Integrand size = 26, antiderivative size = 104 \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=\frac {6 e^2 (b d-a e)^2 x}{b^4}-\frac {(b d-a e)^4}{b^5 (a+b x)}+\frac {2 e^3 (b d-a e) (a+b x)^2}{b^5}+\frac {e^4 (a+b x)^3}{3 b^5}+\frac {4 e (b d-a e)^3 \log (a+b x)}{b^5} \]
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Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2 e^3 (a+b x)^2 (b d-a e)}{b^5}-\frac {(b d-a e)^4}{b^5 (a+b x)}+\frac {4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac {e^4 (a+b x)^3}{3 b^5}+\frac {6 e^2 x (b d-a e)^2}{b^4} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^4}{(a+b x)^2} \, dx \\ & = \int \left (\frac {6 e^2 (b d-a e)^2}{b^4}+\frac {(b d-a e)^4}{b^4 (a+b x)^2}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)}+\frac {4 e^3 (b d-a e) (a+b x)}{b^4}+\frac {e^4 (a+b x)^2}{b^4}\right ) \, dx \\ & = \frac {6 e^2 (b d-a e)^2 x}{b^4}-\frac {(b d-a e)^4}{b^5 (a+b x)}+\frac {2 e^3 (b d-a e) (a+b x)^2}{b^5}+\frac {e^4 (a+b x)^3}{3 b^5}+\frac {4 e (b d-a e)^3 \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=\frac {-3 a^4 e^4+3 a^3 b e^3 (4 d+3 e x)+6 a^2 b^2 e^2 \left (-3 d^2-4 d e x+e^2 x^2\right )-2 a b^3 e \left (-6 d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+b^4 \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )-12 e (-b d+a e)^3 (a+b x) \log (a+b x)}{3 b^5 (a+b x)} \]
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Time = 2.58 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.68
method | result | size |
default | \(\frac {e^{2} \left (\frac {1}{3} b^{2} e^{2} x^{3}-x^{2} a b \,e^{2}+2 b^{2} d e \,x^{2}+3 a^{2} e^{2} x -8 a b d e x +6 b^{2} d^{2} x \right )}{b^{4}}-\frac {4 e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{b^{5} \left (b x +a \right )}\) | \(175\) |
norman | \(\frac {\frac {\left (4 e^{4} a^{4}-12 b \,e^{3} d \,a^{3}+12 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{a \,b^{4}}+\frac {e^{4} x^{4}}{3 b}+\frac {2 e^{2} \left (a^{2} e^{2}-3 a b d e +3 b^{2} d^{2}\right ) x^{2}}{b^{3}}-\frac {2 e^{3} \left (a e -3 b d \right ) x^{3}}{3 b^{2}}}{b x +a}-\frac {4 e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(180\) |
risch | \(\frac {e^{4} x^{3}}{3 b^{2}}-\frac {e^{4} x^{2} a}{b^{3}}+\frac {2 e^{3} d \,x^{2}}{b^{2}}+\frac {3 e^{4} a^{2} x}{b^{4}}-\frac {8 e^{3} a d x}{b^{3}}+\frac {6 e^{2} d^{2} x}{b^{2}}-\frac {4 e^{4} \ln \left (b x +a \right ) a^{3}}{b^{5}}+\frac {12 e^{3} \ln \left (b x +a \right ) a^{2} d}{b^{4}}-\frac {12 e^{2} \ln \left (b x +a \right ) a \,d^{2}}{b^{3}}+\frac {4 e \ln \left (b x +a \right ) d^{3}}{b^{2}}-\frac {e^{4} a^{4}}{b^{5} \left (b x +a \right )}+\frac {4 e^{3} d \,a^{3}}{b^{4} \left (b x +a \right )}-\frac {6 e^{2} d^{2} a^{2}}{b^{3} \left (b x +a \right )}+\frac {4 a \,d^{3} e}{b^{2} \left (b x +a \right )}-\frac {d^{4}}{b \left (b x +a \right )}\) | \(230\) |
parallelrisch | \(-\frac {-b^{4} x^{4} e^{4}+2 x^{3} a \,b^{3} e^{4}-6 x^{3} b^{4} d \,e^{3}+12 \ln \left (b x +a \right ) x \,a^{3} b \,e^{4}-36 \ln \left (b x +a \right ) x \,a^{2} b^{2} d \,e^{3}+36 \ln \left (b x +a \right ) x a \,b^{3} d^{2} e^{2}-12 \ln \left (b x +a \right ) x \,b^{4} d^{3} e -6 x^{2} a^{2} b^{2} e^{4}+18 x^{2} a \,b^{3} d \,e^{3}-18 x^{2} b^{4} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}-36 \ln \left (b x +a \right ) a^{3} b d \,e^{3}+36 \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e^{2}-12 \ln \left (b x +a \right ) a \,b^{3} d^{3} e +12 e^{4} a^{4}-36 b \,e^{3} d \,a^{3}+36 b^{2} e^{2} d^{2} a^{2}-12 a \,b^{3} d^{3} e +3 b^{4} d^{4}}{3 b^{5} \left (b x +a \right )}\) | \(276\) |
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (102) = 204\).
Time = 0.34 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.58 \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=\frac {b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 2 \, {\left (3 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (3 \, b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (6 \, a b^{3} d^{2} e^{2} - 8 \, a^{2} b^{2} d e^{3} + 3 \, a^{3} b e^{4}\right )} x + 12 \, {\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \, {\left (b^{6} x + a b^{5}\right )}} \]
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Time = 0.39 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=x^{2} \left (- \frac {a e^{4}}{b^{3}} + \frac {2 d e^{3}}{b^{2}}\right ) + x \left (\frac {3 a^{2} e^{4}}{b^{4}} - \frac {8 a d e^{3}}{b^{3}} + \frac {6 d^{2} e^{2}}{b^{2}}\right ) + \frac {- a^{4} e^{4} + 4 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} + 4 a b^{3} d^{3} e - b^{4} d^{4}}{a b^{5} + b^{6} x} + \frac {e^{4} x^{3}}{3 b^{2}} - \frac {4 e \left (a e - b d\right )^{3} \log {\left (a + b x \right )}}{b^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.77 \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{b^{6} x + a b^{5}} + \frac {b^{2} e^{4} x^{3} + 3 \, {\left (2 \, b^{2} d e^{3} - a b e^{4}\right )} x^{2} + 3 \, {\left (6 \, b^{2} d^{2} e^{2} - 8 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} x}{3 \, b^{4}} + \frac {4 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left (b x + a\right )}{b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.81 \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=\frac {4 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{4} e^{4} x^{3} + 6 \, b^{4} d e^{3} x^{2} - 3 \, a b^{3} e^{4} x^{2} + 18 \, b^{4} d^{2} e^{2} x - 24 \, a b^{3} d e^{3} x + 9 \, a^{2} b^{2} e^{4} x}{3 \, b^{6}} - \frac {b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{{\left (b x + a\right )} b^{5}} \]
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Time = 9.60 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.95 \[ \int \frac {(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx=x\,\left (\frac {2\,a\,\left (\frac {2\,a\,e^4}{b^3}-\frac {4\,d\,e^3}{b^2}\right )}{b}-\frac {a^2\,e^4}{b^4}+\frac {6\,d^2\,e^2}{b^2}\right )-x^2\,\left (\frac {a\,e^4}{b^3}-\frac {2\,d\,e^3}{b^2}\right )+\frac {e^4\,x^3}{3\,b^2}-\frac {\ln \left (a+b\,x\right )\,\left (4\,a^3\,e^4-12\,a^2\,b\,d\,e^3+12\,a\,b^2\,d^2\,e^2-4\,b^3\,d^3\,e\right )}{b^5}-\frac {a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}{b\,\left (x\,b^5+a\,b^4\right )} \]
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