Integrand size = 22, antiderivative size = 111 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^3}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{2 e^4 (d+e x)^2}+\frac {3 B c d-b B e-A c e}{e^4 (d+e x)}+\frac {B c \log (d+e x)}{e^4} \]
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Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {785} \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^3}+\frac {-A c e-b B e+3 B c d}{e^4 (d+e x)}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{2 e^4 (d+e x)^2}+\frac {B c \log (d+e x)}{e^4} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)^4}+\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 (d+e x)^3}+\frac {-3 B c d+b B e+A c e}{e^3 (d+e x)^2}+\frac {B c}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^3}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{2 e^4 (d+e x)^2}+\frac {3 B c d-b B e-A c e}{e^4 (d+e x)}+\frac {B c \log (d+e x)}{e^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {-A e \left (b e (d+3 e x)+2 c \left (d^2+3 d e x+3 e^2 x^2\right )\right )+B \left (-2 b e \left (d^2+3 d e x+3 e^2 x^2\right )+c d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )+6 B c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]
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Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {-\frac {d \left (A b \,e^{2}+2 A c d e +2 B b d e -11 B c \,d^{2}\right )}{6 e^{4}}-\frac {\left (A c e +B b e -3 B c d \right ) x^{2}}{e^{2}}-\frac {\left (A b \,e^{2}+2 A c d e +2 B b d e -9 B c \,d^{2}\right ) x}{2 e^{3}}}{\left (e x +d \right )^{3}}+\frac {B c \ln \left (e x +d \right )}{e^{4}}\) | \(109\) |
risch | \(\frac {-\frac {d \left (A b \,e^{2}+2 A c d e +2 B b d e -11 B c \,d^{2}\right )}{6 e^{4}}-\frac {\left (A c e +B b e -3 B c d \right ) x^{2}}{e^{2}}-\frac {\left (A b \,e^{2}+2 A c d e +2 B b d e -9 B c \,d^{2}\right ) x}{2 e^{3}}}{\left (e x +d \right )^{3}}+\frac {B c \ln \left (e x +d \right )}{e^{4}}\) | \(109\) |
default | \(\frac {d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right )}{3 e^{4} \left (e x +d \right )^{3}}-\frac {A c e +B b e -3 B c d}{e^{4} \left (e x +d \right )}+\frac {B c \ln \left (e x +d \right )}{e^{4}}-\frac {A b \,e^{2}-2 A c d e -2 B b d e +3 B c \,d^{2}}{2 e^{4} \left (e x +d \right )^{2}}\) | \(116\) |
parallelrisch | \(-\frac {-6 B \ln \left (e x +d \right ) x^{3} c \,e^{3}-18 B \ln \left (e x +d \right ) x^{2} c d \,e^{2}+6 A c \,e^{3} x^{2}-18 B \ln \left (e x +d \right ) x c \,d^{2} e +6 B \,x^{2} b \,e^{3}-18 B \,x^{2} c d \,e^{2}+3 A b \,e^{3} x +6 A c d \,e^{2} x -6 B \ln \left (e x +d \right ) c \,d^{3}+6 B x b d \,e^{2}-27 B c \,d^{2} e x +A b d \,e^{2}+2 A c \,d^{2} e +2 B b \,d^{2} e -11 B c \,d^{3}}{6 e^{4} \left (e x +d \right )^{3}}\) | \(171\) |
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Time = 0.34 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.51 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {11 \, B c d^{3} - A b d e^{2} - 2 \, {\left (B b + A c\right )} d^{2} e + 6 \, {\left (3 \, B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} + 3 \, {\left (9 \, B c d^{2} e - A b e^{3} - 2 \, {\left (B b + A c\right )} d e^{2}\right )} x + 6 \, {\left (B c e^{3} x^{3} + 3 \, B c d e^{2} x^{2} + 3 \, B c d^{2} e x + B c d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
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Time = 1.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {B c \log {\left (d + e x \right )}}{e^{4}} + \frac {- A b d e^{2} - 2 A c d^{2} e - 2 B b d^{2} e + 11 B c d^{3} + x^{2} \left (- 6 A c e^{3} - 6 B b e^{3} + 18 B c d e^{2}\right ) + x \left (- 3 A b e^{3} - 6 A c d e^{2} - 6 B b d e^{2} + 27 B c d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.23 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {11 \, B c d^{3} - A b d e^{2} - 2 \, {\left (B b + A c\right )} d^{2} e + 6 \, {\left (3 \, B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} + 3 \, {\left (9 \, B c d^{2} e - A b e^{3} - 2 \, {\left (B b + A c\right )} d e^{2}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac {B c \log \left (e x + d\right )}{e^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {B c \log \left ({\left | e x + d \right |}\right )}{e^{4}} + \frac {6 \, {\left (3 \, B c d e - B b e^{2} - A c e^{2}\right )} x^{2} + 3 \, {\left (9 \, B c d^{2} - 2 \, B b d e - 2 \, A c d e - A b e^{2}\right )} x + \frac {11 \, B c d^{3} - 2 \, B b d^{2} e - 2 \, A c d^{2} e - A b d e^{2}}{e}}{6 \, {\left (e x + d\right )}^{3} e^{3}} \]
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Time = 10.80 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^4} \, dx=\frac {B\,c\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {A\,b\,d\,e^2-11\,B\,c\,d^3+2\,A\,c\,d^2\,e+2\,B\,b\,d^2\,e}{6\,e^4}+\frac {x\,\left (A\,b\,e^2-9\,B\,c\,d^2+2\,A\,c\,d\,e+2\,B\,b\,d\,e\right )}{2\,e^3}+\frac {x^2\,\left (A\,c\,e+B\,b\,e-3\,B\,c\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]
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