Integrand size = 20, antiderivative size = 138 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {c (3 c d-2 b e) x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {\left (c d^2-b d e+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^5 (d+e x)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \log (d+e x)}{e^5} \]
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Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5}-\frac {\left (a e^2-b d e+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)}-\frac {c x (3 c d-2 b e)}{e^4}+\frac {c^2 x^2}{2 e^3} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {c (3 c d-2 b e)}{e^4}+\frac {c^2 x}{e^3}+\frac {\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^3}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^2}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)}\right ) \, dx \\ & = -\frac {c (3 c d-2 b e) x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {\left (c d^2-b d e+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^5 (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^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {e^2 (b d-a e) (3 b d+a e+4 b e x)+c^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+2 c e \left (a d e (3 d+4 e x)+b \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )\right )+2 \left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right ) (d+e x)^2 \log (d+e x)}{2 e^5 (d+e x)^2} \]
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Time = 2.86 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {c \left (\frac {1}{2} c e \,x^{2}+2 b e x -3 c d x \right )}{e^{4}}-\frac {2 a b \,e^{3}-4 d \,e^{2} a c -2 b^{2} d \,e^{2}+6 b c e \,d^{2}-4 c^{2} d^{3}}{e^{5} \left (e x +d \right )}-\frac {a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{2 e^{5} \left (e x +d \right )^{2}}+\frac {\left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(180\) |
norman | \(\frac {-\frac {a^{2} e^{4}+2 a b d \,e^{3}-6 a c \,d^{2} e^{2}-3 b^{2} d^{2} e^{2}+18 d^{3} e b c -18 c^{2} d^{4}}{2 e^{5}}+\frac {c^{2} x^{4}}{2 e}-\frac {2 \left (a b \,e^{3}-2 d \,e^{2} a c -b^{2} d \,e^{2}+6 b c e \,d^{2}-6 c^{2} d^{3}\right ) x}{e^{4}}+\frac {2 \left (b e -c d \right ) c \,x^{3}}{e^{2}}}{\left (e x +d \right )^{2}}+\frac {\left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(182\) |
risch | \(\frac {c^{2} x^{2}}{2 e^{3}}+\frac {2 c b x}{e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\left (-2 a b \,e^{3}+4 d \,e^{2} a c +2 b^{2} d \,e^{2}-6 b c e \,d^{2}+4 c^{2} d^{3}\right ) x -\frac {a^{2} e^{4}+2 a b d \,e^{3}-6 a c \,d^{2} e^{2}-3 b^{2} d^{2} e^{2}+10 d^{3} e b c -7 c^{2} d^{4}}{2 e}}{e^{4} \left (e x +d \right )^{2}}+\frac {2 \ln \left (e x +d \right ) a c}{e^{3}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 \ln \left (e x +d \right ) b c d}{e^{4}}+\frac {6 \ln \left (e x +d \right ) c^{2} d^{2}}{e^{5}}\) | \(201\) |
parallelrisch | \(\frac {-2 a b d \,e^{3}+3 b^{2} d^{2} e^{2}-a^{2} e^{4}+12 \ln \left (e x +d \right ) c^{2} d^{4}+24 x \,c^{2} d^{3} e +2 \ln \left (e x +d \right ) x^{2} b^{2} e^{4}-24 \ln \left (e x +d \right ) x b c \,d^{2} e^{2}+18 c^{2} d^{4}+c^{2} x^{4} e^{4}-12 \ln \left (e x +d \right ) x^{2} b c d \,e^{3}+8 \ln \left (e x +d \right ) x a c d \,e^{3}+24 \ln \left (e x +d \right ) x \,c^{2} d^{3} e -18 d^{3} e b c -4 x^{3} c^{2} d \,e^{3}+8 x a c d \,e^{3}+4 \ln \left (e x +d \right ) x^{2} a c \,e^{4}+4 x^{3} b c \,e^{4}+2 \ln \left (e x +d \right ) b^{2} d^{2} e^{2}-4 x a b \,e^{4}+4 x \,b^{2} d \,e^{3}+4 \ln \left (e x +d \right ) a c \,d^{2} e^{2}-12 \ln \left (e x +d \right ) b c \,d^{3} e -24 x b c \,d^{2} e^{2}+12 \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{2}+4 \ln \left (e x +d \right ) x \,b^{2} d \,e^{3}+6 a c \,d^{2} e^{2}}{2 e^{5} \left (e x +d \right )^{2}}\) | \(342\) |
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Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (134) = 268\).
Time = 0.39 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {c^{2} e^{4} x^{4} + 7 \, c^{2} d^{4} - 10 \, b c d^{3} e - 2 \, a b d e^{3} - a^{2} e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 4 \, {\left (c^{2} d e^{3} - b c e^{4}\right )} x^{3} - {\left (11 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3}\right )} x^{2} + 2 \, {\left (c^{2} d^{3} e - 4 \, b c d^{2} e^{2} - 2 \, a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x + 2 \, {\left (6 \, c^{2} d^{4} - 6 \, b c d^{3} e + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 2 \, {\left (6 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]
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Time = 1.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {c^{2} x^{2}}{2 e^{3}} + x \left (\frac {2 b c}{e^{3}} - \frac {3 c^{2} d}{e^{4}}\right ) + \frac {- a^{2} e^{4} - 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 10 b c d^{3} e + 7 c^{2} d^{4} + x \left (- 4 a b e^{4} + 8 a c d e^{3} + 4 b^{2} d e^{3} - 12 b c d^{2} e^{2} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} + \frac {\left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} \]
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Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {7 \, c^{2} d^{4} - 10 \, b c d^{3} e - 2 \, a b d e^{3} - a^{2} e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} - a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac {c^{2} e x^{2} - 2 \, {\left (3 \, c^{2} d - 2 \, b c e\right )} x}{2 \, e^{4}} + \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} + \frac {c^{2} e^{3} x^{2} - 6 \, c^{2} d e^{2} x + 4 \, b c e^{3} x}{2 \, e^{6}} + \frac {7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} - 2 \, a b d e^{3} - a^{2} e^{4} + 4 \, {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3} + 2 \, a c d e^{3} - a b e^{4}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{5}} \]
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Time = 0.09 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{e^5}-\frac {\frac {a^2\,e^4+2\,a\,b\,d\,e^3-6\,a\,c\,d^2\,e^2-3\,b^2\,d^2\,e^2+10\,b\,c\,d^3\,e-7\,c^2\,d^4}{2\,e}-x\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e-2\,a\,b\,e^3+4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )}{d^2\,e^4+2\,d\,e^5\,x+e^6\,x^2}-x\,\left (\frac {3\,c^2\,d}{e^4}-\frac {2\,b\,c}{e^3}\right )+\frac {c^2\,x^2}{2\,e^3} \]
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