Optimal. Leaf size=179 \[ \frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2} \]
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Rubi [A] time = 0.167009, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1855, 1876, 275, 208, 1167, 205} \[ \frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1855
Rule 1876
Rule 275
Rule 208
Rule 1167
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2}{\left (a-b x^4\right )^3} \, dx &=\frac{x \left (c+d x+e x^2\right )}{8 a \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 (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{\int \frac{21 c+12 d x+5 e x^2}{a-b x^4} \, dx}{32 a^2}\\ &=\frac{x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\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 (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\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 (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\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) \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{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}+5 e\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2}\\ &=\frac{x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac{\left (\frac{21 \sqrt{b} c}{\sqrt{a}}-5 e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c+5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.213492, size = 244, normalized size = 1.36 \[ \frac{-\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt{b} c+12 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt{b} c-12 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{16 a^2 x (c+x (d+e x))}{\left (a-b x^4\right )^2}+\frac{2 \sqrt [4]{a} \left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac{12 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{128 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 286, normalized size = 1.6 \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}{128\,{a}^{3}}\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{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \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}{32\,{a}^{2}}\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}}{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}{64\,b{a}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e}{128\,b{a}^{2}}\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 [B] time = 10.2057, size = 563, normalized size = 3.15 \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{- 11 a c x - 10 a d x^{2} - 9 a e x^{3} + 7 b c x^{5} + 6 b d x^{6} + 5 b e x^{7}}{32 a^{4} - 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10124, size = 481, normalized size = 2.69 \begin{align*} -\frac{5 \, b x^{7} e + 6 \, b d x^{6} + 7 \, b c x^{5} - 9 \, a x^{3} e - 10 \, a d x^{2} - 11 \, a c x}{32 \,{\left (b x^{4} - a\right )}^{2} a^{2}} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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