∫ 1_____________ (df+efx)3(a+b(d+ex)2+c(d+ex)4)2 dx [652]

Optimal. Leaf size=228 \[ -\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2} e f^3}-\frac {2 b \log (d+e x)}{a^3 e f^3}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3} \]

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 228, 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 \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3}-\frac {2 b \log (d+e x)}{a^3 e f^3}-\frac {b^2-3 a c}{a^2 e f^3 \left (b^2-4 a c\right ) (d+e x)^2}-\frac {\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{a^3 e f^3 \left (b^2-4 a c\right )^{3/2}}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f^3 \left (b^2-4 a c\right ) (d+e x)^2 \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)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e f^3}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{2 e f^3}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-2 \left (b^2-3 a c\right )-2 b c x}{x^2 \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^3}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \left (\frac {2 \left (-b^2+3 a c\right )}{a x^2}-\frac {2 b \left (-b^2+4 a c\right )}{a^2 x}+\frac {2 \left (-b^4+5 a b^2 c-3 a^2 c^2-b c \left (b^2-4 a c\right ) x\right )}{a^2 \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^3}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {2 b \log (d+e x)}{a^3 e f^3}-\frac {\text {Subst}\left (\int \frac {-b^4+5 a b^2 c-3 a^2 c^2-b c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{a^3 \left (b^2-4 a c\right ) e f^3}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {2 b \log (d+e x)}{a^3 e f^3}+\frac {b \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^3 e f^3}+\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^3 \left (b^2-4 a c\right ) e f^3}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {2 b \log (d+e x)}{a^3 e f^3}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{a^3 \left (b^2-4 a c\right ) e f^3}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) e f^3 (d+e x)^2}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^3 (d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2} e f^3}-\frac {2 b \log (d+e x)}{a^3 e f^3}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a^3 e f^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.34, size = 287, normalized size = 1.26 \begin {gather*} \frac {-\frac {a}{(d+e x)^2}+\frac {a \left (b^3-3 a b c+b^2 c (d+e x)^2-2 a c^2 (d+e x)^2\right )}{\left (-b^2+4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-4 b \log (d+e x)+\frac {\left (b^4-6 a b^2 c+6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b 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^4+6 a b^2 c-6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b 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}}}{2 a^3 e f^3} \end {gather*}

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.25, size = 466, normalized size = 2.04


method	result	size

default	\(\frac {-\frac {1}{2 a^{2} e \left (e x +d \right )^{2}}-\frac {2 b \ln \left (e x +d \right )}{a^{3} e}-\frac {\frac {\frac {a c e \left (2 a c -b^{2}\right ) x^{2}}{8 a c -2 b^{2}}+\frac {c d a \left (2 a c -b^{2}\right ) x}{4 a c -b^{2}}+\frac {a \left (2 a \,c^{2} d^{2}-b^{2} c \,d^{2}+3 a b c -b^{3}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (e^{3} b c \left (-4 a c +b^{2}\right ) \textit {\_R}^{3}+3 d \,e^{2} b c \left (-4 a c +b^{2}\right ) \textit {\_R}^{2}+e \left (-12 a b \,c^{2} d^{2}+3 b^{3} c \,d^{2}+3 a^{2} c^{2}-5 a \,b^{2} c +b^{4}\right ) \textit {\_R} -4 a b \,c^{2} d^{3}+b^{3} c \,d^{3}+3 a^{2} c^{2} d -5 a \,b^{2} c d +b^{4} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{\left (4 a c -b^{2}\right ) e}}{a^{3}}}{f^{3}}\)	\(466\)
risch	\(\frac {-\frac {\left (3 a c -b^{2}\right ) e^{3} c \,x^{4}}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {4 \left (3 a c -b^{2}\right ) c d \,e^{2} x^{3}}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {\left (36 a \,c^{2} d^{2}-12 b^{2} c \,d^{2}+7 a b c -2 b^{3}\right ) e \,x^{2}}{2 a^{2} \left (4 a c -b^{2}\right )}-\frac {d \left (12 a \,c^{2} d^{2}-4 b^{2} c \,d^{2}+7 a b c -2 b^{3}\right ) x}{a^{2} \left (4 a c -b^{2}\right )}-\frac {6 a \,c^{2} d^{4}-2 b^{2} c \,d^{4}+7 a b c \,d^{2}-2 b^{3} d^{2}+4 a^{2} c -a \,b^{2}}{2 e \,a^{2} \left (4 a c -b^{2}\right )}}{f^{3} \left (e x +d \right )^{2} \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 {2 b \ln \left (e x +d \right )}{a^{3} e \,f^{3}}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (64 a^{6} c^{3} e^{2} f^{6}-48 a^{5} b^{2} c^{2} e^{2} f^{6}+12 a^{4} b^{4} c \,e^{2} f^{6}-a^{3} b^{6} e^{2} f^{6}\right ) \textit {\_Z}^{2}+\left (-64 a^{3} b \,c^{3} e \,f^{3}+48 a^{2} b^{3} c^{2} e \,f^{3}-12 a \,b^{5} c e \,f^{3}+b^{7} e \,f^{3}\right ) \textit {\_Z} +9 a \,c^{4}-2 b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 a^{7} c^{3} e^{4} f^{6}-128 a^{6} b^{2} c^{2} e^{4} f^{6}+34 a^{5} b^{4} c \,e^{4} f^{6}-3 a^{4} b^{6} e^{4} f^{6}\right ) \textit {\_R}^{2}+\left (-68 a^{4} b \,c^{3} e^{3} f^{3}+33 a^{3} b^{3} c^{2} e^{3} f^{3}-4 a^{2} b^{5} c \,e^{3} f^{3}\right ) \textit {\_R} +18 a^{2} c^{4} e^{2}-12 a \,b^{2} c^{3} e^{2}+2 b^{4} c^{2} e^{2}\right ) x^{2}+\left (\left (320 a^{7} c^{3} d \,e^{3} f^{6}-256 a^{6} b^{2} c^{2} d \,e^{3} f^{6}+68 a^{5} b^{4} c d \,e^{3} f^{6}-6 a^{4} b^{6} d \,e^{3} f^{6}\right ) \textit {\_R}^{2}+\left (-136 a^{4} b \,c^{3} d \,e^{2} f^{3}+66 a^{3} b^{3} c^{2} d \,e^{2} f^{3}-8 a^{2} b^{5} c d \,e^{2} f^{3}\right ) \textit {\_R} +36 a^{2} c^{4} d e -24 a \,b^{2} c^{3} d e +4 b^{4} c^{2} d e \right ) x +\left (160 a^{7} c^{3} d^{2} e^{2} f^{6}-128 a^{6} b^{2} c^{2} d^{2} e^{2} f^{6}+34 a^{5} b^{4} c \,d^{2} e^{2} f^{6}-3 a^{4} b^{6} d^{2} e^{2} f^{6}-16 a^{7} b \,c^{2} e^{2} f^{6}+8 a^{6} b^{3} c \,e^{2} f^{6}-a^{5} b^{5} e^{2} f^{6}\right ) \textit {\_R}^{2}+\left (-68 a^{4} b \,c^{3} d^{2} e \,f^{3}+33 a^{3} b^{3} c^{2} d^{2} e \,f^{3}-4 a^{2} b^{5} c \,d^{2} e \,f^{3}+12 a^{5} c^{3} e \,f^{3}-39 a^{4} b^{2} c^{2} e \,f^{3}+17 a^{3} b^{4} c e \,f^{3}-2 a^{2} b^{6} e \,f^{3}\right ) \textit {\_R} +18 a^{2} c^{4} d^{2}-12 a \,b^{2} c^{3} d^{2}+2 b^{4} c^{2} d^{2}+24 a^{2} b \,c^{3}-14 a \,b^{3} c^{2}+2 b^{5} c \right )\right )\)	\(1064\)

________________________________________________________________________________________

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. 2217 vs. \(2 (224) = 448\).
time = 1.34, size = 4560, normalized size = 20.00 \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. 687 vs. \(2 (224) = 448\).
time = 3.06, size = 687, normalized size = 3.01 \begin {gather*} \frac {{\left (a^{3} b^{4} c f^{3} e^{3} - 6 \, a^{4} b^{2} c^{2} f^{3} e^{3} + 6 \, a^{5} c^{3} f^{3} 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^{3} b^{4} c f^{3} e^{3} - 6 \, a^{4} b^{2} c^{2} f^{3} e^{3} + 6 \, a^{5} c^{3} f^{3} 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 )}{2 \, {\left (a^{6} b^{4} c f^{6} e^{4} - 8 \, a^{7} b^{2} c^{2} f^{6} e^{4} + 16 \, a^{8} c^{3} f^{6} e^{4}\right )}} + \frac {b 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 )}{2 \, a^{3} f^{3}} - \frac {2 \, b e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{3} f^{3}} - \frac {{\left (2 \, a b^{2} c d^{4} - 6 \, a^{2} c^{2} d^{4} + 2 \, a b^{3} d^{2} - 7 \, a^{2} b c d^{2} + 2 \, {\left (a b^{2} c e^{4} - 3 \, a^{2} c^{2} e^{4}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 8 \, {\left (a b^{2} c d e^{3} - 3 \, a^{2} c^{2} d e^{3}\right )} x^{3} + {\left (12 \, a b^{2} c d^{2} e^{2} - 36 \, a^{2} c^{2} d^{2} e^{2} + 2 \, a b^{3} e^{2} - 7 \, a^{2} b c e^{2}\right )} x^{2} + 2 \, {\left (4 \, a b^{2} c d^{3} e - 12 \, a^{2} c^{2} d^{3} e + 2 \, a b^{3} d e - 7 \, a^{2} b c d e\right )} x\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 )} {\left (x e + d\right )}^{2} a^{3} f^{3}} \end {gather*}

________________________________________________________________________________________

Mupad [B]
time = 13.52, size = 2500, normalized size = 10.96 \begin {gather*} \text {Too large to display} \end {gather*}

________________________________________________________________________________________

3.7.52 \(\int \frac {1}{(d f+e f x)^3 (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\) [652]