Optimal. Leaf size=44 \[ -\frac{f \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.0631629, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1142, 1107, 618, 206} \[ -\frac{f \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 1142
Rule 1107
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{d f+e f x}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac{f \operatorname{Subst}\left (\int \frac{x}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac{f \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=-\frac{f \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{e}\\ &=-\frac{f \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} e}\\ \end{align*}
Mathematica [A] time = 0.0172062, size = 47, normalized size = 1.07 \[ \frac{f \tan ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right )}{e \sqrt{4 a c-b^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.005, size = 130, normalized size = 3. \begin{align*}{\frac{f}{2\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ({\it \_R}\,e+d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,c{d}^{2}e{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e f x + d f}{{\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55573, size = 603, normalized size = 13.7 \begin{align*} \left [\frac{f \log \left (\frac{2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \,{\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \,{\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c -{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c} e}, -\frac{\sqrt{-b^{2} + 4 \, a c} f \arctan \left (-\frac{{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{{\left (b^{2} - 4 \, a c\right )} e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.3556, size = 189, normalized size = 4.3 \begin{align*} - \frac{f \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 4 a c f \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} f \sqrt{- \frac{1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} + \frac{f \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{4 a c f \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} f \sqrt{- \frac{1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.43048, size = 250, normalized size = 5.68 \begin{align*} \frac{\sqrt{b^{2} - 4 \, a c} f e \log \left ({\left |{\left (b + \sqrt{b^{2} - 4 \, a c}\right )} x^{2} e^{2} + 2 \,{\left (b + \sqrt{b^{2} - 4 \, a c}\right )} d x e +{\left (b + \sqrt{b^{2} - 4 \, a c}\right )} d^{2} + 2 \, a \right |}\right )}{2 \,{\left (b^{2} e^{2} - 4 \, a c e^{2}\right )}} - \frac{\sqrt{b^{2} - 4 \, a c} f e \log \left ({\left | -{\left (b - \sqrt{b^{2} - 4 \, a c}\right )} x^{2} e^{2} - 2 \,{\left (b - \sqrt{b^{2} - 4 \, a c}\right )} d x e -{\left (b - \sqrt{b^{2} - 4 \, a c}\right )} d^{2} - 2 \, a \right |}\right )}{2 \,{\left (b^{2} e^{2} - 4 \, a c e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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