Optimal. Leaf size=266 \[ \frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}-\frac{21 d \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 d \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}-\frac{21 d \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{3 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}+\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.249303, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ \frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}-\frac{21 d \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 d \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}-\frac{21 d \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{3 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}+\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2} \]
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{d+e x}{\left (a+c x^4\right )^3} \, dx &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}-\frac{\int \frac{-7 d-6 e x}{\left (a+c x^4\right )^2} \, dx}{8 a}\\ &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac{\int \frac{21 d+12 e x}{a+c x^4} \, dx}{32 a^2}\\ &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac{\int \left (\frac{21 d}{a+c x^4}+\frac{12 e x}{a+c x^4}\right ) \, dx}{32 a^2}\\ &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac{(21 d) \int \frac{1}{a+c x^4} \, dx}{32 a^2}+\frac{(3 e) \int \frac{x}{a+c x^4} \, dx}{8 a^2}\\ &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac{(21 d) \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{64 a^{5/2}}+\frac{(21 d) \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{64 a^{5/2}}+\frac{(3 e) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{16 a^2}\\ &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac{3 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}+\frac{(21 d) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{5/2} \sqrt{c}}+\frac{(21 d) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{5/2} \sqrt{c}}-\frac{(21 d) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}-\frac{(21 d) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}\\ &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac{3 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}-\frac{21 d \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 d \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{(21 d) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}-\frac{(21 d) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}\\ &=\frac{x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac{x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac{3 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}-\frac{21 d \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 d \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{c}}-\frac{21 d \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}+\frac{21 d \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{c}}\\ \end{align*}
Mathematica [A] time = 0.19068, size = 249, normalized size = 0.94 \[ \frac{\frac{32 a^{7/4} x (d+e x)}{\left (a+c x^4\right )^2}+\frac{8 a^{3/4} x (7 d+6 e x)}{a+c x^4}-\frac{6 \left (8 \sqrt [4]{a} e+7 \sqrt{2} \sqrt [4]{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt{c}}+\frac{6 \left (7 \sqrt{2} \sqrt [4]{c} d-8 \sqrt [4]{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt{c}}-\frac{21 \sqrt{2} d \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{c}}+\frac{21 \sqrt{2} d \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{c}}}{256 a^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 222, normalized size = 0.8 \begin{align*}{\frac{dx}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{7\,dx}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{21\,d\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{21\,d\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{21\,d\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{e{x}^{2}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{3\,e{x}^{2}}{16\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{3\,e}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 2.29965, size = 192, normalized size = 0.72 \begin{align*} \operatorname{RootSum}{\left (268435456 t^{4} a^{11} c^{2} + 4718592 t^{2} a^{6} c e^{2} - 2709504 t a^{3} c d^{2} e + 20736 a e^{4} + 194481 c d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 67108864 t^{3} a^{9} c e^{2} - 9633792 t^{2} a^{6} c d^{2} e - 589824 t a^{4} e^{4} - 2765952 t a^{3} c d^{4} + 423360 a d^{2} e^{3}}{193536 a d e^{4} - 453789 c d^{5}} \right )} \right )\right )} + \frac{11 a d x + 10 a e x^{2} + 7 c d x^{5} + 6 c e x^{6}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21667, size = 351, normalized size = 1.32 \begin{align*} \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} d \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c} - \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} d \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a c} c e + 7 \, \left (a c^{3}\right )^{\frac{1}{4}} c d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{2}} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a c} c e + 7 \, \left (a c^{3}\right )^{\frac{1}{4}} c d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{2}} + \frac{6 \, c x^{6} e + 7 \, c d x^{5} + 10 \, a x^{2} e + 11 \, a d x}{32 \,{\left (c x^{4} + a\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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