Optimal. Leaf size=123 \[ \frac{\left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2}+\frac{e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c} \]
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Rubi [A] time = 0.0968408, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {702, 635, 205, 260} \[ \frac{\left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2}+\frac{e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 702
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{a+c x^2} \, dx &=\int \left (\frac{e^2 \left (6 c d^2-a e^2\right )}{c^2}+\frac{4 d e^3 x}{c}+\frac{e^4 x^2}{c}+\frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c}+\frac{\int \frac{c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x}{a+c x^2} \, dx}{c^2}\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c}+\frac{\left (4 d e \left (c d^2-a e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c}+\frac{\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \int \frac{1}{a+c x^2} \, dx}{c^2}\\ &=\frac{e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c}+\frac{\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2}\\ \end{align*}
Mathematica [A] time = 0.0872038, size = 111, normalized size = 0.9 \[ \frac{\left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{e \left (6 \left (c d^3-a d e^2\right ) \log \left (a+c x^2\right )-3 a e^3 x+c e x \left (18 d^2+6 d e x+e^2 x^2\right )\right )}{3 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 150, normalized size = 1.2 \begin{align*}{\frac{{e}^{4}{x}^{3}}{3\,c}}+2\,{\frac{d{e}^{3}{x}^{2}}{c}}-{\frac{{e}^{4}ax}{{c}^{2}}}+6\,{\frac{{d}^{2}{e}^{2}x}{c}}-2\,{\frac{\ln \left ( c{x}^{2}+a \right ) ad{e}^{3}}{{c}^{2}}}+2\,{\frac{\ln \left ( c{x}^{2}+a \right ){d}^{3}e}{c}}+{\frac{{a}^{2}{e}^{4}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-6\,{\frac{a{d}^{2}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{{d}^{4}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \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 [A] time = 2.01712, size = 595, normalized size = 4.84 \begin{align*} \left [\frac{2 \, a c^{2} e^{4} x^{3} + 12 \, a c^{2} d e^{3} x^{2} - 3 \,{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 6 \,{\left (6 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x + 12 \,{\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{6 \, a c^{3}}, \frac{a c^{2} e^{4} x^{3} + 6 \, a c^{2} d e^{3} x^{2} + 3 \,{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 3 \,{\left (6 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x + 6 \,{\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{3 \, a c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.24195, size = 401, normalized size = 3.26 \begin{align*} \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log{\left (x + \frac{4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log{\left (x + \frac{4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \frac{2 d e^{3} x^{2}}{c} + \frac{e^{4} x^{3}}{3 c} - \frac{x \left (a e^{4} - 6 c d^{2} e^{2}\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25517, size = 153, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (c d^{3} e - a d e^{3}\right )} \log \left (c x^{2} + a\right )}{c^{2}} + \frac{{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{c^{2} x^{3} e^{4} + 6 \, c^{2} d x^{2} e^{3} + 18 \, c^{2} d^{2} x e^{2} - 3 \, a c x e^{4}}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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