Optimal. Leaf size=394 \[ -\frac{3 d \left (7 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}+\frac{3 d \left (7 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}-\frac{3 d \left (5 \sqrt{a} e^2+7 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{3 d \left (5 \sqrt{a} e^2+7 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{x \left (18 d^2 e x+7 d^3+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{9 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}-\frac{a e^3-c x \left (3 d^2 e x+d^3+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.346238, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647, Rules used = {1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{3 d \left (7 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}+\frac{3 d \left (7 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}-\frac{3 d \left (5 \sqrt{a} e^2+7 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{3 d \left (5 \sqrt{a} e^2+7 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{x \left (18 d^2 e x+7 d^3+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{9 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}-\frac{a e^3-c x \left (3 d^2 e x+d^3+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1854
Rule 1855
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+c x^4\right )^3} \, dx &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}-\frac{\int \frac{-7 d^3-18 d^2 e x-15 d e^2 x^2}{\left (a+c x^4\right )^2} \, dx}{8 a}\\ &=\frac{x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac{\int \frac{21 d^3+36 d^2 e x+15 d e^2 x^2}{a+c x^4} \, dx}{32 a^2}\\ &=\frac{x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac{\int \left (\frac{36 d^2 e x}{a+c x^4}+\frac{21 d^3+15 d e^2 x^2}{a+c x^4}\right ) \, dx}{32 a^2}\\ &=\frac{x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac{\int \frac{21 d^3+15 d e^2 x^2}{a+c x^4} \, dx}{32 a^2}+\frac{\left (9 d^2 e\right ) \int \frac{x}{a+c x^4} \, dx}{8 a^2}\\ &=\frac{x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac{\left (9 d^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{16 a^2}+\frac{\left (3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{64 a^2 c}+\frac{\left (3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}+5 e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{64 a^2 c}\\ &=\frac{x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac{9 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}-\frac{\left (3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right )\right ) \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^{9/4} c^{3/4}}-\frac{\left (3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right )\right ) \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^{9/4} c^{3/4}}+\frac{\left (3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}+5 e^2\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c}+\frac{\left (3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}+5 e^2\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c}\\ &=\frac{x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac{9 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}-\frac{3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}+\frac{3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}+\frac{\left (3 d \left (7 \sqrt{c} d^2+5 \sqrt{a} e^2\right )\right ) \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} c^{3/4}}-\frac{\left (3 d \left (7 \sqrt{c} d^2+5 \sqrt{a} e^2\right )\right ) \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} c^{3/4}}\\ &=\frac{x \left (7 d^3+18 d^2 e x+15 d e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{8 a c \left (a+c x^4\right )^2}+\frac{9 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}-\frac{3 d \left (7 \sqrt{c} d^2+5 \sqrt{a} e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{3 d \left (7 \sqrt{c} d^2+5 \sqrt{a} e^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}-\frac{3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}+\frac{3 d \left (\frac{7 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.35593, size = 388, normalized size = 0.98 \[ \frac{\frac{3 \sqrt{2} \left (5 a^{3/4} d e^2-7 \sqrt [4]{a} \sqrt{c} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}+\frac{3 \sqrt{2} \left (7 \sqrt [4]{a} \sqrt{c} d^3-5 a^{3/4} d e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{32 a^2 \left (a e^3-c d x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{c \left (a+c x^4\right )^2}-\frac{6 \sqrt [4]{a} d \left (24 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt{2} \sqrt{a} e^2+7 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{6 \sqrt [4]{a} d \left (-24 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt{2} \sqrt{a} e^2+7 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 a d x \left (7 d^2+18 d e x+15 e^2 x^2\right )}{a+c x^4}}{256 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 470, normalized size = 1.2 \begin{align*}{\frac{{d}^{3}x}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{7\,{d}^{3}x}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{21\,{d}^{3}\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}^{3}\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}^{3}\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{3\,e{d}^{2}{x}^{2}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{9\,e{d}^{2}{x}^{2}}{16\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{9\,e{d}^{2}}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,d{e}^{2}{x}^{3}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{15\,d{e}^{2}{x}^{3}}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{15\,d{e}^{2}\sqrt{2}}{256\,{a}^{2}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{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{15\,d{e}^{2}\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{15\,d{e}^{2}\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{3}{x}^{4}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{{e}^{3}{x}^{4}}{8\,{a}^{2} \left ( c{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 [A] time = 9.52886, size = 413, normalized size = 1.05 \begin{align*} \operatorname{RootSum}{\left (268435456 t^{4} a^{11} c^{3} + 63111168 t^{2} a^{6} c^{2} d^{4} e^{2} + t \left (4147200 a^{4} c d^{4} e^{5} - 8128512 a^{3} c^{2} d^{8} e\right ) + 50625 a^{2} d^{4} e^{8} + 245106 a c d^{8} e^{4} + 194481 c^{2} d^{12}, \left ( t \mapsto t \log{\left (x + \frac{262144000 t^{3} a^{10} c^{2} e^{6} + 3714056192 t^{3} a^{9} c^{3} d^{4} e^{2} - 539688960 t^{2} a^{7} c^{2} d^{4} e^{5} + 202309632 t^{2} a^{6} c^{3} d^{8} e + 77328000 t a^{5} c d^{4} e^{8} + 660699648 t a^{4} c^{2} d^{8} e^{4} + 19361664 t a^{3} c^{3} d^{12} + 3037500 a^{3} d^{4} e^{11} - 26360640 a^{2} c d^{8} e^{7} - 60566940 a c^{2} d^{12} e^{3}}{421875 a^{3} d^{3} e^{12} - 29598075 a^{2} c d^{7} e^{8} - 58012227 a c^{2} d^{11} e^{4} + 3176523 c^{3} d^{15}} \right )} \right )\right )} + \frac{- 4 a^{2} e^{3} + 11 a c d^{3} x + 30 a c d^{2} e x^{2} + 27 a c d e^{2} x^{3} + 7 c^{2} d^{3} x^{5} + 18 c^{2} d^{2} e x^{6} + 15 c^{2} d e^{2} x^{7}}{32 a^{4} c + 64 a^{3} c^{2} x^{4} + 32 a^{2} c^{3} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.211, size = 525, normalized size = 1.33 \begin{align*} \frac{3 \, \sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + 7 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\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^{3}} + \frac{3 \, \sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + 7 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\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^{3}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} - \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} + \frac{15 \, c^{2} d x^{7} e^{2} + 18 \, c^{2} d^{2} x^{6} e + 7 \, c^{2} d^{3} x^{5} + 27 \, a c d x^{3} e^{2} + 30 \, a c d^{2} x^{2} e + 11 \, a c d^{3} x - 4 \, a^{2} e^{3}}{32 \,{\left (c x^{4} + a\right )}^{2} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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