Integrand size = 24, antiderivative size = 37 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {696} \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rule 696
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d^3 \left (b^2+8 b c x+4 c \left (a+2 c x^2\right )\right )}{2 (a+x (b+c x))^2} \]
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Time = 2.79 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05
method | result | size |
gosper | \(-\frac {d^{3} \left (8 c^{2} x^{2}+8 b c x +4 a c +b^{2}\right )}{2 \left (c \,x^{2}+b x +a \right )^{2}}\) | \(39\) |
default | \(\frac {d^{3} \left (-4 c^{2} x^{2}-4 b c x -2 a c -\frac {1}{2} b^{2}\right )}{\left (c \,x^{2}+b x +a \right )^{2}}\) | \(40\) |
risch | \(\frac {-4 c^{2} d^{3} x^{2}-4 b c \,d^{3} x -\frac {d^{3} \left (4 a c +b^{2}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}\) | \(47\) |
norman | \(\frac {-4 c^{2} d^{3} x^{2}+\frac {-4 a \,c^{3} d^{3}-b^{2} c^{2} d^{3}}{2 c^{2}}-4 b c \,d^{3} x}{\left (c \,x^{2}+b x +a \right )^{2}}\) | \(60\) |
parallelrisch | \(\frac {-8 c^{4} d^{3} x^{2}-8 b \,c^{3} d^{3} x -4 a \,c^{3} d^{3}-b^{2} c^{2} d^{3}}{2 c^{2} \left (c \,x^{2}+b x +a \right )^{2}}\) | \(60\) |
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (35) = 70\).
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.92 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + {\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (34) = 68\).
Time = 0.95 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.16 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=\frac {- 4 a c d^{3} - b^{2} d^{3} - 8 b c d^{3} x - 8 c^{2} d^{3} x^{2}}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (35) = 70\).
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.92 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + {\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.35 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {b^{2} d^{5} + 4 \, a c d^{5} + 8 \, {\left (c d x^{2} + b d x\right )} c d^{4}}{2 \, {\left (c d x^{2} + b d x + a d\right )}^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.00 \[ \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\frac {b^2\,d^3}{2}+4\,b\,c\,d^3\,x+4\,c^2\,d^3\,x^2+2\,a\,c\,d^3}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \]
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