Integrand size = 19, antiderivative size = 93 \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=-\frac {d (c d-b e)^2 x}{e^4}+\frac {(c d-b e)^2 x^2}{2 e^3}-\frac {c (c d-2 b e) x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {d^2 (c d-b e)^2 \log (d+e x)}{e^5} \]
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Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=\frac {d^2 (c d-b e)^2 \log (d+e x)}{e^5}-\frac {d x (c d-b e)^2}{e^4}+\frac {x^2 (c d-b e)^2}{2 e^3}-\frac {c x^3 (c d-2 b e)}{3 e^2}+\frac {c^2 x^4}{4 e} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d (c d-b e)^2}{e^4}+\frac {(-c d+b e)^2 x}{e^3}-\frac {c (c d-2 b e) x^2}{e^2}+\frac {c^2 x^3}{e}+\frac {d^2 (c d-b e)^2}{e^4 (d+e x)}\right ) \, dx \\ & = -\frac {d (c d-b e)^2 x}{e^4}+\frac {(c d-b e)^2 x^2}{2 e^3}-\frac {c (c d-2 b e) x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {d^2 (c d-b e)^2 \log (d+e x)}{e^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.14 \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=-\frac {d (c d-b e)^2 x}{e^4}+\frac {(-c d+b e)^2 x^2}{2 e^3}-\frac {c (c d-2 b e) x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {\left (c^2 d^4-2 b c d^3 e+b^2 d^2 e^2\right ) \log (d+e x)}{e^5} \]
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Time = 1.91 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.31
method | result | size |
norman | \(\frac {c^{2} x^{4}}{4 e}+\frac {\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) x^{2}}{2 e^{3}}-\frac {d \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) x}{e^{4}}+\frac {c \left (2 b e -c d \right ) x^{3}}{3 e^{2}}+\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(122\) |
default | \(-\frac {-\frac {c^{2} x^{4} e^{3}}{4}+\frac {\left (-\left (b e -c d \right ) e^{2} c -e^{3} b c \right ) x^{3}}{3}+\frac {\left (-\left (b e -c d \right ) e^{2} b +c e \left (b d e -c \,d^{2}\right )\right ) x^{2}}{2}+\left (b e -c d \right ) \left (b d e -c \,d^{2}\right ) x}{e^{4}}+\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(135\) |
risch | \(\frac {c^{2} x^{4}}{4 e}+\frac {2 x^{3} b c}{3 e}-\frac {c^{2} d \,x^{3}}{3 e^{2}}+\frac {x^{2} b^{2}}{2 e}-\frac {x^{2} b c d}{e^{2}}+\frac {x^{2} d^{2} c^{2}}{2 e^{3}}-\frac {b^{2} d x}{e^{2}}+\frac {2 b c \,d^{2} x}{e^{3}}-\frac {c^{2} d^{3} x}{e^{4}}+\frac {d^{2} \ln \left (e x +d \right ) b^{2}}{e^{3}}-\frac {2 d^{3} \ln \left (e x +d \right ) b c}{e^{4}}+\frac {d^{4} \ln \left (e x +d \right ) c^{2}}{e^{5}}\) | \(152\) |
parallelrisch | \(\frac {3 c^{2} x^{4} e^{4}+8 x^{3} b c \,e^{4}-4 x^{3} c^{2} d \,e^{3}+6 x^{2} b^{2} e^{4}-12 x^{2} b c d \,e^{3}+6 x^{2} c^{2} d^{2} e^{2}+12 \ln \left (e x +d \right ) b^{2} d^{2} e^{2}-24 \ln \left (e x +d \right ) b c \,d^{3} e +12 \ln \left (e x +d \right ) c^{2} d^{4}-12 x \,b^{2} d \,e^{3}+24 x b c \,d^{2} e^{2}-12 x \,c^{2} d^{3} e}{12 e^{5}}\) | \(152\) |
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Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.43 \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=\frac {3 \, c^{2} e^{4} x^{4} - 4 \, {\left (c^{2} d e^{3} - 2 \, b c e^{4}\right )} x^{3} + 6 \, {\left (c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} - 12 \, {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x + 12 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \]
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Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.25 \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=\frac {c^{2} x^{4}}{4 e} + \frac {d^{2} \left (b e - c d\right )^{2} \log {\left (d + e x \right )}}{e^{5}} + x^{3} \cdot \left (\frac {2 b c}{3 e} - \frac {c^{2} d}{3 e^{2}}\right ) + x^{2} \left (\frac {b^{2}}{2 e} - \frac {b c d}{e^{2}} + \frac {c^{2} d^{2}}{2 e^{3}}\right ) + x \left (- \frac {b^{2} d}{e^{2}} + \frac {2 b c d^{2}}{e^{3}} - \frac {c^{2} d^{3}}{e^{4}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.41 \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=\frac {3 \, c^{2} e^{3} x^{4} - 4 \, {\left (c^{2} d e^{2} - 2 \, b c e^{3}\right )} x^{3} + 6 \, {\left (c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3}\right )} x^{2} - 12 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} x}{12 \, e^{4}} + \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.49 \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=\frac {3 \, c^{2} e^{3} x^{4} - 4 \, c^{2} d e^{2} x^{3} + 8 \, b c e^{3} x^{3} + 6 \, c^{2} d^{2} e x^{2} - 12 \, b c d e^{2} x^{2} + 6 \, b^{2} e^{3} x^{2} - 12 \, c^{2} d^{3} x + 24 \, b c d^{2} e x - 12 \, b^{2} d e^{2} x}{12 \, e^{4}} + \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} \]
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Time = 9.55 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.52 \[ \int \frac {\left (b x+c x^2\right )^2}{d+e x} \, dx=x^2\,\left (\frac {b^2}{2\,e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{2\,e}\right )-x^3\,\left (\frac {c^2\,d}{3\,e^2}-\frac {2\,b\,c}{3\,e}\right )+\frac {c^2\,x^4}{4\,e}+\frac {\ln \left (d+e\,x\right )\,\left (b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4\right )}{e^5}-\frac {d\,x\,\left (\frac {b^2}{e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{e}\right )}{e} \]
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