Integrand size = 29, antiderivative size = 14 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=\frac {(c+d x)^4}{4 d} \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 32} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=\frac {(c+d x)^4}{4 d} \]
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Rule 32
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (c+d x)^3 \, dx \\ & = \frac {(c+d x)^4}{4 d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=\frac {(c+d x)^4}{4 d} \]
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Time = 2.35 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (d x +c \right )^{4}}{4 d}\) | \(13\) |
parallelrisch | \(\frac {1}{4} d^{3} x^{4}+c \,d^{2} x^{3}+\frac {3}{2} c^{2} d \,x^{2}+c^{3} x\) | \(32\) |
gosper | \(\frac {x \left (d^{3} x^{3}+4 c \,d^{2} x^{2}+6 c^{2} d x +4 c^{3}\right )}{4}\) | \(33\) |
risch | \(\frac {d^{3} x^{4}}{4}+c \,d^{2} x^{3}+\frac {3 c^{2} d \,x^{2}}{2}+c^{3} x +\frac {c^{4}}{4 d}\) | \(40\) |
norman | \(\frac {\left (\frac {1}{2} a b \,d^{3}+b^{2} c \,d^{2}\right ) x^{5}+\left (\frac {1}{4} a^{2} d^{3}+2 a b c \,d^{2}+\frac {3}{2} b^{2} c^{2} d \right ) x^{4}+\left (c \,a^{2} d^{2}+3 a b \,c^{2} d +b^{2} c^{3}\right ) x^{3}-\frac {a^{2} \left (3 a^{2} c^{2} d +4 a b \,c^{3}\right )}{2 b^{2}}+\frac {b^{2} d^{3} x^{6}}{4}-\frac {a \left (3 a^{2} c^{2} d +3 a b \,c^{3}\right ) x}{b}}{\left (b x +a \right )^{2}}\) | \(148\) |
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=\frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (8) = 16\).
Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=\frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=\frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \]
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Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx=c^3\,x+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {d^3\,x^4}{4} \]
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