Integrand size = 31, antiderivative size = 102 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^3 (4 b d-3 a e) x}{b^4}+\frac {e^4 x^2}{2 b^3}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}+\frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5} \]
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Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 45} \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}+\frac {e^3 x (4 b d-3 a e)}{b^4}+\frac {e^4 x^2}{2 b^3} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^4}{(a+b x)^3} \, dx \\ & = \int \left (\frac {e^3 (4 b d-3 a e)}{b^4}+\frac {e^4 x}{b^3}+\frac {(b d-a e)^4}{b^4 (a+b x)^3}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)^2}+\frac {6 e^2 (b d-a e)^2}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {e^3 (4 b d-3 a e) x}{b^4}+\frac {e^4 x^2}{2 b^3}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}+\frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {7 a^4 e^4+2 a^3 b e^3 (-10 d+e x)+a^2 b^2 e^2 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a b^3 e \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+b^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )+12 e^2 (b d-a e)^2 (a+b x)^2 \log (a+b x)}{2 b^5 (a+b x)^2} \]
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Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.70
method | result | size |
default | \(-\frac {e^{3} \left (-\frac {1}{2} b e \,x^{2}+3 a e x -4 b d x \right )}{b^{4}}+\frac {6 e^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}{2 b^{5} \left (b x +a \right )^{2}}+\frac {4 e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{5} \left (b x +a \right )}\) | \(173\) |
risch | \(\frac {e^{4} x^{2}}{2 b^{3}}-\frac {3 e^{4} a x}{b^{4}}+\frac {4 e^{3} d x}{b^{3}}+\frac {\left (4 a^{3} e^{4}-12 b \,a^{2} e^{3} d +12 b^{2} a \,d^{2} e^{2}-4 b^{3} d^{3} e \right ) x +\frac {7 e^{4} a^{4}-20 b d \,e^{3} a^{3}+18 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -b^{4} d^{4}}{2 b}}{b^{4} \left (b x +a \right )^{2}}+\frac {6 e^{4} \ln \left (b x +a \right ) a^{2}}{b^{5}}-\frac {12 e^{3} \ln \left (b x +a \right ) a d}{b^{4}}+\frac {6 e^{2} \ln \left (b x +a \right ) d^{2}}{b^{3}}\) | \(193\) |
norman | \(\frac {\frac {\left (18 a^{3} e^{4}-36 b \,a^{2} e^{3} d +12 b^{2} a \,d^{2} e^{2}-4 b^{3} d^{3} e \right ) x^{2}}{b^{3}}+\frac {e^{4} x^{5}}{2}+\frac {a \left (22 e^{4} a^{4}-44 b d \,e^{3} a^{3}+18 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -b^{4} d^{4}\right )}{2 b^{5}}+\frac {\left (54 e^{4} a^{4}-108 b d \,e^{3} a^{3}+42 b^{2} d^{2} e^{2} a^{2}-12 b^{3} d^{3} e a -b^{4} d^{4}\right ) x}{2 b^{4}}-\frac {e^{3} \left (3 a e -8 b d \right ) x^{4}}{2 b}}{\left (b x +a \right )^{3}}+\frac {6 e^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(235\) |
parallelrisch | \(\frac {e^{4} x^{4} b^{4}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{4}-24 \ln \left (b x +a \right ) x^{2} a \,b^{3} d \,e^{3}+12 \ln \left (b x +a \right ) x^{2} b^{4} d^{2} e^{2}-4 x^{3} a \,b^{3} e^{4}+8 x^{3} b^{4} d \,e^{3}+24 \ln \left (b x +a \right ) x \,a^{3} b \,e^{4}-48 \ln \left (b x +a \right ) x \,a^{2} b^{2} d \,e^{3}+24 \ln \left (b x +a \right ) x a \,b^{3} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}-24 \ln \left (b x +a \right ) a^{3} b d \,e^{3}+12 \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e^{2}+24 x \,a^{3} b \,e^{4}-48 x \,a^{2} b^{2} d \,e^{3}+24 x a \,b^{3} d^{2} e^{2}-8 x \,b^{4} d^{3} e +18 e^{4} a^{4}-36 b d \,e^{3} a^{3}+18 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -b^{4} d^{4}}{2 b^{5} \left (b^{2} x^{2}+2 a b x +a^{2}\right )}\) | \(318\) |
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (98) = 196\).
Time = 0.41 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.86 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {b^{4} e^{4} x^{4} - b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 20 \, a^{3} b d e^{3} + 7 \, a^{4} e^{4} + 4 \, {\left (2 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + {\left (16 \, a b^{3} d e^{3} - 11 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \, {\left (4 \, b^{4} d^{3} e - 12 \, a b^{3} d^{2} e^{2} + 8 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
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Time = 0.67 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.81 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=x \left (- \frac {3 a e^{4}}{b^{4}} + \frac {4 d e^{3}}{b^{3}}\right ) + \frac {7 a^{4} e^{4} - 20 a^{3} b d e^{3} + 18 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - b^{4} d^{4} + x \left (8 a^{3} b e^{4} - 24 a^{2} b^{2} d e^{3} + 24 a b^{3} d^{2} e^{2} - 8 b^{4} d^{3} e\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {e^{4} x^{2}}{2 b^{3}} + \frac {6 e^{2} \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b^{4} d^{4} + 4 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 7 \, a^{4} e^{4} + 8 \, {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {b e^{4} x^{2} + 2 \, {\left (4 \, b d e^{3} - 3 \, a e^{4}\right )} x}{2 \, b^{4}} + \frac {6 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left (b x + a\right )}{b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {6 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} e^{4} x^{2} + 8 \, b^{3} d e^{3} x - 6 \, a b^{2} e^{4} x}{2 \, b^{6}} - \frac {b^{4} d^{4} + 4 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 7 \, a^{4} e^{4} + 8 \, {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \]
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Time = 10.73 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.93 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^4\,x^2}{2\,b^3}-x\,\left (\frac {3\,a\,e^4}{b^4}-\frac {4\,d\,e^3}{b^3}\right )-\frac {\frac {-7\,a^4\,e^4+20\,a^3\,b\,d\,e^3-18\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4}{2\,b}-x\,\left (4\,a^3\,e^4-12\,a^2\,b\,d\,e^3+12\,a\,b^2\,d^2\,e^2-4\,b^3\,d^3\,e\right )}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+\frac {\ln \left (a+b\,x\right )\,\left (6\,a^2\,e^4-12\,a\,b\,d\,e^3+6\,b^2\,d^2\,e^2\right )}{b^5} \]
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