Integrand size = 29, antiderivative size = 78 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=\frac {b^3 x}{e^3}+\frac {(b d-a e)^3}{2 e^4 (d+e x)^2}-\frac {3 b (b d-a e)^2}{e^4 (d+e x)}-\frac {3 b^2 (b d-a e) \log (d+e x)}{e^4} \]
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Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 45} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=-\frac {3 b^2 (b d-a e) \log (d+e x)}{e^4}-\frac {3 b (b d-a e)^2}{e^4 (d+e x)}+\frac {(b d-a e)^3}{2 e^4 (d+e x)^2}+\frac {b^3 x}{e^3} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^3}{(d+e x)^3} \, dx \\ & = \int \left (\frac {b^3}{e^3}+\frac {(-b d+a e)^3}{e^3 (d+e x)^3}+\frac {3 b (b d-a e)^2}{e^3 (d+e x)^2}-\frac {3 b^2 (b d-a e)}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {b^3 x}{e^3}+\frac {(b d-a e)^3}{2 e^4 (d+e x)^2}-\frac {3 b (b d-a e)^2}{e^4 (d+e x)}-\frac {3 b^2 (b d-a e) \log (d+e x)}{e^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=\frac {-a^3 e^3-3 a^2 b e^2 (d+2 e x)+3 a b^2 d e (3 d+4 e x)+b^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )-6 b^2 (b d-a e) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2} \]
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Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {b^{3} x}{e^{3}}-\frac {3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{4} \left (e x +d \right )}+\frac {3 b^{2} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{4}}-\frac {a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}{2 e^{4} \left (e x +d \right )^{2}}\) | \(114\) |
norman | \(\frac {\frac {b^{3} x^{3}}{e}-\frac {a^{3} e^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +9 b^{3} d^{3}}{2 e^{4}}-\frac {\left (3 a^{2} b \,e^{2}-6 a \,b^{2} d e +6 b^{3} d^{2}\right ) x}{e^{3}}}{\left (e x +d \right )^{2}}+\frac {3 b^{2} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{4}}\) | \(116\) |
risch | \(\frac {b^{3} x}{e^{3}}+\frac {\left (-3 a^{2} b \,e^{2}+6 a \,b^{2} d e -3 b^{3} d^{2}\right ) x -\frac {a^{3} e^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}}{2 e}}{e^{3} \left (e x +d \right )^{2}}+\frac {3 b^{2} \ln \left (e x +d \right ) a}{e^{3}}-\frac {3 b^{3} \ln \left (e x +d \right ) d}{e^{4}}\) | \(121\) |
parallelrisch | \(\frac {6 \ln \left (e x +d \right ) x^{2} a \,b^{2} e^{3}-6 \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{2}+2 b^{3} x^{3} e^{3}+12 \ln \left (e x +d \right ) x a \,b^{2} d \,e^{2}-12 \ln \left (e x +d \right ) x \,b^{3} d^{2} e +6 \ln \left (e x +d \right ) a \,b^{2} d^{2} e -6 \ln \left (e x +d \right ) b^{3} d^{3}-6 x \,a^{2} b \,e^{3}+12 x a \,b^{2} d \,e^{2}-12 x \,b^{3} d^{2} e -a^{3} e^{3}-3 a^{2} b d \,e^{2}+9 a \,b^{2} d^{2} e -9 b^{3} d^{3}}{2 e^{4} \left (e x +d \right )^{2}}\) | \(191\) |
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (76) = 152\).
Time = 0.36 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=\frac {2 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d e^{2} x^{2} - 5 \, b^{3} d^{3} + 9 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 2 \, {\left (2 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} d^{3} - a b^{2} d^{2} e + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (b^{3} d^{2} e - a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
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Time = 0.44 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.64 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=\frac {b^{3} x}{e^{3}} + \frac {3 b^{2} \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- a^{3} e^{3} - 3 a^{2} b d e^{2} + 9 a b^{2} d^{2} e - 5 b^{3} d^{3} + x \left (- 6 a^{2} b e^{3} + 12 a b^{2} d e^{2} - 6 b^{3} d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=\frac {b^{3} x}{e^{3}} - \frac {5 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac {3 \, {\left (b^{3} d - a b^{2} e\right )} \log \left (e x + d\right )}{e^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=\frac {b^{3} x}{e^{3}} - \frac {3 \, {\left (b^{3} d - a b^{2} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{4}} - \frac {5 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{4}} \]
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Time = 10.85 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.67 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^3} \, dx=\frac {b^3\,x}{e^3}-\frac {\ln \left (d+e\,x\right )\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{e^4}-\frac {\frac {a^3\,e^3+3\,a^2\,b\,d\,e^2-9\,a\,b^2\,d^2\,e+5\,b^3\,d^3}{2\,e}+x\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2} \]
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