Optimal. Leaf size=1046 \[ \text{result too large to display} \]
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Rubi [A] time = 1.55713, antiderivative size = 1046, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {472, 583, 523, 220, 409, 1217, 1707} \[ -\frac{b (7 b c-9 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}{32 a^3 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b \sqrt [4]{d} (7 b c-9 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 )}{16 (-a)^{5/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{16 (-a)^{11/4} (b c-a d)^{3/2}}-\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{16 (-a)^{11/4} (a d-b c)^{3/2}}-\frac{d^{3/4} (7 b c-4 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 )}{24 a^2 c^{5/4} (b c-a d) \sqrt{d x^4+c}}-\frac{b \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (7 b c-9 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 )}{16 (-a)^{5/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{b \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (7 b c-9 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 )}{32 a^3 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{b \sqrt{d x^4+c}}{4 a (b c-a d) x^3 \left (b x^4+a\right )}-\frac{(7 b c-4 a d) \sqrt{d x^4+c}}{12 a^2 c (b c-a d) x^3} \]
Antiderivative was successfully verified.
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Rule 472
Rule 583
Rule 523
Rule 220
Rule 409
Rule 1217
Rule 1707
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx &=\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^3 \left (a+b x^4\right )}-\frac{\int \frac{-7 b c+4 a d-5 b d x^4}{x^4 \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{4 a (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^3}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^3 \left (a+b x^4\right )}+\frac{\int \frac{-21 b^2 c^2+20 a b c d+4 a^2 d^2-b d (7 b c-4 a d) x^4}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{12 a^2 c (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^3}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^3 \left (a+b x^4\right )}-\frac{(b (7 b c-9 a d)) \int \frac{1}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{4 a^2 (b c-a d)}-\frac{(d (7 b c-4 a d)) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{12 a^2 c (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^3}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^3 \left (a+b x^4\right )}-\frac{d^{3/4} (7 b c-4 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 )}{24 a^2 c^{5/4} (b c-a d) \sqrt{c+d x^4}}-\frac{(b (7 b c-9 a d)) \int \frac{1}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx}{8 a^3 (b c-a d)}-\frac{(b (7 b c-9 a d)) \int \frac{1}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx}{8 a^3 (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^3}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^3 \left (a+b x^4\right )}-\frac{d^{3/4} (7 b c-4 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 )}{24 a^2 c^{5/4} (b c-a d) \sqrt{c+d x^4}}-\frac{\left (b^{3/2} \sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (7 b c-9 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}{8 a^3 (b c-a d) (b c+a d)}-\frac{\left (b^{3/2} \sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (7 b c-9 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}{8 a^3 (b c-a d) (b c+a d)}-\frac{\left (b \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt{d} (7 b c-9 a d)\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{8 a^2 (b c-a d) (b c+a d)}+\frac{\left (b \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d} (7 b c-9 a d)\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{8 (-a)^{5/2} (b c-a d) (b c+a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^3}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^3 \left (a+b x^4\right )}+\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{16 (-a)^{11/4} (b c-a d)^{3/2}}-\frac{b^{5/4} (7 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{16 (-a)^{11/4} (-b c+a d)^{3/2}}-\frac{d^{3/4} (7 b c-4 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 )}{24 a^2 c^{5/4} (b c-a d) \sqrt{c+d x^4}}-\frac{b \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt [4]{d} (7 b c-9 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 )}{16 a^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^4}}+\frac{b \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (7 b c-9 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 )}{16 (-a)^{5/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^4}}-\frac{b \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (7 b c-9 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 )}{32 a^3 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^4}}-\frac{b \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 (7 b c-9 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 )}{32 a^3 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^4}}\\ \end{align*}
Mathematica [C] time = 0.562522, size = 408, normalized size = 0.39 \[ \frac{\frac{a \left (25 a c \left (4 a^2 d \left (c+2 d x^4\right )+4 a b \left (-c^2+5 c d x^4+d^2 x^8\right )-7 b^2 c x^4 \left (4 c+d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+10 x^4 \left (c+d x^4\right ) \left (-4 a^2 d+4 a b \left (c-d x^4\right )+7 b^2 c x^4\right ) \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )\right )}{\left (a+b x^4\right ) \left (2 x^4 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}+b d x^8 \sqrt{\frac{d x^4}{c}+1} (7 b c-4 a d) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{60 a^3 c x^3 \sqrt{c+d x^4} (a d-b c)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.013, size = 626, normalized size = 0.6 \begin{align*} \text{result too large to display} \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}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c} 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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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