Optimal. Leaf size=460 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{1}{a c x} \]
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Rubi [A] time = 0.445054, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {480, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{1}{a c x} \]
Antiderivative was successfully verified.
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Rule 480
Rule 584
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=-\frac{1}{a c x}+\frac{\int \frac{x^2 \left (-b c-a d-b d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx}{a c}\\ &=-\frac{1}{a c x}+\frac{\int \left (-\frac{b^2 c x^2}{(b c-a d) \left (a+b x^4\right )}-\frac{a d^2 x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx}{a c}\\ &=-\frac{1}{a c x}-\frac{b^2 \int \frac{x^2}{a+b x^4} \, dx}{a (b c-a d)}+\frac{d^2 \int \frac{x^2}{c+d x^4} \, dx}{c (b c-a d)}\\ &=-\frac{1}{a c x}+\frac{b^{3/2} \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{2 a (b c-a d)}-\frac{b^{3/2} \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{2 a (b c-a d)}-\frac{d^{3/2} \int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx}{2 c (b c-a d)}+\frac{d^{3/2} \int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx}{2 c (b c-a d)}\\ &=-\frac{1}{a c x}-\frac{b \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 a (b c-a d)}-\frac{b \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 a (b c-a d)}-\frac{b^{5/4} \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}{4 \sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \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}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 c (b c-a d)}+\frac{d \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 c (b c-a d)}+\frac{d^{5/4} \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}{4 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \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}{4 \sqrt{2} c^{5/4} (b c-a d)}\\ &=-\frac{1}{a c x}-\frac{b^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{b^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}\\ &=-\frac{1}{a c x}+\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}-\frac{b^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4} (b c-a d)}-\frac{d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}+\frac{d^{5/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{5/4} (b c-a d)}-\frac{b^{5/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{b^{5/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{5/4} (b c-a d)}+\frac{d^{5/4} \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}-\frac{d^{5/4} \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{5/4} (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.209379, size = 385, normalized size = 0.84 \[ \frac{\frac{\sqrt{2} b^{5/4} x \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{5/4}}-\frac{\sqrt{2} b^{5/4} x \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{5/4}}-\frac{2 \sqrt{2} b^{5/4} x \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac{2 \sqrt{2} b^{5/4} x \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac{8 b}{a}-\frac{\sqrt{2} d^{5/4} x \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{5/4}}+\frac{\sqrt{2} d^{5/4} x \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{5/4}}+\frac{2 \sqrt{2} d^{5/4} x \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{2 \sqrt{2} d^{5/4} x \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac{8 d}{c}}{8 a d x-8 b c x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 331, normalized size = 0.7 \begin{align*} -{\frac{d\sqrt{2}}{8\,c \left ( ad-bc \right ) }\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{2}}{4\,c \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{d\sqrt{2}}{4\,c \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-{\frac{1}{acx}}+{\frac{b\sqrt{2}}{8\,a \left ( ad-bc \right ) }\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{2}}{4\,a \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{2}}{4\,a \left ( ad-bc \right ) }\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \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 [B] time = 2.62832, size = 2730, normalized size = 5.93 \begin{align*} \text{result too large to display} \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 [C] time = 4.77351, size = 2279, normalized size = 4.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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