Optimal. Leaf size=220 \[ \frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3} \]
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Rubi [A] time = 0.188727, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{\left (15 \sqrt{a} e+77 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3} \]
Antiderivative was successfully verified.
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Rule 1854
Rule 1855
Rule 1876
Rule 275
Rule 208
Rule 1167
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3}{\left (a-b x^4\right )^4} \, dx &=\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac{\int \frac{-11 c-10 d x-9 e x^2}{\left (a-b x^4\right )^3} \, dx}{12 a}\\ &=\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}+\frac{\int \frac{77 c+60 d x+45 e x^2}{\left (a-b x^4\right )^2} \, dx}{96 a^2}\\ &=\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac{\int \frac{-231 c-120 d x-45 e x^2}{a-b x^4} \, dx}{384 a^3}\\ &=\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac{\int \left (-\frac{120 d x}{a-b x^4}+\frac{-231 c-45 e x^2}{a-b x^4}\right ) \, dx}{384 a^3}\\ &=\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac{\int \frac{-231 c-45 e x^2}{a-b x^4} \, dx}{384 a^3}+\frac{(5 d) \int \frac{x}{a-b x^4} \, dx}{16 a^3}\\ &=\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3}-\frac{\left (\frac{77 \sqrt{b} c}{\sqrt{a}}-15 e\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{256 a^3}+\frac{\left (\frac{77 \sqrt{b} c}{\sqrt{a}}+15 e\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{256 a^3}\\ &=\frac{x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac{x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac{a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}+\frac{\left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac{\left (77 \sqrt{b} c+15 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\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 = 0.307724, size = 286, normalized size = 1.3 \[ \frac{-\frac{3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt{b} c+40 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt{b} c-40 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}-\frac{128 a^3 (a f+b x (c+x (d+e x)))}{b \left (b x^4-a\right )^3}+\frac{16 a^2 x (11 c+x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac{6 \sqrt [4]{a} \left (77 \sqrt{b} c-15 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (77 c+15 x (4 d+3 e x))}{a-b x^4}+\frac{120 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{1536 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 280, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}-{\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}-{\frac{77\,{b}^{2}c{x}^{9}}{384\,{a}^{3}}}+{\frac{21\,be{x}^{7}}{64\,{a}^{2}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}+{\frac{33\,bc{x}^{5}}{64\,{a}^{2}}}-{\frac{113\,e{x}^{3}}{384\,a}}-{\frac{11\,d{x}^{2}}{32\,a}}-{\frac{51\,cx}{128\,a}}-{\frac{f}{12\,b}} \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}}}}-{\frac{15\,e}{256\,{a}^{3}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e}{512\,{a}^{3}b}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-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 [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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Giac [B] time = 1.08736, size = 555, normalized size = 2.52 \begin{align*} \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \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 )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 15 \, \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 )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \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 )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 15 \, \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 )}{1024 \, a^{4} b^{3}} - \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} - 126 \, a b^{2} x^{7} e - 160 \, a b^{2} d x^{6} - 198 \, a b^{2} c x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x + 32 \, a^{3} f}{384 \,{\left (b x^{4} - a\right )}^{3} a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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