Optimal. Leaf size=308 \[ -\frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{4 c \left (a+b x^4\right )^{3/4} (b c-a d)}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-3 a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-3 a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )} \]
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Rubi [A] time = 0.203448, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {412, 529, 237, 335, 275, 231, 407, 409, 1218} \[ -\frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 c \left (a+b x^4\right )^{3/4} (b c-a d)}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-3 a d) \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (2 b c-3 a d) \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )} \]
Antiderivative was successfully verified.
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Rule 412
Rule 529
Rule 237
Rule 335
Rule 275
Rule 231
Rule 407
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^4}}{\left (c+d x^4\right )^2} \, dx &=\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}-\frac{\int \frac{-3 a-2 b x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{4 c}\\ &=\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}+\frac{(a b) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{4 c (b c-a d)}+\frac{(2 b c-3 a d) \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{4 c (b c-a d)}\\ &=\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}+\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{4 c (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac{\left ((2 b c-3 a d) \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 )}{4 c (b c-a d)}\\ &=\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}-\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{4 c (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac{\left ((2 b c-3 a d) \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 )}{8 c^2 (b c-a d)}+\frac{\left ((2 b c-3 a d) \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 )}{8 c^2 (b c-a d)}\\ &=\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}+\frac{(2 b c-3 a d) \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 )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac{(2 b c-3 a d) \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 )}{8 \sqrt [4]{b} c^2 (b c-a d)}-\frac{\left (a b \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{8 c (b c-a d) \left (a+b x^4\right )^{3/4}}\\ &=\frac{x \sqrt [4]{a+b x^4}}{4 c \left (c+d x^4\right )}-\frac{\sqrt{a} b^{3/2} \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 c (b c-a d) \left (a+b x^4\right )^{3/4}}+\frac{(2 b c-3 a d) \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 )}{8 \sqrt [4]{b} c^2 (b c-a d)}+\frac{(2 b c-3 a d) \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 )}{8 \sqrt [4]{b} c^2 (b c-a d)}\\ \end{align*}
Mathematica [C] time = 0.211929, size = 233, normalized size = 0.76 \[ \frac{x \left (\frac{5 \left (\frac{a+b x^4}{c}-\frac{15 a^2 F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}\right )}{c+d x^4}+\frac{2 b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{c^2}\right )}{20 \left (a+b x^4\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.435, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}}\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}}}{{\left (d x^{4} + c\right )}^{2}}\,{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}}}{\left (c + d x^{4}\right )^{2}}\, 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}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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