Optimal. Leaf size=162 \[ \frac{x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac{77 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{77 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (c+d x)}{12 a \left (a-b x^4\right )^3} \]
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Rubi [A] time = 0.130489, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1855, 1876, 212, 208, 205, 275} \[ \frac{x (77 c+60 d x)}{384 a^3 \left (a-b x^4\right )}+\frac{x (11 c+10 d x)}{96 a^2 \left (a-b x^4\right )^2}+\frac{77 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{77 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x (c+d x)}{12 a \left (a-b x^4\right )^3} \]
Antiderivative was successfully verified.
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Rule 1855
Rule 1876
Rule 212
Rule 208
Rule 205
Rule 275
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{1}{\sqrt{a}-\sqrt{b} x^2} \, dx}{256 a^{7/2}}+\frac{(77 c) \int \frac{1}{\sqrt{a}+\sqrt{b} x^2} \, 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{77 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{77 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} \sqrt [4]{b}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.177661, size = 217, normalized size = 1.34 \[ \frac{\frac{128 a^3 x (c+d x)}{\left (a-b x^4\right )^3}+\frac{16 a^2 x (11 c+10 d x)}{\left (a-b x^4\right )^2}+\frac{4 a x (77 c+60 d x)}{a-b x^4}-\frac{3 \left (77 \sqrt [4]{a} \sqrt [4]{b} c+40 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{3 \left (77 \sqrt [4]{a} \sqrt [4]{b} c-40 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{462 \sqrt [4]{a} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{120 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{1536 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 177, normalized size = 1.1 \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}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c}{256\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,d}{64\,{a}^{3}}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \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 = 6.64574, size = 231, normalized size = 1.43 \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 [B] time = 1.0728, size = 400, normalized size = 2.47 \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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