Optimal. Leaf size=703 \[ -\frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} (2 b c-a d) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{3 a \sqrt [4]{c} \sqrt{c+d x^4} (a d+b c)}-\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{\frac{\sqrt{-a} \left (\frac{b c}{a}-d\right )}{\sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a^2 \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}{\sqrt{c+d x^4}}\right )}{4 a^2 \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}}}-\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right ) (b c-a d) \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}-\frac{\left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-a d) \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^4} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )}-\frac{\sqrt{c+d x^4}}{3 a x^3} \]
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Rubi [A] time = 1.19046, antiderivative size = 893, normalized size of antiderivative = 1.27, number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {475, 523, 220, 409, 1217, 1707} \[ -\frac{(b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\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]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right ) \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt [4]{d} (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right ) \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}{4 (-a)^{3/2} \sqrt [4]{c} (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt [4]{b} \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{4 (-a)^{7/4}}+\frac{\sqrt [4]{b} \sqrt{a d-b c} \tan ^{-1}\left (\frac{\sqrt{a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{4 (-a)^{7/4}}-\frac{d^{3/4} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{6 a \sqrt [4]{c} \sqrt{d x^4+c}}-\frac{\left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 a \sqrt [4]{c} (b c+a d) \sqrt{d x^4+c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\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]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^4+c}}-\frac{\sqrt{d x^4+c}}{3 a x^3} \]
Warning: Unable to verify antiderivative.
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Rule 475
Rule 523
Rule 220
Rule 409
Rule 1217
Rule 1707
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^4}}{x^4 \left (a+b x^4\right )} \, dx &=-\frac{\sqrt{c+d x^4}}{3 a x^3}+\frac{\int \frac{-3 b c+2 a d-b d x^4}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{3 a}\\ &=-\frac{\sqrt{c+d x^4}}{3 a x^3}-\frac{d \int \frac{1}{\sqrt{c+d x^4}} \, dx}{3 a}-\frac{(b c-a d) \int \frac{1}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{a}\\ &=-\frac{\sqrt{c+d x^4}}{3 a x^3}-\frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{6 a \sqrt [4]{c} \sqrt{c+d x^4}}-\frac{(b c-a d) \int \frac{1}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx}{2 a^2}-\frac{(b c-a d) \int \frac{1}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx}{2 a^2}\\ &=-\frac{\sqrt{c+d x^4}}{3 a x^3}-\frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{6 a \sqrt [4]{c} \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-a d)\right ) \int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx}{2 a^2 (b c+a d)}-\frac{\left (\sqrt{b} \sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (b c-a d)\right ) \int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx}{2 a^2 (b c+a d)}-\frac{\left (\left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt{d} (b c-a d)\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{2 a (b c+a d)}-\frac{\left (\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d} (b c-a d)\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{2 (-a)^{3/2} (b c+a d)}\\ &=-\frac{\sqrt{c+d x^4}}{3 a x^3}-\frac{\sqrt [4]{b} \sqrt{b c-a d} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 (-a)^{7/4}}+\frac{\sqrt [4]{b} \sqrt{-b c+a d} \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 (-a)^{7/4}}-\frac{d^{3/4} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{6 a \sqrt [4]{c} \sqrt{c+d x^4}}-\frac{\left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 a \sqrt [4]{c} (b c+a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 (-a)^{3/2} \sqrt [4]{c} (b c+a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (b c-a d) \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} 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]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 (b c-a d) \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} 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]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a^2 \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^4}}\\ \end{align*}
Mathematica [C] time = 0.266061, size = 333, normalized size = 0.47 \[ \frac{\frac{a \left (25 a c \left (a c-a d x^4+4 b c x^4+b d x^8\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-10 x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{15 a^2 x^3 \sqrt{c+d x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.016, size = 370, normalized size = 0.5 \begin{align*} -{\frac{b}{a} \left ({\frac{d}{b}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}}-{\frac{1}{8\,{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{ad-bc}{{{\it \_alpha}}^{3}} \left ( -{{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \right ) }+{\frac{1}{a} \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{d{x}^{4}+c}}+{\frac{2\,d}{3}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \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{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{4}}\,{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{c + d x^{4}}}{x^{4} \left (a + b 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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