Optimal. Leaf size=162 \[ \frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.108515, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {382, 377, 212, 208, 205} \[ \frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 382
Rule 377
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{4 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c}-\sqrt{b c-a d} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)}+\frac{(4 b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c}+\sqrt{b c-a d} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)}\\ &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}\\ \end{align*}
Mathematica [C] time = 0.105946, size = 99, normalized size = 0.61 \[ \frac{x \left (\left (c+d x^4\right ) (4 b c-3 a d) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-c d \left (a+b x^4\right )\right )}{4 c^2 \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.41, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}}{\frac{1}{\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{1}{{\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 )}^{\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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