Integrand size = 27, antiderivative size = 38 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=-\frac {b c-a d}{3 b^2 (a+b x)^3}-\frac {d}{2 b^2 (a+b x)^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {24, 45} \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=-\frac {b c-a d}{3 b^2 (a+b x)^3}-\frac {d}{2 b^2 (a+b x)^2} \]
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Rule 24
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {b^2 c+b^2 d x}{(a+b x)^4} \, dx}{b^2} \\ & = \frac {\int \left (\frac {b (b c-a d)}{(a+b x)^4}+\frac {b d}{(a+b x)^3}\right ) \, dx}{b^2} \\ & = -\frac {b c-a d}{3 b^2 (a+b x)^3}-\frac {d}{2 b^2 (a+b x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=-\frac {2 b c+a d+3 b d x}{6 b^2 (a+b x)^3} \]
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Time = 2.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {3 b d x +a d +2 b c}{6 b^{2} \left (b x +a \right )^{3}}\) | \(26\) |
risch | \(\frac {-\frac {d x}{2 b}-\frac {a d +2 b c}{6 b^{2}}}{\left (b x +a \right )^{3}}\) | \(30\) |
parallelrisch | \(\frac {-3 b^{2} d x -a b d -2 b^{2} c}{6 b^{3} \left (b x +a \right )^{3}}\) | \(32\) |
default | \(-\frac {-a d +b c}{3 b^{2} \left (b x +a \right )^{3}}-\frac {d}{2 b^{2} \left (b x +a \right )^{2}}\) | \(35\) |
norman | \(\frac {\frac {a \left (-a \,b^{2} d -2 b^{3} c \right )}{6 b^{4}}-\frac {d \,x^{2}}{2}+\frac {\left (-2 a \,b^{2} d -b^{3} c \right ) x}{3 b^{3}}}{\left (b x +a \right )^{4}}\) | \(56\) |
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Time = 0.37 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=-\frac {3 \, b d x + 2 \, b c + a d}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
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Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=\frac {- a d - 2 b c - 3 b d x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]
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Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=-\frac {3 \, b d x + 2 \, b c + a d}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=-\frac {c}{3 \, {\left (b x + a\right )}^{3} b} - \frac {d}{2 \, {\left (b x + a\right )}^{2} b^{2}} + \frac {a d}{3 \, {\left (b x + a\right )}^{3} b^{2}} \]
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Time = 9.75 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx=-\frac {\frac {a\,d+2\,b\,c}{6\,b^2}+\frac {d\,x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
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