Optimal. Leaf size=308 \[ -\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )} \]
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Rubi [A] time = 0.254637, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1855
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2}{\left (a+b x^4\right )^2} \, dx &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}-\frac{\int \frac{-3 c-2 d x-e x^2}{a+b x^4} \, dx}{4 a}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}-\frac{\int \left (-\frac{2 d x}{a+b x^4}+\frac{-3 c-e x^2}{a+b x^4}\right ) \, dx}{4 a}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}-\frac{\int \frac{-3 c-e x^2}{a+b x^4} \, dx}{4 a}+\frac{d \int \frac{x}{a+b x^4} \, dx}{2 a}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}+\frac{d \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{4 a}+\frac{\left (\frac{3 \sqrt{b} c}{\sqrt{a}}-e\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{8 a b}+\frac{\left (\frac{3 \sqrt{b} c}{\sqrt{a}}+e\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{8 a b}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{\left (\frac{3 \sqrt{b} c}{\sqrt{a}}+e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b}+\frac{\left (\frac{3 \sqrt{b} c}{\sqrt{a}}+e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a b}-\frac{\left (3 \sqrt{b} c-\sqrt{a} e\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^{3/4}}-\frac{\left (3 \sqrt{b} c-\sqrt{a} e\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^{3/4}}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}-\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \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^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \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^{3/4}}+\frac{\left (3 \sqrt{b} c+\sqrt{a} e\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^{3/4}}-\frac{\left (3 \sqrt{b} c+\sqrt{a} e\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^{3/4}}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a+b x^4\right )}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}-\frac{\left (3 \sqrt{b} c+\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c+\sqrt{a} e\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \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^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \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^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.330297, size = 305, normalized size = 0.99 \[ \frac{\frac{\sqrt{2} \left (a^{3/4} e-3 \sqrt [4]{a} \sqrt{b} c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{\sqrt{2} \left (3 \sqrt [4]{a} \sqrt{b} c-a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}-\frac{2 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{2 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{8 a x (c+x (d+e x))}{a+b x^4}}{32 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 344, normalized size = 1.1 \begin{align*}{\frac{cx}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{3\,c\sqrt{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\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{3\,c\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,c\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{d}{4\,a}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e{x}^{3}}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{e\sqrt{2}}{32\,ab}\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{e\sqrt{2}}{16\,ab}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e\sqrt{2}}{16\,ab}\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 [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 [A] time = 7.28354, size = 505, normalized size = 1.64 \begin{align*} \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{3} + t^{2} \left (3072 a^{4} b^{2} c e + 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} - 1152 a^{2} b^{2} c^{2} d\right ) + a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} + 81 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} b^{2} e^{3} - 36864 t^{3} a^{6} b^{3} c^{2} e + 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e + 13824 t^{2} a^{4} b^{3} c^{3} d + 144 t a^{4} b c e^{4} + 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} + 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} + 120 a^{2} b c d^{3} e^{2} - 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} - 9 a^{2} b c^{2} e^{4} + 96 a^{2} b c d^{2} e^{3} - 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} + 729 b^{3} c^{6}} \right )} \right )\right )} + \frac{c x + d x^{2} + e x^{3}}{4 a^{2} + 4 a b x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09527, size = 413, normalized size = 1.34 \begin{align*} \frac{x^{3} e + d x^{2} + c x}{4 \,{\left (b x^{4} + a\right )} a} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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