Optimal. Leaf size=596 \[ -\frac{b^{7/4} (3 b c-11 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{b x}{4 a \left (a+b x^4\right ) \left (c+d x^4\right ) (b c-a d)}+\frac{d x (a d+b c)}{4 a c \left (c+d x^4\right ) (b c-a d)^2} \]
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Rubi [A] time = 0.738621, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {414, 527, 522, 211, 1165, 628, 1162, 617, 204} \[ -\frac{b^{7/4} (3 b c-11 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{b x}{4 a \left (a+b x^4\right ) \left (c+d x^4\right ) (b c-a d)}+\frac{d x (a d+b c)}{4 a c \left (c+d x^4\right ) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 414
Rule 527
Rule 522
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx &=\frac{b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac{\int \frac{-3 b c+4 a d-7 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx}{4 a (b c-a d)}\\ &=\frac{d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac{b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac{\int \frac{-4 \left (3 b^2 c^2-8 a b c d+3 a^2 d^2\right )-12 b d (b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx}{16 a c (b c-a d)^2}\\ &=\frac{d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac{b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}+\frac{\left (b^2 (3 b c-11 a d)\right ) \int \frac{1}{a+b x^4} \, dx}{4 a (b c-a d)^3}+\frac{\left (d^2 (11 b c-3 a d)\right ) \int \frac{1}{c+d x^4} \, dx}{4 c (b c-a d)^3}\\ &=\frac{d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac{b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}+\frac{\left (b^2 (3 b c-11 a d)\right ) \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{8 a^{3/2} (b c-a d)^3}+\frac{\left (b^2 (3 b c-11 a d)\right ) \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{8 a^{3/2} (b c-a d)^3}+\frac{\left (d^2 (11 b c-3 a d)\right ) \int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx}{8 c^{3/2} (b c-a d)^3}+\frac{\left (d^2 (11 b c-3 a d)\right ) \int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx}{8 c^{3/2} (b c-a d)^3}\\ &=\frac{d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac{b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}+\frac{\left (b^{3/2} (3 b c-11 a d)\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} (b c-a d)^3}+\frac{\left (b^{3/2} (3 b c-11 a d)\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} (b c-a d)^3}-\frac{\left (b^{7/4} (3 b c-11 a d)\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{\left (b^{7/4} (3 b c-11 a d)\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{\left (d^{3/2} (11 b c-3 a d)\right ) \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{16 c^{3/2} (b c-a d)^3}+\frac{\left (d^{3/2} (11 b c-3 a d)\right ) \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{16 c^{3/2} (b c-a d)^3}-\frac{\left (d^{7/4} (11 b c-3 a d)\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{16 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{\left (d^{7/4} (11 b c-3 a d)\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{16 \sqrt{2} c^{7/4} (b c-a d)^3}\\ &=\frac{d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac{b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{\left (b^{7/4} (3 b c-11 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{\left (b^{7/4} (3 b c-11 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{\left (d^{7/4} (11 b c-3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{\left (d^{7/4} (11 b c-3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}\\ &=\frac{d (b c+a d) x}{4 a c (b c-a d)^2 \left (c+d x^4\right )}+\frac{b x}{4 a (b c-a d) \left (a+b x^4\right ) \left (c+d x^4\right )}-\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} (b c-a d)^3}-\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{7/4} (3 b c-11 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}+\frac{d^{7/4} (11 b c-3 a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 1.4117, size = 561, normalized size = 0.94 \[ \frac{1}{32} \left (\frac{\sqrt{2} b^{7/4} (11 a d-3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4} (b c-a d)^3}+\frac{\sqrt{2} b^{7/4} (11 a d-3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{7/4} (11 a d-3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4} (b c-a d)^3}+\frac{2 \sqrt{2} b^{7/4} (11 a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4} (a d-b c)^3}+\frac{8 b^2 x}{a \left (a+b x^4\right ) (b c-a d)^2}+\frac{\sqrt{2} d^{7/4} (11 b c-3 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4} (a d-b c)^3}+\frac{\sqrt{2} d^{7/4} (11 b c-3 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{7/4} (3 a d-11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{7/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{7/4} (b c-a d)^3}+\frac{8 d^2 x}{c \left (c+d x^4\right ) (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 784, normalized size = 1.3 \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(-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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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(-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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