Integrand size = 31, antiderivative size = 111 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e^3 (b d-a e)}{b^5 (a+b x)}+\frac {e^4 \log (a+b x)}{b^5} \]
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Time = 0.06 (sec) , antiderivative size = 111, 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} \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac {e^4 \log (a+b x)}{b^5} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^4}{(a+b x)^5} \, dx \\ & = \int \left (\frac {(b d-a e)^4}{b^4 (a+b x)^5}+\frac {4 e (b d-a e)^3}{b^4 (a+b x)^4}+\frac {6 e^2 (b d-a e)^2}{b^4 (a+b x)^3}+\frac {4 e^3 (b d-a e)}{b^4 (a+b x)^2}+\frac {e^4}{b^4 (a+b x)}\right ) \, dx \\ & = -\frac {(b d-a e)^4}{4 b^5 (a+b x)^4}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac {4 e^3 (b d-a e)}{b^5 (a+b x)}+\frac {e^4 \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )}{12 b^5 (a+b x)^4}+\frac {e^4 \log (a+b x)}{b^5} \]
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Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}}{4 b^{5} \left (b x +a \right )^{4}}+\frac {4 e \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 b^{5} \left (b x +a \right )^{3}}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {3 e^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{5} \left (b x +a \right )^{2}}+\frac {4 e^{3} \left (a e -b d \right )}{b^{5} \left (b x +a \right )}\) | \(184\) |
risch | \(\frac {\frac {4 e^{3} \left (a e -b d \right ) x^{3}}{b^{2}}+\frac {3 e^{2} \left (3 e^{2} a^{2}-2 a b d e -b^{2} d^{2}\right ) x^{2}}{b^{3}}+\frac {2 e \left (11 a^{3} e^{3}-6 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) x}{3 b^{4}}+\frac {25 e^{4} a^{4}-12 b d \,e^{3} a^{3}-6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -3 b^{4} d^{4}}{12 b^{5}}}{\left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}}\) | \(195\) |
norman | \(\frac {\frac {\left (4 a \,e^{4}-4 b d \,e^{3}\right ) x^{4}}{b}+\frac {\left (13 a^{2} e^{4}-10 a b d \,e^{3}-3 b^{2} d^{2} e^{2}\right ) x^{3}}{b^{2}}+\frac {a \left (25 e^{4} a^{4}-12 b d \,e^{3} a^{3}-6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -3 b^{4} d^{4}\right )}{12 b^{5}}+\frac {\left (49 a^{3} e^{4}-30 b \,a^{2} e^{3} d -15 b^{2} a \,d^{2} e^{2}-4 b^{3} d^{3} e \right ) x^{2}}{3 b^{3}}+\frac {\left (113 e^{4} a^{4}-60 b d \,e^{3} a^{3}-30 b^{2} d^{2} e^{2} a^{2}-20 b^{3} d^{3} e a -3 b^{4} d^{4}\right ) x}{12 b^{4}}}{\left (b x +a \right )^{5}}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}}\) | \(242\) |
parallelrisch | \(\frac {12 \ln \left (b x +a \right ) x^{4} b^{4} e^{4}+48 \ln \left (b x +a \right ) x^{3} a \,b^{3} e^{4}+72 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{4}+48 x^{3} a \,b^{3} e^{4}-48 x^{3} b^{4} d \,e^{3}+48 \ln \left (b x +a \right ) x \,a^{3} b \,e^{4}+108 x^{2} a^{2} b^{2} e^{4}-72 x^{2} a \,b^{3} d \,e^{3}-36 x^{2} b^{4} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}+88 x \,a^{3} b \,e^{4}-48 x \,a^{2} b^{2} d \,e^{3}-24 x a \,b^{3} d^{2} e^{2}-16 x \,b^{4} d^{3} e +25 e^{4} a^{4}-12 b d \,e^{3} a^{3}-6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a -3 b^{4} d^{4}}{12 b^{5} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )^{2}}\) | \(278\) |
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Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (107) = 214\).
Time = 0.30 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (100) = 200\).
Time = 10.66 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {25 a^{4} e^{4} - 12 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - 3 b^{4} d^{4} + x^{3} \cdot \left (48 a b^{3} e^{4} - 48 b^{4} d e^{3}\right ) + x^{2} \cdot \left (108 a^{2} b^{2} e^{4} - 72 a b^{3} d e^{3} - 36 b^{4} d^{2} e^{2}\right ) + x \left (88 a^{3} b e^{4} - 48 a^{2} b^{2} d e^{3} - 24 a b^{3} d^{2} e^{2} - 16 b^{4} d^{3} e\right )}{12 a^{4} b^{5} + 48 a^{3} b^{6} x + 72 a^{2} b^{7} x^{2} + 48 a b^{8} x^{3} + 12 b^{9} x^{4}} + \frac {e^{4} \log {\left (a + b x \right )}}{b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (107) = 214\).
Time = 0.21 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} + \frac {e^{4} \log \left (b x + a\right )}{b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {48 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 36 \, {\left (b^{3} d^{2} e^{2} + 2 \, a b^{2} d e^{3} - 3 \, a^{2} b e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2} + 6 \, a^{2} b d e^{3} - 11 \, a^{3} e^{4}\right )} x + \frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4}}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4}} \]
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Time = 0.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^4\,\ln \left (a+b\,x\right )}{b^5}-\frac {\frac {-25\,a^4\,e^4+12\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+3\,b^4\,d^4}{12\,b^5}+\frac {3\,x^2\,\left (-3\,a^2\,e^4+2\,a\,b\,d\,e^3+b^2\,d^2\,e^2\right )}{b^3}+\frac {2\,x\,\left (-11\,a^3\,e^4+6\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2+2\,b^3\,d^3\,e\right )}{3\,b^4}-\frac {4\,e^3\,x^3\,\left (a\,e-b\,d\right )}{b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]
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