3.7.0 ∫ 1 _____________________ (d+ex)(a+b(d+ex)2+c(d+ex)4)2 dx [626]

Integrand size = 30, antiderivative size = 162 \[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} e}+\frac {\log (d+e x)}{a^2 e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e} \]

Rubi [A] (veriﬁed)

Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1156, 1128, 754, 814, 648, 632, 212, 642} \[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e \left (b^2-4 a c\right )^{3/2}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}+\frac {\log (d+e x)}{a^2 e}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

Rubi steps \begin{align*} \text {integral}& = \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} \\ & = \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} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \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} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \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} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e}-\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} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e}-\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}-\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} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\log (d+e x)}{a^2 e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e}+\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} \\ & = \frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e \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}+\frac {\log (d+e x)}{a^2 e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.26 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {\frac {2 a \left (b^2-2 a c+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+4 \log (d+e x)-\frac {\left (b^3-6 a b c+b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (b^3-6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^2 e} \]

Maple [C] (veriﬁed)

Time = 0.74 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.46


method	result	size

default	\(-\frac {\frac {\frac {a e b c \,x^{2}}{8 a c -2 b^{2}}+\frac {b c d a x}{4 a c -b^{2}}-\frac {a \left (-b c \,d^{2}+2 a c -b^{2}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (e^{3} c \left (4 a c -b^{2}\right ) \textit {\_R}^{3}+3 c d \,e^{2} \left (4 a c -b^{2}\right ) \textit {\_R}^{2}+e \left (12 a \,c^{2} d^{2}-3 b^{2} c \,d^{2}+5 a b c -b^{3}\right ) \textit {\_R} +4 a \,c^{2} d^{3}-b^{2} c \,d^{3}+5 a b c d -b^{3} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 \left (4 a c -b^{2}\right ) e}}{a^{2}}+\frac {\ln \left (e x +d \right )}{a^{2} e}\)	\(399\)
risch	\(\frac {-\frac {c \,x^{2} b e}{2 a \left (4 a c -b^{2}\right )}-\frac {x b c d}{\left (4 a c -b^{2}\right ) a}+\frac {-b c \,d^{2}+2 a c -b^{2}}{2 e a \left (4 a c -b^{2}\right )}}{c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a}+\frac {\ln \left (e x +d \right )}{a^{2} e}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3} e^{2}-48 a^{4} b^{2} c^{2} e^{2}+12 a^{3} b^{4} c \,e^{2}-a^{2} b^{6} e^{2}\right ) \textit {\_Z}^{2}+\left (64 c^{3} a^{3} e -48 a^{2} b^{2} c^{2} e +12 a \,b^{4} c e -b^{6} e \right ) \textit {\_Z} +16 a \,c^{3}-3 b^{2} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 a^{5} c^{3} e^{4}-128 a^{4} b^{2} c^{2} e^{4}+34 a^{3} b^{4} c \,e^{4}-3 a^{2} b^{6} e^{4}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} e^{3}-36 a^{2} b^{2} c^{2} e^{3}+4 a \,b^{4} c \,e^{3}\right ) \textit {\_R} +2 b^{2} c^{2} e^{2}\right ) x^{2}+\left (\left (320 a^{5} c^{3} d \,e^{3}-256 a^{4} b^{2} c^{2} d \,e^{3}+68 a^{3} b^{4} c d \,e^{3}-6 a^{2} b^{6} d \,e^{3}\right ) \textit {\_R}^{2}+\left (160 a^{3} c^{3} d \,e^{2}-72 a^{2} b^{2} c^{2} d \,e^{2}+8 a \,b^{4} c d \,e^{2}\right ) \textit {\_R} +4 b^{2} c^{2} d e \right ) x +\left (160 a^{5} c^{3} d^{2} e^{2}-128 a^{4} b^{2} c^{2} d^{2} e^{2}+34 a^{3} b^{4} c \,d^{2} e^{2}-3 a^{2} b^{6} d^{2} e^{2}-16 a^{5} b \,c^{2} e^{2}+8 a^{4} b^{3} c \,e^{2}-a^{3} b^{5} e^{2}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} d^{2} e -36 a^{2} b^{2} c^{2} d^{2} e +4 a \,b^{4} c \,d^{2} e +36 a^{3} b \,c^{2} e -17 a^{2} b^{3} c e +2 a \,b^{5} e \right ) \textit {\_R} +2 d^{2} c^{2} b^{2}-8 a b \,c^{2}+2 b^{3} c \right )\right )}{2}\)	\(690\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (152) = 304\).

Time = 0.45 (sec) , antiderivative size = 2476, normalized size of antiderivative = 15.28 \[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Too large to display} \]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Timed out} \]

Maxima [F]

\[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\int { \frac {1}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{2} {\left (e x + d\right )}} \,d x } \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (152) = 304\).

Time = 0.40 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.88 \[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=-\frac {{\left (a^{2} b^{3} c e^{3} - 6 \, a^{3} b c^{2} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} + 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{2} b^{3} c e^{3} - 6 \, a^{3} b c^{2} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} - 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{4 \, {\left (a^{4} b^{4} c e^{4} - 8 \, a^{5} b^{2} c^{2} e^{4} + 16 \, a^{6} c^{3} e^{4}\right )}} - \frac {\log \left ({\left | c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4} + b e^{2} x^{2} + 2 \, b d e x + b d^{2} + a \right |}\right )}{4 \, a^{2} e} + \frac {\log \left ({\left | e x + d \right |}\right )}{a^{2} e} + \frac {a b c e^{2} x^{2} + 2 \, a b c d e x + a b c d^{2} + a b^{2} - 2 \, a^{2} c}{2 \, {\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4} + b e^{2} x^{2} + 2 \, b d e x + b d^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{2} e} \]

Mupad [B] (veriﬁcation not implemented)

Time = 14.69 (sec) , antiderivative size = 11072, normalized size of antiderivative = 68.35 \[ \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Too large to display} \]

\(\int \frac {1}{(d+e x) (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\) [626]

Optimal result