Optimal. Leaf size=44 \[ -\frac{f \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]
[Out]
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Rubi [A] time = 0.134701, 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 \[ -\frac{f \tanh ^{-1}\left (\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In] Int[(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]
[Out]
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Rubi in Sympy [A] time = 18.7569, size = 41, normalized size = 0.93 \[ - \frac{f \operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{e \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*f*x+d*f)/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)
[Out]
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Mathematica [A] time = 0.0271992, 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.
[In] Integrate[(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]
[Out]
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Maple [C] time = 0.002, size = 130, normalized size = 3. \[{\frac{f}{2\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( e{\it \_R}+d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{\it \_R}\,c{d}^{2}e+2\,c{d}^{3}+be{\it \_R}+bd}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e f x + d f}{{\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x + d*f)/((e*x + d)^4*c + (e*x + d)^2*b + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.319564, size = 1, normalized size = 0.02 \[ \left [\frac{f \log \left (-\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e x + b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \,{\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \,{\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} +{\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \,{\left (2 \, c d^{3} + b d\right )} e x + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c} e}, \frac{f \arctan \left (-\frac{{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{\sqrt{-b^{2} + 4 \, a c} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x + d*f)/((e*x + d)^4*c + (e*x + d)^2*b + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.23891, size = 189, normalized size = 4.3 \[ - \frac{f \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{- 4 a c f \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} f \sqrt{- \frac{1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} + \frac{f \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (\frac{2 d x}{e} + x^{2} + \frac{4 a c f \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} f \sqrt{- \frac{1}{4 a c - b^{2}}} + b f + 2 c d^{2} f}{2 c e^{2} f} \right )}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x+d*f)/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e f x + d f}{{\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x + d*f)/((e*x + d)^4*c + (e*x + d)^2*b + a),x, algorithm="giac")
[Out]