Optimal. Leaf size=291 \[ \frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}-\frac{77 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{77 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3} \]
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Rubi [A] time = 0.267577, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {1855, 1876, 211, 1165, 628, 1162, 617, 204, 275, 205} \[ \frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}-\frac{77 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{77 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3} \]
Antiderivative was successfully verified.
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Rule 1855
Rule 1876
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 275
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x}{\left (a+b x^4\right )^4} \, dx &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}-\frac{\int \frac{-11 c-10 d x}{\left (a+b x^4\right )^3} \, dx}{12 a}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{\int \frac{77 c+60 d x}{\left (a+b x^4\right )^2} \, dx}{96 a^2}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}-\frac{\int \frac{-231 c-120 d x}{a+b x^4} \, dx}{384 a^3}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}-\frac{\int \left (-\frac{231 c}{a+b x^4}-\frac{120 d x}{a+b x^4}\right ) \, dx}{384 a^3}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{(77 c) \int \frac{1}{a+b x^4} \, dx}{128 a^3}+\frac{(5 d) \int \frac{x}{a+b x^4} \, dx}{16 a^3}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{(77 c) \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{256 a^{7/2}}+\frac{(77 c) \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{256 a^{7/2}}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{(77 c) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} \sqrt{b}}+\frac{(77 c) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} \sqrt{b}}-\frac{(77 c) \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}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(77 c) \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}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}-\frac{77 c \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{(77 c) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(77 c) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}\\ &=\frac{x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac{x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac{x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}-\frac{77 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{77 c \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{77 c \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} \sqrt [4]{b}}\\ \end{align*}
Mathematica [A] time = 0.284213, size = 274, normalized size = 0.94 \[ \frac{\frac{256 a^{11/4} x (c+d x)}{\left (a+b x^4\right )^3}+\frac{32 a^{7/4} x (11 c+10 d x)}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} x (77 c+60 d x)}{a+b x^4}-\frac{6 \left (80 \sqrt [4]{a} d+77 \sqrt{2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{b}}+\frac{6 \left (77 \sqrt{2} \sqrt [4]{b} c-80 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt{b}}-\frac{231 \sqrt{2} c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}+\frac{231 \sqrt{2} c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}}{3072 a^{15/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 225, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( b{x}^{4}+a \right ) ^{3}} \left ({\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}+{\frac{77\,{b}^{2}c{x}^{9}}{384\,{a}^{3}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}+{\frac{33\,bc{x}^{5}}{64\,{a}^{2}}}+{\frac{11\,d{x}^{2}}{32\,a}}+{\frac{51\,cx}{128\,a}} \right ) }+{\frac{77\,c\sqrt{2}}{1024\,{a}^{4}}\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{77\,c\sqrt{2}}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{77\,c\sqrt{2}}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{5\,d}{32\,{a}^{3}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 6.6255, size = 231, normalized size = 0.79 \begin{align*} \operatorname{RootSum}{\left (68719476736 t^{4} a^{15} b^{2} + 838860800 t^{2} a^{8} b d^{2} - 485703680 t a^{4} b c^{2} d + 2560000 a d^{4} + 35153041 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 429496729600 t^{3} a^{12} b d^{2} - 62170071040 t^{2} a^{8} b c^{2} d - 2621440000 t a^{5} d^{4} - 17998356992 t a^{4} b c^{4} + 1897280000 a c^{2} d^{3}}{788480000 a c d^{4} - 2706784157 b c^{5}} \right )} \right )\right )} + \frac{153 a^{2} c x + 132 a^{2} d x^{2} + 198 a b c x^{5} + 160 a b d x^{6} + 77 b^{2} c x^{9} + 60 b^{2} d x^{10}}{384 a^{6} + 1152 a^{5} b x^{4} + 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07999, size = 378, normalized size = 1.3 \begin{align*} \frac{77 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b} - \frac{77 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\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 )}{512 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\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 )}{512 \, a^{4} b^{2}} + \frac{60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} + 160 \, a b d x^{6} + 198 \, a b c x^{5} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \,{\left (b x^{4} + a\right )}^{3} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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