Integrand size = 24, antiderivative size = 102 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=-\frac {c (4 c d-3 b e) x}{e^3}+\frac {c^2 x^2}{e^2}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \log (d+e x)}{e^4} \]
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Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=\frac {\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac {c x (4 c d-3 b e)}{e^3}+\frac {c^2 x^2}{e^2} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {c (4 c d-3 b e)}{e^3}+\frac {2 c^2 x}{e^2}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^2}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)}\right ) \, dx \\ & = -\frac {c (4 c d-3 b e) x}{e^3}+\frac {c^2 x^2}{e^2}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \log (d+e x)}{e^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=\frac {-c e (4 c d-3 b e) x+c^2 e^2 x^2+\frac {(2 c d-b e) \left (c d^2+e (-b d+a e)\right )}{d+e x}+\left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right ) \log (d+e x)}{e^4} \]
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Time = 0.53 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {c \left (c e \,x^{2}+3 b e x -4 c d x \right )}{e^{3}}-\frac {a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}}{e^{4} \left (e x +d \right )}+\frac {\left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(115\) |
norman | \(\frac {\frac {c^{2} x^{3}}{e}+\frac {\left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+6 b c \,d^{2} e -6 c^{2} d^{3}\right ) x}{e^{3} d}+\frac {3 \left (b e -c d \right ) c \,x^{2}}{e^{2}}}{e x +d}+\frac {\left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(125\) |
risch | \(\frac {c^{2} x^{2}}{e^{2}}+\frac {3 c b x}{e^{2}}-\frac {4 c^{2} d x}{e^{3}}-\frac {a b}{e \left (e x +d \right )}+\frac {2 a c d}{e^{2} \left (e x +d \right )}+\frac {b^{2} d}{e^{2} \left (e x +d \right )}-\frac {3 b c \,d^{2}}{e^{3} \left (e x +d \right )}+\frac {2 c^{2} d^{3}}{e^{4} \left (e x +d \right )}+\frac {2 \ln \left (e x +d \right ) a c}{e^{2}}+\frac {\ln \left (e x +d \right ) b^{2}}{e^{2}}-\frac {6 \ln \left (e x +d \right ) b c d}{e^{3}}+\frac {6 \ln \left (e x +d \right ) c^{2} d^{2}}{e^{4}}\) | \(166\) |
parallelrisch | \(\frac {c^{2} x^{3} e^{3}+2 \ln \left (e x +d \right ) x a c \,e^{3}+\ln \left (e x +d \right ) x \,b^{2} e^{3}-6 \ln \left (e x +d \right ) x b c d \,e^{2}+6 \ln \left (e x +d \right ) x \,c^{2} d^{2} e +3 x^{2} b c \,e^{3}-3 x^{2} c^{2} d \,e^{2}+2 \ln \left (e x +d \right ) a c d \,e^{2}+\ln \left (e x +d \right ) b^{2} d \,e^{2}-6 \ln \left (e x +d \right ) b c \,d^{2} e +6 \ln \left (e x +d \right ) c^{2} d^{3}-a b \,e^{3}+2 a c d \,e^{2}+b^{2} d \,e^{2}-6 b c \,d^{2} e +6 c^{2} d^{3}}{e^{4} \left (e x +d \right )}\) | \(199\) |
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Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.69 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=\frac {c^{2} e^{3} x^{3} + 2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (c^{2} d e^{2} - b c e^{3}\right )} x^{2} - {\left (4 \, c^{2} d^{2} e - 3 \, b c d e^{2}\right )} x + {\left (6 \, c^{2} d^{3} - 6 \, b c d^{2} e + {\left (b^{2} + 2 \, a c\right )} d e^{2} + {\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{5} x + d e^{4}} \]
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Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.24 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=\frac {c^{2} x^{2}}{e^{2}} + x \left (\frac {3 b c}{e^{2}} - \frac {4 c^{2} d}{e^{3}}\right ) + \frac {- a b e^{3} + 2 a c d e^{2} + b^{2} d e^{2} - 3 b c d^{2} e + 2 c^{2} d^{3}}{d e^{4} + e^{5} x} + \frac {\left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.15 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=\frac {2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}}{e^{5} x + d e^{4}} + \frac {c^{2} e x^{2} - {\left (4 \, c^{2} d - 3 \, b c e\right )} x}{e^{3}} + \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.76 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=\frac {{\left (c^{2} - \frac {3 \, {\left (2 \, c^{2} d e - b c e^{2}\right )}}{{\left (e x + d\right )} e}\right )} {\left (e x + d\right )}^{2}}{e^{4}} - \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{4}} + \frac {\frac {2 \, c^{2} d^{3} e^{2}}{e x + d} - \frac {3 \, b c d^{2} e^{3}}{e x + d} + \frac {b^{2} d e^{4}}{e x + d} + \frac {2 \, a c d e^{4}}{e x + d} - \frac {a b e^{5}}{e x + d}}{e^{6}} \]
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Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.25 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx=\frac {b^2\,d\,e^2-3\,b\,c\,d^2\,e-a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2}{e\,\left (x\,e^4+d\,e^3\right )}-x\,\left (\frac {4\,c^2\,d}{e^3}-\frac {3\,b\,c}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{e^4}+\frac {c^2\,x^2}{e^2} \]
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