Integrand size = 19, antiderivative size = 57 \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=\frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3} \]
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Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {45} \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=\frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 a^2}{c^6 (a-b x)^6}-\frac {4 a}{c^6 (a-b x)^5}+\frac {1}{c^6 (a-b x)^4}\right ) \, dx \\ & = \frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=-\frac {2 a^2+5 a b x+5 b^2 x^2}{15 b c^6 (-a+b x)^5} \]
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Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.56
method | result | size |
risch | \(\frac {\frac {b \,x^{2}}{3}+\frac {a x}{3}+\frac {2 a^{2}}{15 b}}{c^{6} \left (-b x +a \right )^{5}}\) | \(32\) |
gosper | \(\frac {5 b^{2} x^{2}+5 a b x +2 a^{2}}{15 \left (-b x +a \right )^{5} c^{6} b}\) | \(36\) |
norman | \(\frac {\frac {2 a^{2}}{15 b c}+\frac {b \,x^{2}}{3 c}+\frac {a x}{3 c}}{c^{5} \left (-b x +a \right )^{5}}\) | \(41\) |
parallelrisch | \(\frac {-5 x^{2} b^{6}-5 x a \,b^{5}-2 a^{2} b^{4}}{15 b^{5} c^{6} \left (b x -a \right )^{5}}\) | \(42\) |
default | \(\frac {-\frac {a}{b \left (-b x +a \right )^{4}}+\frac {1}{3 b \left (-b x +a \right )^{3}}+\frac {4 a^{2}}{5 b \left (-b x +a \right )^{5}}}{c^{6}}\) | \(49\) |
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none
Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.67 \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=-\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (46) = 92\).
Time = 0.65 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.75 \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=\frac {- 2 a^{2} - 5 a b x - 5 b^{2} x^{2}}{- 15 a^{5} b c^{6} + 75 a^{4} b^{2} c^{6} x - 150 a^{3} b^{3} c^{6} x^{2} + 150 a^{2} b^{4} c^{6} x^{3} - 75 a b^{5} c^{6} x^{4} + 15 b^{6} c^{6} x^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.67 \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=-\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]
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none
Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=-\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b x - a\right )}^{5} b c^{6}} \]
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Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x)^2}{(a c-b c x)^6} \, dx=\frac {\frac {a\,x}{3}+\frac {b\,x^2}{3}+\frac {2\,a^2}{15\,b}}{a^5\,c^6-5\,a^4\,b\,c^6\,x+10\,a^3\,b^2\,c^6\,x^2-10\,a^2\,b^3\,c^6\,x^3+5\,a\,b^4\,c^6\,x^4-b^5\,c^6\,x^5} \]
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