Optimal. Leaf size=263 \[ -\frac{f^2 (d+e x) \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{\sqrt{c} f^2 \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} e \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} f^2 \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} e \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.967378, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{f^2 (d+e x) \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{\sqrt{c} f^2 \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} e \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} f^2 \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} e \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
[In] Int[(d*f + e*f*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 72.5918, size = 236, normalized size = 0.9 \[ - \frac{\sqrt{2} \sqrt{c} f^{2} \left (b + \frac{\sqrt{- 4 a c + b^{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{e \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} \sqrt{c} f^{2} \left (b - \frac{\sqrt{- 4 a c + b^{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \left (d + e x\right )}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{e \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{f^{2} \left (b + 2 c \left (d + e x\right )^{2}\right ) \left (d + e x\right )}{2 e \left (- 4 a c + b^{2}\right ) \left (a + b \left (d + e x\right )^{2} + c \left (d + e x\right )^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*f*x+d*f)**2/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 1.72752, size = 250, normalized size = 0.95 \[ -\frac{f^2 \left (\frac{b (d+e x)+2 c (d+e x)^3}{\left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}-2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In] Integrate[(d*f + e*f*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.008, size = 693, normalized size = 2.6 \[{\frac{c{e}^{2}{f}^{2}{x}^{3}}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+3\,{\frac{cde{f}^{2}{x}^{2}}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+3\,{\frac{{f}^{2}xc{d}^{2}}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{b{f}^{2}x}{ \left ( 2\,c{e}^{4}{x}^{4}+8\,cd{e}^{3}{x}^{3}+12\,c{d}^{2}{e}^{2}{x}^{2}+8\,c{d}^{3}ex+2\,b{e}^{2}{x}^{2}+2\,c{d}^{4}+4\,bdex+2\,b{d}^{2}+2\,a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{f}^{2}{d}^{3}c}{ \left ( c{e}^{4}{x}^{4}+4\,cd{e}^{3}{x}^{3}+6\,c{d}^{2}{e}^{2}{x}^{2}+4\,c{d}^{3}ex+b{e}^{2}{x}^{2}+c{d}^{4}+2\,bdex+b{d}^{2}+a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{d{f}^{2}b}{ \left ( 2\,c{e}^{4}{x}^{4}+8\,cd{e}^{3}{x}^{3}+12\,c{d}^{2}{e}^{2}{x}^{2}+8\,c{d}^{3}ex+2\,b{e}^{2}{x}^{2}+2\,c{d}^{4}+4\,bdex+2\,b{d}^{2}+2\,a \right ) e \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{f}^{2}}{4\,e}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( 2\,{{\it \_R}}^{2}c{e}^{2}+4\,{\it \_R}\,cde+2\,c{d}^{2}-b \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,{\it \_R}\,c{d}^{2}e+2\,c{d}^{3}+be{\it \_R}+bd \right ) }}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{1}{2} \, f^{2} \int -\frac{2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} - b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{4} x^{4} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e^{3} x^{3} +{\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} +{\left (b^{3} - 4 \, a b c + 6 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{2} x^{2} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} d^{2} + 2 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} +{\left (b^{3} - 4 \, a b c\right )} d\right )} e x}\,{d x} - \frac{2 \, c e^{3} f^{2} x^{3} + 6 \, c d e^{2} f^{2} x^{2} +{\left (6 \, c d^{2} + b\right )} e f^{2} x +{\left (2 \, c d^{3} + b d\right )} f^{2}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{5} x^{4} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d e^{4} x^{3} +{\left (b^{3} - 4 \, a b c + 6 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{3} x^{2} + 2 \,{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} +{\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2} x +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x + d*f)^2/((e*x + d)^4*c + (e*x + d)^2*b + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.308677, size = 3510, normalized size = 13.35 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x + d*f)^2/((e*x + d)^4*c + (e*x + d)^2*b + a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 157.283, size = 646, normalized size = 2.46 \[ \frac{b d f^{2} + 2 c d^{3} f^{2} + 6 c d e^{2} f^{2} x^{2} + 2 c e^{3} f^{2} x^{3} + x \left (b e f^{2} + 6 c d^{2} e f^{2}\right )}{8 a^{2} c e - 2 a b^{2} e + 8 a b c d^{2} e + 8 a c^{2} d^{4} e - 2 b^{3} d^{2} e - 2 b^{2} c d^{4} e + x^{4} \left (8 a c^{2} e^{5} - 2 b^{2} c e^{5}\right ) + x^{3} \left (32 a c^{2} d e^{4} - 8 b^{2} c d e^{4}\right ) + x^{2} \left (8 a b c e^{3} + 48 a c^{2} d^{2} e^{3} - 2 b^{3} e^{3} - 12 b^{2} c d^{2} e^{3}\right ) + x \left (16 a b c d e^{2} + 32 a c^{2} d^{3} e^{2} - 4 b^{3} d e^{2} - 8 b^{2} c d^{3} e^{2}\right )} + \operatorname{RootSum}{\left (t^{4} \left (1048576 a^{7} c^{6} e^{4} - 1572864 a^{6} b^{2} c^{5} e^{4} + 983040 a^{5} b^{4} c^{4} e^{4} - 327680 a^{4} b^{6} c^{3} e^{4} + 61440 a^{3} b^{8} c^{2} e^{4} - 6144 a^{2} b^{10} c e^{4} + 256 a b^{12} e^{4}\right ) + t^{2} \left (- 12288 a^{4} b c^{4} e^{2} f^{4} + 8192 a^{3} b^{3} c^{3} e^{2} f^{4} - 1536 a^{2} b^{5} c^{2} e^{2} f^{4} + 16 b^{9} e^{2} f^{4}\right ) + 16 a^{2} c^{3} f^{8} + 24 a b^{2} c^{2} f^{8} + 9 b^{4} c f^{8}, \left ( t \mapsto t \log{\left (x + \frac{16384 t^{3} a^{5} c^{4} e^{3} - 8192 t^{3} a^{4} b^{2} c^{3} e^{3} + 512 t^{3} a^{2} b^{6} c e^{3} - 64 t^{3} a b^{8} e^{3} - 128 t a^{2} b c^{2} e f^{4} - 16 t a b^{3} c e f^{4} - 4 t b^{5} e f^{4} + 4 a c^{2} d f^{6} + 3 b^{2} c d f^{6}}{4 a c^{2} e f^{6} + 3 b^{2} c e f^{6}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x+d*f)**2/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e f x + d f\right )}^{2}}{{\left ({\left (e x + d\right )}^{4} c +{\left (e x + d\right )}^{2} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*f*x + d*f)^2/((e*x + d)^4*c + (e*x + d)^2*b + a)^2,x, algorithm="giac")
[Out]