+\frac{\sqrt [4]{b} \left (\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} (b c+a d) \sqrt{a+b x^4}}+\frac{\sqrt [4]{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} c (b c+a d) \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2 \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c+a d) \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2 \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c+a d) \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0505894, size = 161, normalized size = 0.22 \[ -\frac{5 a c x F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\sqrt{a+b x^4} \left (c+d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.016, size = 191, normalized size = 0.3 \begin{align*}{\frac{1}{8\,d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d+c \right ) }{\frac{1}{{{\it \_alpha}}^{3}} \left ( -{{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \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{1}{\sqrt{b x^{4} + a}{\left (d x^{4} + c\right )}}\,{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{1}{\sqrt{a + b x^{4}} \left (c + d x^{4}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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