Optimal. Leaf size=291 \[ -\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]
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Rubi [A] time = 0.198006, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]
Antiderivative was successfully verified.
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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)^2}{a+c x^4} \, dx &=\int \left (\frac{2 d e x}{a+c x^4}+\frac{d^2+e^2 x^2}{a+c x^4}\right ) \, dx\\ &=(2 d e) \int \frac{x}{a+c x^4} \, dx+\int \frac{d^2+e^2 x^2}{a+c x^4} \, dx\\ &=(d e) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )+\frac{\left (\frac{\sqrt{c} d^2}{\sqrt{a}}-e^2\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{2 c}+\frac{\left (\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{2 c}\\ &=\frac{d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{\left (\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}+\frac{\left (\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\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}{4 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{c} d^2-\sqrt{a} e^2\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}{4 \sqrt{2} a^{3/4} c^{3/4}}\\ &=\frac{d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{\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 )}{4 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\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 )}{4 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{3/4}}\\ &=\frac{d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{\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 )}{2 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\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 )}{2 \sqrt{2} a^{3/4} c^{3/4}}-\frac{\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 )}{4 \sqrt{2} a^{3/4} c^{3/4}}+\frac{\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 )}{4 \sqrt{2} a^{3/4} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.103288, size = 243, normalized size = 0.84 \[ \frac{-\sqrt{2} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )\right )-2 \left (4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (-4 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 a^{3/4} c^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 292, normalized size = 1. \begin{align*}{\frac{{d}^{2}\sqrt{2}}{8\,a}\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{{d}^{2}\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{{d}^{2}\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{de\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{2}\sqrt{2}}{8\,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{{e}^{2}\sqrt{2}}{4\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{2}\sqrt{2}}{4\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \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 = 2.56115, size = 277, normalized size = 0.95 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} c^{3} + 192 t^{2} a^{2} c^{2} d^{2} e^{2} + t \left (32 a^{2} c d e^{5} - 32 a c^{2} d^{5} e\right ) + a^{2} e^{8} + 2 a c d^{4} e^{4} + c^{2} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{64 t^{3} a^{4} c^{2} e^{6} + 448 t^{3} a^{3} c^{3} d^{4} e^{2} - 160 t^{2} a^{3} c^{2} d^{3} e^{5} + 32 t^{2} a^{2} c^{3} d^{7} e + 60 t a^{3} c d^{2} e^{8} + 256 t a^{2} c^{2} d^{6} e^{4} + 4 t a c^{3} d^{10} + 6 a^{3} d e^{11} - 24 a^{2} c d^{5} e^{7} - 30 a c^{2} d^{9} e^{3}}{a^{3} e^{12} - 33 a^{2} c d^{4} e^{8} - 33 a c^{2} d^{8} e^{4} + c^{3} d^{12}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11494, size = 385, normalized size = 1.32 \begin{align*} \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a c} c^{2} d e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac{3}{4}} 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 )}{4 \, a c^{3}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a c} c^{2} d e + \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac{3}{4}} 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 )}{4 \, a c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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