Optimal. Leaf size=155 \[ \frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )} \]
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Rubi [A] time = 0.118335, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1854, 1876, 275, 208, 1167, 205} \[ \frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1854
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 )^2} \, dx &=\frac{a f+b x \left (c+d x+e x^2\right )}{4 a b \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{a f+b x \left (c+d x+e x^2\right )}{4 a b \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{a f+b x \left (c+d x+e x^2\right )}{4 a b \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{a f+b x \left (c+d x+e x^2\right )}{4 a b \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{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a}+\frac{\left (3 \sqrt{b} c+\sqrt{a} e\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a^{3/2}}\\ &=\frac{a f+b x \left (c+d x+e x^2\right )}{4 a b \left (a-b x^4\right )}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c+\sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.139363, size = 220, normalized size = 1.42 \[ \frac{-\sqrt [4]{b} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{3/4} e+3 \sqrt [4]{a} \sqrt{b} c+2 \sqrt{a} \sqrt [4]{b} d\right )+\sqrt [4]{b} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (a^{3/4} e+3 \sqrt [4]{a} \sqrt{b} c-2 \sqrt{a} \sqrt [4]{b} d\right )+\frac{4 a (a f+b x (c+x (d+e x)))}{a-b x^4}-2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a} e-3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt{a} \sqrt{b} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{16 a^2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 248, normalized size = 1.6 \begin{align*} -{\frac{cx}{4\,a \left ( b{x}^{4}-a \right ) }}+{\frac{3\,c}{16\,{a}^{2}}\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{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{d}{8\,a}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e{x}^{3}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{16\,ab}\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}}}}}}-{\frac{f{x}^{4}}{4\,a \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 [B] time = 15.1272, size = 518, normalized size = 3.34 \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{a f + b c x + b d x^{2} + b e x^{3}}{- 4 a^{2} b + 4 a b^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0842, size = 452, normalized size = 2.92 \begin{align*} -\frac{b x^{3} e + b d x^{2} + b c x + a f}{4 \,{\left (b x^{4} - a\right )} a b} + \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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