Optimal. Leaf size=349 \[ -\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac{x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]
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Rubi [A] time = 0.265617, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {413, 385, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac{x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 413
Rule 385
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx &=-\frac{(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}+\frac{\int \frac{a (b c+7 a d)+b (5 b c+3 a d) x^4}{\left (c+d x^4\right )^2} \, dx}{8 c d}\\ &=-\frac{(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac{(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac{1}{c+d x^4} \, dx}{32 c^2 d^2}\\ &=-\frac{(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac{(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d^2}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d^2}\\ &=-\frac{(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac{(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{5/2}}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{5/2}}-\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\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}{128 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\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}{128 \sqrt{2} c^{11/4} d^{9/4}}\\ &=-\frac{(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac{(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}-\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}\\ &=-\frac{(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac{(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}-\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{d} x^2\right )}{128 \sqrt{2} c^{11/4} d^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.186252, size = 319, normalized size = 0.91 \[ \frac{-\frac{8 c^{3/4} \sqrt [4]{d} x \left (-7 a^2 d^2-2 a b c d+9 b^2 c^2\right )}{c+d x^4}-\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )+\frac{32 c^{7/4} \sqrt [4]{d} x (b c-a d)^2}{\left (c+d x^4\right )^2}}{256 c^{11/4} d^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 499, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ({\frac{ \left ( 7\,{a}^{2}{d}^{2}+2\,cabd-9\,{b}^{2}{c}^{2} \right ){x}^{5}}{32\,{c}^{2}d}}+{\frac{ \left ( 11\,{a}^{2}{d}^{2}-6\,cabd-5\,{b}^{2}{c}^{2} \right ) x}{32\,{d}^{2}c}} \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}ab}{64\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{128\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{256\,{c}^{3}}\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}ab}{128\,{c}^{2}d}\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{5\,\sqrt{2}{b}^{2}}{256\,{d}^{2}c}\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{21\,\sqrt{2}{a}^{2}}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}ab}{64\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{128\,{d}^{2}c}\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 = 1.5754, size = 3302, normalized size = 9.46 \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 [A] time = 8.0575, size = 264, normalized size = 0.76 \begin{align*} \frac{x^{5} \left (7 a^{2} d^{3} + 2 a b c d^{2} - 9 b^{2} c^{2} d\right ) + x \left (11 a^{2} c d^{2} - 6 a b c^{2} d - 5 b^{2} c^{3}\right )}{32 c^{4} d^{2} + 64 c^{3} d^{3} x^{4} + 32 c^{2} d^{4} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} c^{11} d^{9} + 194481 a^{8} d^{8} + 222264 a^{7} b c d^{7} + 280476 a^{6} b^{2} c^{2} d^{6} + 176904 a^{5} b^{3} c^{3} d^{5} + 112806 a^{4} b^{4} c^{4} d^{4} + 42120 a^{3} b^{5} c^{5} d^{3} + 15900 a^{2} b^{6} c^{6} d^{2} + 3000 a b^{7} c^{7} d + 625 b^{8} c^{8}, \left ( t \mapsto t \log{\left (\frac{128 t c^{3} d^{2}}{21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14108, size = 549, normalized size = 1.57 \begin{align*} \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\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 )}{128 \, c^{3} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\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 )}{128 \, c^{3} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac{9 \, b^{2} c^{2} d x^{5} - 2 \, a b c d^{2} x^{5} - 7 \, a^{2} d^{3} x^{5} + 5 \, b^{2} c^{3} x + 6 \, a b c^{2} d x - 11 \, a^{2} c d^{2} x}{32 \,{\left (d x^{4} + c\right )}^{2} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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