Integrand size = 24, antiderivative size = 55 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{12} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^3+\frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {706, 643} \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]
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Rule 643
Rule 706
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3+\frac {1}{4} \left (\left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx \\ & = \frac {1}{12} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^3+\frac {1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.76 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} d^3 x (b+c x) \left (3 a^2 \left (b^2+2 b c x+2 c^2 x^2\right )+x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )+a x \left (3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right )\right ) \]
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Time = 2.55 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84
method | result | size |
default | \(-d^{3} \left (-c \left (c \,x^{2}+b x +a \right )^{4}+\frac {\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{3}}{3}\right )\) | \(46\) |
gosper | \(\frac {x \left (3 c^{5} x^{7}+12 b \,c^{4} x^{6}+8 x^{5} a \,c^{4}+19 x^{5} b^{2} c^{3}+24 x^{4} a b \,c^{3}+15 c^{2} b^{3} x^{4}+6 a^{2} c^{3} x^{3}+27 a \,b^{2} c^{2} x^{3}+6 b^{4} c \,x^{3}+12 a^{2} b \,c^{2} x^{2}+14 a \,b^{3} c \,x^{2}+b^{5} x^{2}+9 c x \,a^{2} b^{2}+3 b^{4} x a +3 a^{2} b^{3}\right ) d^{3}}{3}\) | \(152\) |
norman | \(\left (\frac {8}{3} a \,c^{4} d^{3}+\frac {19}{3} b^{2} c^{3} d^{3}\right ) x^{6}+\left (4 a^{2} b \,c^{2} d^{3}+\frac {14}{3} a \,b^{3} c \,d^{3}+\frac {1}{3} b^{5} d^{3}\right ) x^{3}+\left (8 a b \,c^{3} d^{3}+5 d^{3} c^{2} b^{3}\right ) x^{5}+\left (3 a^{2} b^{2} c \,d^{3}+a \,b^{4} d^{3}\right ) x^{2}+\left (2 a^{2} c^{3} d^{3}+9 a \,b^{2} c^{2} d^{3}+2 b^{4} c \,d^{3}\right ) x^{4}+c^{5} d^{3} x^{8}+a^{2} b^{3} d^{3} x +4 b \,c^{4} d^{3} x^{7}\) | \(183\) |
parallelrisch | \(a^{2} b^{3} d^{3} x +3 a^{2} b^{2} c \,d^{3} x^{2}+d^{3} b^{4} x^{2} a +4 d^{3} a^{2} b \,c^{2} x^{3}+\frac {14}{3} a \,b^{3} c \,d^{3} x^{3}+\frac {1}{3} d^{3} x^{3} b^{5}+2 d^{3} a^{2} c^{3} x^{4}+9 d^{3} a \,b^{2} c^{2} x^{4}+2 b^{4} c \,d^{3} x^{4}+8 d^{3} a b \,c^{3} x^{5}+5 b^{3} c^{2} d^{3} x^{5}+\frac {8}{3} d^{3} a \,c^{4} x^{6}+\frac {19}{3} d^{3} b^{2} c^{3} x^{6}+4 b \,c^{4} d^{3} x^{7}+c^{5} d^{3} x^{8}\) | \(194\) |
risch | \(\frac {1}{3} a^{3} b^{2} d^{3}+d^{3} b^{4} x^{2} a +2 d^{3} a^{2} c^{3} x^{4}+2 b^{4} c \,d^{3} x^{4}-\frac {1}{3} a^{4} c \,d^{3}+4 d^{3} a^{2} b \,c^{2} x^{3}+c^{5} d^{3} x^{8}+3 a^{2} b^{2} c \,d^{3} x^{2}+\frac {1}{3} d^{3} x^{3} b^{5}+9 d^{3} a \,b^{2} c^{2} x^{4}+a^{2} b^{3} d^{3} x +\frac {19}{3} d^{3} b^{2} c^{3} x^{6}+\frac {8}{3} d^{3} a \,c^{4} x^{6}+4 b \,c^{4} d^{3} x^{7}+8 d^{3} a b \,c^{3} x^{5}+\frac {14}{3} a \,b^{3} c \,d^{3} x^{3}+5 b^{3} c^{2} d^{3} x^{5}\) | \(214\) |
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (51) = 102\).
Time = 0.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.93 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx=c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac {1}{3} \, {\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{6} + a^{2} b^{3} d^{3} x + {\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{5} + {\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{3} + {\left (a b^{4} + 3 \, a^{2} b^{2} c\right )} d^{3} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (49) = 98\).
Time = 0.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 3.53 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx=a^{2} b^{3} d^{3} x + 4 b c^{4} d^{3} x^{7} + c^{5} d^{3} x^{8} + x^{6} \cdot \left (\frac {8 a c^{4} d^{3}}{3} + \frac {19 b^{2} c^{3} d^{3}}{3}\right ) + x^{5} \cdot \left (8 a b c^{3} d^{3} + 5 b^{3} c^{2} d^{3}\right ) + x^{4} \cdot \left (2 a^{2} c^{3} d^{3} + 9 a b^{2} c^{2} d^{3} + 2 b^{4} c d^{3}\right ) + x^{3} \cdot \left (4 a^{2} b c^{2} d^{3} + \frac {14 a b^{3} c d^{3}}{3} + \frac {b^{5} d^{3}}{3}\right ) + x^{2} \cdot \left (3 a^{2} b^{2} c d^{3} + a b^{4} d^{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (51) = 102\).
Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.93 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx=c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac {1}{3} \, {\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{6} + a^{2} b^{3} d^{3} x + {\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{5} + {\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{3} + {\left (a b^{4} + 3 \, a^{2} b^{2} c\right )} d^{3} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (51) = 102\).
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.25 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx={\left (c d x^{2} + b d x\right )} a^{2} b^{2} d^{2} + \frac {3 \, {\left (c d x^{2} + b d x\right )}^{2} a b^{2} d^{2} + 6 \, {\left (c d x^{2} + b d x\right )}^{2} a^{2} c d^{2} + {\left (c d x^{2} + b d x\right )}^{3} b^{2} d + 8 \, {\left (c d x^{2} + b d x\right )}^{3} a c d + 3 \, {\left (c d x^{2} + b d x\right )}^{4} c}{3 \, d} \]
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Time = 0.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.76 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^2 \, dx=c^5\,d^3\,x^8+a^2\,b^3\,d^3\,x+4\,b\,c^4\,d^3\,x^7+\frac {b\,d^3\,x^3\,\left (12\,a^2\,c^2+14\,a\,b^2\,c+b^4\right )}{3}+\frac {c^3\,d^3\,x^6\,\left (19\,b^2+8\,a\,c\right )}{3}+c\,d^3\,x^4\,\left (2\,a^2\,c^2+9\,a\,b^2\,c+2\,b^4\right )+a\,b^2\,d^3\,x^2\,\left (b^2+3\,a\,c\right )+b\,c^2\,d^3\,x^5\,\left (5\,b^2+8\,a\,c\right ) \]
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