Optimal. Leaf size=351 \[ -\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.31771, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1854
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+f x^3}{\left (a+b x^4\right )^3} \, dx &=-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}-\frac{\int \frac{-7 c-6 d x-5 e x^2}{\left (a+b x^4\right )^2} \, dx}{8 a}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac{\int \frac{21 c+12 d x+5 e x^2}{a+b x^4} \, dx}{32 a^2}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac{\int \left (\frac{12 d x}{a+b x^4}+\frac{21 c+5 e x^2}{a+b x^4}\right ) \, dx}{32 a^2}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac{\int \frac{21 c+5 e x^2}{a+b x^4} \, dx}{32 a^2}+\frac{(3 d) \int \frac{x}{a+b x^4} \, dx}{8 a^2}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{16 a^2}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{64 a^2 b}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}+5 e\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{64 a^2 b}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 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}{128 \sqrt{2} a^{9/4} b^{3/4}}-\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 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}{128 \sqrt{2} a^{9/4} b^{3/4}}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}+5 e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}+5 e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{9/4} b^{3/4}}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{9/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c+5 \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 )}{64 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (21 \sqrt{b} c+5 \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 )}{64 \sqrt{2} a^{11/4} b^{3/4}}\\ &=\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}-\frac{a f-b x \left (c+d x+e x^2\right )}{8 a b \left (a+b x^4\right )^2}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{\left (21 \sqrt{b} c+5 \sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c+5 \sqrt{a} e\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{9/4} b^{3/4}}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{9/4} b^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.328364, size = 347, normalized size = 0.99 \[ \frac{\frac{\sqrt{2} \left (5 a^{3/4} e-21 \sqrt [4]{a} \sqrt{b} c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{\sqrt{2} \left (21 \sqrt [4]{a} \sqrt{b} c-5 a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}-\frac{32 a^2 (a f-b x (c+x (d+e x)))}{b \left (a+b x^4\right )^2}-\frac{2 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{2 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{8 a x (7 c+x (6 d+5 e x))}{a+b x^4}}{256 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 432, normalized size = 1.2 \begin{align*}{\frac{cx}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{21\,c\sqrt{2}}{256\,{a}^{3}}\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{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{3\,d}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e{x}^{3}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{5\,e{x}^{3}}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{5\,e\sqrt{2}}{256\,b{a}^{2}}\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{5\,e\sqrt{2}}{128\,b{a}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e\sqrt{2}}{128\,b{a}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{f{x}^{4}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{f{x}^{4}}{8\,{a}^{2} \left ( b{x}^{4}+a \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 [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 = 68.5143, size = 578, normalized size = 1.65 \begin{align*} \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (6881280 a^{6} b^{2} c e + 4718592 a^{6} b^{2} d^{2}\right ) + t \left (153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) + 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} + 194481 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e + 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} - 118540800 t a^{4} b^{2} c^{3} e^{2} + 365783040 t a^{4} b^{2} c^{2} d^{2} e + 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} + 4536000 a^{2} b c d^{3} e^{2} - 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} - 275625 a^{2} b c^{2} e^{4} + 3024000 a^{2} b c d^{2} e^{3} - 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} + 85766121 b^{3} c^{6}} \right )} \right )\right )} + \frac{- 4 a^{2} f + 11 a b c x + 10 a b d x^{2} + 9 a b e x^{3} + 7 b^{2} c x^{5} + 6 b^{2} d x^{6} + 5 b^{2} e x^{7}}{32 a^{4} b + 64 a^{3} b^{2} x^{4} + 32 a^{2} b^{3} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1009, size = 478, normalized size = 1.36 \begin{align*} \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 5 \, \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 )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 5 \, \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 )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \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 )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \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 )}{256 \, a^{3} b^{3}} + \frac{5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + 9 \, a b x^{3} e + 10 \, a b d x^{2} + 11 \, a b c x - 4 \, a^{2} f}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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