∫ 1____________ (df+efx)(a+b(d+ex)2+c(d+ex)4)2 dx [650]

Optimal. Leaf size=174 \[ \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} e f}+\frac {\log (d+e x)}{a^2 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f} \]

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1156, 1128, 754, 814, 648, 632, 212, 642} \begin {gather*} \frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e f \left (b^2-4 a c\right )^{3/2}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f}+\frac {\log (d+e x)}{a^2 e f}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \end {gather*}

\begin {align*} \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{2 e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-b^2+4 a c-b c x}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 a \left (b^2-4 a c\right ) e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \left (\frac {-b^2+4 a c}{a x}+\frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,(d+e x)^2\right )}{2 a \left (b^2-4 a c\right ) e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e f}-\frac {\text {Subst}\left (\int \frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^2 \left (b^2-4 a c\right ) e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e f}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 e f}-\frac {\left (b \left (b^2-6 a c\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 \left (b^2-4 a c\right ) e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f}+\frac {\left (b \left (b^2-6 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a^2 \left (b^2-4 a c\right ) e f}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} e f}+\frac {\log (d+e x)}{a^2 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.30, size = 238, normalized size = 1.37 \begin {gather*} \frac {\frac {2 a \left (b^2-2 a c+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+4 \log (d+e x)-\frac {\left (b^3-6 a b c+b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (b^3-6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^2 e f} \end {gather*}

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.29, size = 403, normalized size = 2.32


method	result	size

default	\(\frac {\frac {\ln \left (e x +d \right )}{e \,a^{2}}-\frac {\frac {\frac {a b c e \,x^{2}}{8 a c -2 b^{2}}+\frac {b c d a x}{4 a c -b^{2}}-\frac {a \left (-b c \,d^{2}+2 a c -b^{2}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (e^{3} c \left (4 a c -b^{2}\right ) \textit {\_R}^{3}+3 d \,e^{2} c \left (4 a c -b^{2}\right ) \textit {\_R}^{2}+e \left (12 a \,c^{2} d^{2}-3 b^{2} c \,d^{2}+5 a b c -b^{3}\right ) \textit {\_R} +4 a \,c^{2} d^{3}-b^{2} c \,d^{3}+5 a b c d -b^{3} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 \left (4 a c -b^{2}\right ) e}}{a^{2}}}{f}\)	\(403\)
risch	\(\frac {-\frac {c \,x^{2} b e}{2 a \left (4 a c -b^{2}\right )}-\frac {x b c d}{\left (4 a c -b^{2}\right ) a}+\frac {-b c \,d^{2}+2 a c -b^{2}}{2 e a \left (4 a c -b^{2}\right )}}{f \left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )}+\frac {\ln \left (e x +d \right )}{a^{2} e f}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (64 a^{5} c^{3} e^{2} f^{2}-48 a^{4} b^{2} c^{2} e^{2} f^{2}+12 a^{3} b^{4} c \,e^{2} f^{2}-a^{2} b^{6} e^{2} f^{2}\right ) \textit {\_Z}^{2}+\left (64 a^{3} c^{3} e f -48 a^{2} b^{2} c^{2} e f +12 a \,b^{4} c e f -b^{6} e f \right ) \textit {\_Z} +16 a \,c^{3}-3 b^{2} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 a^{5} c^{3} e^{4} f^{2}-128 a^{4} b^{2} c^{2} e^{4} f^{2}+34 a^{3} b^{4} c \,e^{4} f^{2}-3 a^{2} b^{6} e^{4} f^{2}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} e^{3} f -36 a^{2} b^{2} c^{2} e^{3} f +4 a \,b^{4} c \,e^{3} f \right ) \textit {\_R} +2 b^{2} c^{2} e^{2}\right ) x^{2}+\left (\left (320 a^{5} c^{3} d \,e^{3} f^{2}-256 a^{4} b^{2} c^{2} d \,e^{3} f^{2}+68 a^{3} b^{4} c d \,e^{3} f^{2}-6 a^{2} b^{6} d \,e^{3} f^{2}\right ) \textit {\_R}^{2}+\left (160 a^{3} c^{3} d \,e^{2} f -72 a^{2} b^{2} c^{2} d \,e^{2} f +8 a \,b^{4} c d \,e^{2} f \right ) \textit {\_R} +4 b^{2} c^{2} d e \right ) x +\left (160 a^{5} c^{3} d^{2} e^{2} f^{2}-128 a^{4} b^{2} c^{2} d^{2} e^{2} f^{2}+34 a^{3} b^{4} c \,d^{2} e^{2} f^{2}-3 a^{2} b^{6} d^{2} e^{2} f^{2}-16 a^{5} b \,c^{2} e^{2} f^{2}+8 a^{4} b^{3} c \,e^{2} f^{2}-a^{3} b^{5} e^{2} f^{2}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} d^{2} e f -36 a^{2} b^{2} c^{2} d^{2} e f +4 a \,b^{4} c \,d^{2} e f +36 a^{3} b \,c^{2} e f -17 a^{2} b^{3} c e f +2 a \,b^{5} e f \right ) \textit {\_R} +2 d^{2} c^{2} b^{2}-8 a b \,c^{2}+2 b^{3} c \right )\right )}{2}\)	\(769\)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1168 vs. \(2 (167) = 334\).
time = 0.60, size = 2462, normalized size = 14.15 \begin {gather*} \text {Too large to display} \end {gather*}

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (167) = 334\).
time = 3.98, size = 476, normalized size = 2.74 \begin {gather*} -\frac {{\left (a^{2} b^{3} c f e^{3} - 6 \, a^{3} b c^{2} f e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{2} b^{3} c f e^{3} - 6 \, a^{3} b c^{2} f e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{4 \, {\left (a^{4} b^{4} c f^{2} e^{4} - 8 \, a^{5} b^{2} c^{2} f^{2} e^{4} + 16 \, a^{6} c^{3} f^{2} e^{4}\right )}} - \frac {e^{\left (-1\right )} \log \left ({\left | c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a \right |}\right )}{4 \, a^{2} f} + \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{2} f} + \frac {{\left (a b c x^{2} e^{2} + 2 \, a b c d x e + a b c d^{2} + a b^{2} - 2 \, a^{2} c\right )} e^{\left (-1\right )}}{2 \, {\left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{2} f} \end {gather*}

________________________________________________________________________________________

Mupad [B]
time = 11.69, size = 2500, normalized size = 14.37 \begin {gather*} \text {Too large to display} \end {gather*}

________________________________________________________________________________________

3.7.50 \(\int \frac {1}{(d f+e f x) (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\) [650]