Optimal. Leaf size=349 \[ -\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}-\frac{a e^3-c x \left (3 d^2 e x+d^3+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )} \]
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Rubi [A] time = 0.298368, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.706, Rules used = {1854, 27, 12, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}-\frac{a e^3-c x \left (3 d^2 e x+d^3+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1854
Rule 27
Rule 12
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 )^2} \, dx &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}-\frac{\int \frac{-3 d^3-6 d^2 e x-3 d e^2 x^2}{a+c x^4} \, dx}{4 a}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}-\frac{\int -\frac{3 d (d+e x)^2}{a+c x^4} \, dx}{4 a}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{(3 d) \int \frac{(d+e x)^2}{a+c x^4} \, dx}{4 a}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{(3 d) \int \left (\frac{2 d e x}{a+c x^4}+\frac{d^2+e^2 x^2}{a+c x^4}\right ) \, dx}{4 a}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{(3 d) \int \frac{d^2+e^2 x^2}{a+c x^4} \, dx}{4 a}+\frac{\left (3 d^2 e\right ) \int \frac{x}{a+c x^4} \, dx}{2 a}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{\left (3 d^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 a}+\frac{\left (3 d \left (\frac{\sqrt{c} d^2}{\sqrt{a}}-e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a c}+\frac{\left (3 d \left (\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a c}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{\left (3 d \left (\frac{\sqrt{c} d^2}{\sqrt{a}}+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}{16 a c}+\frac{\left (3 d \left (\frac{\sqrt{c} d^2}{\sqrt{a}}+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}{16 a c}-\frac{\left (3 d \left (\sqrt{c} d^2-\sqrt{a} 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}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (3 d \left (\sqrt{c} d^2-\sqrt{a} 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}{16 \sqrt{2} a^{7/4} c^{3/4}}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}-\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (3 d \left (\sqrt{c} d^2+\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 )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (3 d \left (\sqrt{c} d^2+\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 )}{8 \sqrt{2} a^{7/4} c^{3/4}}\\ &=-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{3 d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}-\frac{3 d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{3 d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.386398, size = 347, normalized size = 0.99 \[ \frac{3 \sqrt{2} \sqrt [4]{c} \left (a^{3/4} d e^2-\sqrt [4]{a} \sqrt{c} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+3 \sqrt{2} \sqrt [4]{c} \left (\sqrt [4]{a} \sqrt{c} d^3-a^{3/4} d e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\frac{8 a \left (a e^3-c d x \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{a+c x^4}-6 \sqrt [4]{a} \sqrt [4]{c} d \left (4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt [4]{a} \sqrt [4]{c} d \left (-4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{32 a^2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 390, normalized size = 1.1 \begin{align*}{\frac{{d}^{3}x}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,{d}^{3}\sqrt{2}}{32\,{a}^{2}}\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{3\,{d}^{3}\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,{d}^{3}\sqrt{2}}{16\,{a}^{2}}\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}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,e{d}^{2}}{4\,a}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,d{e}^{2}{x}^{3}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,d{e}^{2}\sqrt{2}}{32\,ac}\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{3\,d{e}^{2}\sqrt{2}}{16\,ac}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,d{e}^{2}\sqrt{2}}{16\,ac}\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}}{4\,a \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 = 6.90911, size = 350, normalized size = 1. \begin{align*} \operatorname{RootSum}{\left (65536 t^{4} a^{7} c^{3} + 27648 t^{2} a^{4} c^{2} d^{4} e^{2} + t \left (3456 a^{3} c d^{4} e^{5} - 3456 a^{2} c^{2} d^{8} e\right ) + 81 a^{2} d^{4} e^{8} + 162 a c d^{8} e^{4} + 81 c^{2} d^{12}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} c^{2} e^{6} + 28672 t^{3} a^{6} c^{3} d^{4} e^{2} - 7680 t^{2} a^{5} c^{2} d^{4} e^{5} + 1536 t^{2} a^{4} c^{3} d^{8} e + 2160 t a^{4} c d^{4} e^{8} + 9216 t a^{3} c^{2} d^{8} e^{4} + 144 t a^{2} c^{3} d^{12} + 162 a^{3} d^{4} e^{11} - 648 a^{2} c d^{8} e^{7} - 810 a c^{2} d^{12} e^{3}}{27 a^{3} d^{3} e^{12} - 891 a^{2} c d^{7} e^{8} - 891 a c^{2} d^{11} e^{4} + 27 c^{3} d^{15}} \right )} \right )\right )} + \frac{- a e^{3} + c d^{3} x + 3 c d^{2} e x^{2} + 3 c d e^{2} x^{3}}{4 a^{2} c + 4 a c^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27711, size = 462, normalized size = 1.32 \begin{align*} \frac{3 \, c d x^{3} e^{2} + 3 \, c d^{2} x^{2} e + c d^{3} x - a e^{3}}{4 \,{\left (c x^{4} + a\right )} a c} + \frac{3 \, \sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + \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 )}{16 \, a^{2} c^{3}} + \frac{3 \, \sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a c} c^{2} d^{2} e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} + \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 )}{16 \, a^{2} c^{3}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \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 )}{32 \, a^{2} c^{3}} - \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \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 )}{32 \, a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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