Integrand size = 30, antiderivative size = 81 \[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} e}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]
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Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1156, 1128, 648, 632, 212, 642} \[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c e \sqrt {b^2-4 a c}}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1128
Rule 1156
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{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 c e}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 c e} \\ & = \frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 c e} \\ & = \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} e}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\frac {-\frac {2 b \arctan \left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.64 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.86
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3} e^{3}+3 \textit {\_R}^{2} d \,e^{2}+3 \textit {\_R} \,d^{2} e +d^{3}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 e}\) | \(151\) |
risch | \(\text {Expression too large to display}\) | \(1002\) |
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Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (73) = 146\).
Time = 0.29 (sec) , antiderivative size = 434, normalized size of antiderivative = 5.36 \[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c + {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right )}{4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right )}{4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (68) = 136\).
Time = 0.92 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.46 \[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} \]
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\[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.56 \[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=-\frac {b \arctan \left (\frac {2 \, c d^{2} + 2 \, {\left (e x^{2} + 2 \, d x\right )} c e + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c e} + \frac {\log \left (c d^{4} + 2 \, {\left (e x^{2} + 2 \, d x\right )} c d^{2} e + {\left (e x^{2} + 2 \, d x\right )}^{2} c e^{2} + b d^{2} + {\left (e x^{2} + 2 \, d x\right )} b e + a\right )}{4 \, c e} \]
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Time = 8.88 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.43 \[ \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\frac {4\,a\,c\,e\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2}-\frac {b^2\,e\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2}-\frac {b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,d^2}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,e^2\,x^2}{\sqrt {4\,a\,c-b^2}}+\frac {4\,c\,d\,e\,x}{\sqrt {4\,a\,c-b^2}}\right )}{2\,c\,e\,\sqrt {4\,a\,c-b^2}} \]
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