Optimal. Leaf size=245 \[ -\frac{(3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} d^{5/4}}-\frac{x (b c-a d)}{4 c d \left (c+d x^4\right )} \]
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Rubi [A] time = 0.146578, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {385, 211, 1165, 628, 1162, 617, 204} \[ -\frac{(3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} d^{5/4}}-\frac{x (b c-a d)}{4 c d \left (c+d x^4\right )} \]
Antiderivative was successfully verified.
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Rule 385
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{a+b x^4}{\left (c+d x^4\right )^2} \, dx &=-\frac{(b c-a d) x}{4 c d \left (c+d x^4\right )}+\frac{(b c+3 a d) \int \frac{1}{c+d x^4} \, dx}{4 c d}\\ &=-\frac{(b c-a d) x}{4 c d \left (c+d x^4\right )}+\frac{(b c+3 a d) \int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx}{8 c^{3/2} d}+\frac{(b c+3 a d) \int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx}{8 c^{3/2} d}\\ &=-\frac{(b c-a d) x}{4 c d \left (c+d x^4\right )}+\frac{(b c+3 a d) \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} d^{3/2}}+\frac{(b c+3 a d) \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} d^{3/2}}-\frac{(b c+3 a d) \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} d^{5/4}}-\frac{(b c+3 a d) \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} d^{5/4}}\\ &=-\frac{(b c-a d) x}{4 c d \left (c+d x^4\right )}-\frac{(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} d^{5/4}}+\frac{(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} d^{5/4}}+\frac{(b c+3 a d) \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} d^{5/4}}-\frac{(b c+3 a d) \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} d^{5/4}}\\ &=-\frac{(b c-a d) x}{4 c d \left (c+d x^4\right )}-\frac{(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} d^{5/4}}+\frac{(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} d^{5/4}}-\frac{(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} d^{5/4}}+\frac{(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} d^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.172137, size = 212, normalized size = 0.87 \[ \frac{-\frac{8 c^{3/4} \sqrt [4]{d} x (b c-a d)}{c+d x^4}-\sqrt{2} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+\sqrt{2} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-2 \sqrt{2} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt{2} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{32 c^{7/4} d^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 295, normalized size = 1.2 \begin{align*}{\frac{ \left ( ad-bc \right ) x}{4\,cd \left ( d{x}^{4}+c \right ) }}+{\frac{3\,\sqrt{2}a}{16\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}b}{16\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}a}{32\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\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{\sqrt{2}b}{32\,cd}\sqrt [4]{{\frac{c}{d}}}\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{3\,\sqrt{2}a}{16\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}b}{16\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \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 = 1.41881, size = 1526, normalized size = 6.23 \begin{align*} \frac{4 \,{\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{c^{5} d^{4} x \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{3}{4}} - c^{5} d^{4} \sqrt{\frac{c^{4} d^{2} \sqrt{-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}} +{\left (b^{2} c^{2} + 6 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{2}}{b^{2} c^{2} + 6 \, a b c d + 9 \, a^{2} d^{2}}} \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{3}{4}}}{b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} + 27 \, a^{3} d^{3}}\right ) +{\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} \log \left (c^{2} d \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} +{\left (b c + 3 \, a d\right )} x\right ) -{\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} \log \left (-c^{2} d \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} +{\left (b c + 3 \, a d\right )} x\right ) - 4 \,{\left (b c - a d\right )} x}{16 \,{\left (c d^{2} x^{4} + c^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.838773, size = 112, normalized size = 0.46 \begin{align*} \frac{x \left (a d - b c\right )}{4 c^{2} d + 4 c d^{2} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} c^{7} d^{5} + 81 a^{4} d^{4} + 108 a^{3} b c d^{3} + 54 a^{2} b^{2} c^{2} d^{2} + 12 a b^{3} c^{3} d + b^{4} c^{4}, \left ( t \mapsto t \log{\left (\frac{16 t c^{2} d}{3 a d + b c} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12676, size = 359, normalized size = 1.47 \begin{align*} \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{16 \, c^{2} d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{16 \, c^{2} d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{32 \, c^{2} d^{2}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{32 \, c^{2} d^{2}} - \frac{b c x - a d x}{4 \,{\left (d x^{4} + c\right )} c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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