Integrand size = 24, antiderivative size = 55 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{20} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^4+\frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4 \]
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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} \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{20} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4+\frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4 \]
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Rule 643
Rule 706
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4+\frac {1}{5} \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 )^3 \, dx \\ & = \frac {1}{20} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^4+\frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(55)=110\).
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.40 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{20} d^3 x (b+c x) \left (20 a^3 \left (b^2+2 b c x+2 c^2 x^2\right )+20 a x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )+x^3 (b+c x)^3 \left (5 b^2+16 b c x+16 c^2 x^2\right )+10 a^2 x \left (3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right )\right ) \]
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Time = 2.53 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84
method | result | size |
default | \(-d^{3} \left (-\frac {4 c \left (c \,x^{2}+b x +a \right )^{5}}{5}+\frac {\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{4}}{4}\right )\) | \(46\) |
gosper | \(\frac {x \left (16 c^{6} x^{9}+80 b \,c^{5} x^{8}+60 x^{7} c^{5} a +165 x^{7} b^{2} c^{4}+240 a b \,c^{4} x^{6}+180 b^{3} c^{3} x^{6}+80 x^{5} a^{2} c^{4}+380 x^{5} b^{2} c^{3} a +110 b^{4} c^{2} x^{5}+240 a^{2} b \,c^{3} x^{4}+300 x^{4} a \,b^{3} c^{2}+36 x^{4} c \,b^{5}+40 x^{3} c^{3} a^{3}+270 x^{3} a^{2} b^{2} c^{2}+120 a \,b^{4} c \,x^{3}+5 x^{3} b^{6}+80 a^{3} b \,c^{2} x^{2}+140 a^{2} b^{3} c \,x^{2}+20 a \,b^{5} x^{2}+60 c x \,a^{3} b^{2}+30 a^{2} b^{4} x +20 a^{3} b^{3}\right ) d^{3}}{20}\) | \(236\) |
norman | \(\left (3 a \,c^{5} d^{3}+\frac {33}{4} b^{2} c^{4} d^{3}\right ) x^{8}+\left (3 a^{3} b^{2} c \,d^{3}+\frac {3}{2} a^{2} b^{4} d^{3}\right ) x^{2}+\left (4 a^{2} c^{4} d^{3}+19 a \,b^{2} c^{3} d^{3}+\frac {11}{2} b^{4} c^{2} d^{3}\right ) x^{6}+\left (12 a^{2} b \,c^{3} d^{3}+15 a \,b^{3} c^{2} d^{3}+\frac {9}{5} b^{5} c \,d^{3}\right ) x^{5}+\left (2 a^{3} c^{3} d^{3}+\frac {27}{2} a^{2} b^{2} c^{2} d^{3}+6 a \,b^{4} c \,d^{3}+\frac {1}{4} b^{6} d^{3}\right ) x^{4}+\left (12 a b \,c^{4} d^{3}+9 b^{3} c^{3} d^{3}\right ) x^{7}+\left (4 a^{3} b \,d^{3} c^{2}+7 a^{2} b^{3} c \,d^{3}+a \,b^{5} d^{3}\right ) x^{3}+a^{3} b^{3} d^{3} x +\frac {4 c^{6} d^{3} x^{10}}{5}+4 b \,c^{5} d^{3} x^{9}\) | \(277\) |
parallelrisch | \(a^{3} b^{3} d^{3} x +3 d^{3} c \,x^{2} a^{3} b^{2}+\frac {3}{2} d^{3} a^{2} b^{4} x^{2}+4 d^{3} a^{3} b \,c^{2} x^{3}+7 d^{3} a^{2} b^{3} c \,x^{3}+d^{3} a \,b^{5} x^{3}+2 d^{3} a^{3} c^{3} x^{4}+\frac {27}{2} d^{3} a^{2} b^{2} c^{2} x^{4}+6 d^{3} a \,b^{4} c \,x^{4}+\frac {1}{4} d^{3} b^{6} x^{4}+12 d^{3} a^{2} b \,c^{3} x^{5}+15 a \,b^{3} c^{2} d^{3} x^{5}+\frac {9}{5} d^{3} b^{5} c \,x^{5}+4 d^{3} a^{2} c^{4} x^{6}+19 d^{3} a \,b^{2} c^{3} x^{6}+\frac {11}{2} d^{3} b^{4} c^{2} x^{6}+12 d^{3} a b \,c^{4} x^{7}+9 d^{3} b^{3} c^{3} x^{7}+3 d^{3} a \,c^{5} x^{8}+\frac {33}{4} d^{3} b^{2} c^{4} x^{8}+4 b \,c^{5} d^{3} x^{9}+\frac {4}{5} c^{6} d^{3} x^{10}\) | \(299\) |
risch | \(-\frac {1}{5} d^{3} c \,a^{5}+\frac {1}{4} d^{3} a^{4} b^{2}+6 d^{3} a \,b^{4} c \,x^{4}+\frac {9}{5} d^{3} b^{5} c \,x^{5}+3 d^{3} c \,x^{2} a^{3} b^{2}+4 b \,c^{5} d^{3} x^{9}+a^{3} b^{3} d^{3} x +12 d^{3} a^{2} b \,c^{3} x^{5}+3 d^{3} a \,c^{5} x^{8}+\frac {33}{4} d^{3} b^{2} c^{4} x^{8}+9 d^{3} b^{3} c^{3} x^{7}+4 d^{3} a^{2} c^{4} x^{6}+2 d^{3} a^{3} c^{3} x^{4}+d^{3} a \,b^{5} x^{3}+\frac {3}{2} d^{3} a^{2} b^{4} x^{2}+12 d^{3} a b \,c^{4} x^{7}+19 d^{3} a \,b^{2} c^{3} x^{6}+\frac {11}{2} d^{3} b^{4} c^{2} x^{6}+\frac {1}{4} d^{3} b^{6} x^{4}+\frac {4}{5} c^{6} d^{3} x^{10}+15 a \,b^{3} c^{2} d^{3} x^{5}+\frac {27}{2} d^{3} a^{2} b^{2} c^{2} x^{4}+4 d^{3} a^{3} b \,c^{2} x^{3}+7 d^{3} a^{2} b^{3} c \,x^{3}\) | \(319\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (51) = 102\).
Time = 0.36 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.42 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac {3}{4} \, {\left (11 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{3} x^{8} + 3 \, {\left (3 \, b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x^{7} + a^{3} b^{3} d^{3} x + \frac {1}{2} \, {\left (11 \, b^{4} c^{2} + 38 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{6} + \frac {3}{5} \, {\left (3 \, b^{5} c + 25 \, a b^{3} c^{2} + 20 \, a^{2} b c^{3}\right )} d^{3} x^{5} + \frac {1}{4} \, {\left (b^{6} + 24 \, a b^{4} c + 54 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{3} x^{4} + {\left (a b^{5} + 7 \, a^{2} b^{3} c + 4 \, a^{3} b c^{2}\right )} d^{3} x^{3} + \frac {3}{2} \, {\left (a^{2} b^{4} + 2 \, a^{3} b^{2} c\right )} d^{3} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (49) = 98\).
Time = 0.04 (sec) , antiderivative size = 299, normalized size of antiderivative = 5.44 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx=a^{3} b^{3} d^{3} x + 4 b c^{5} d^{3} x^{9} + \frac {4 c^{6} d^{3} x^{10}}{5} + x^{8} \cdot \left (3 a c^{5} d^{3} + \frac {33 b^{2} c^{4} d^{3}}{4}\right ) + x^{7} \cdot \left (12 a b c^{4} d^{3} + 9 b^{3} c^{3} d^{3}\right ) + x^{6} \cdot \left (4 a^{2} c^{4} d^{3} + 19 a b^{2} c^{3} d^{3} + \frac {11 b^{4} c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (12 a^{2} b c^{3} d^{3} + 15 a b^{3} c^{2} d^{3} + \frac {9 b^{5} c d^{3}}{5}\right ) + x^{4} \cdot \left (2 a^{3} c^{3} d^{3} + \frac {27 a^{2} b^{2} c^{2} d^{3}}{2} + 6 a b^{4} c d^{3} + \frac {b^{6} d^{3}}{4}\right ) + x^{3} \cdot \left (4 a^{3} b c^{2} d^{3} + 7 a^{2} b^{3} c d^{3} + a b^{5} d^{3}\right ) + x^{2} \cdot \left (3 a^{3} b^{2} c d^{3} + \frac {3 a^{2} b^{4} d^{3}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (51) = 102\).
Time = 0.19 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.42 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac {3}{4} \, {\left (11 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{3} x^{8} + 3 \, {\left (3 \, b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x^{7} + a^{3} b^{3} d^{3} x + \frac {1}{2} \, {\left (11 \, b^{4} c^{2} + 38 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{6} + \frac {3}{5} \, {\left (3 \, b^{5} c + 25 \, a b^{3} c^{2} + 20 \, a^{2} b c^{3}\right )} d^{3} x^{5} + \frac {1}{4} \, {\left (b^{6} + 24 \, a b^{4} c + 54 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{3} x^{4} + {\left (a b^{5} + 7 \, a^{2} b^{3} c + 4 \, a^{3} b c^{2}\right )} d^{3} x^{3} + \frac {3}{2} \, {\left (a^{2} b^{4} + 2 \, a^{3} b^{2} c\right )} d^{3} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (51) = 102\).
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 3.11 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx={\left (c d x^{2} + b d x\right )} a^{3} b^{2} d^{2} + \frac {30 \, {\left (c d x^{2} + b d x\right )}^{2} a^{2} b^{2} d^{3} + 40 \, {\left (c d x^{2} + b d x\right )}^{2} a^{3} c d^{3} + 20 \, {\left (c d x^{2} + b d x\right )}^{3} a b^{2} d^{2} + 80 \, {\left (c d x^{2} + b d x\right )}^{3} a^{2} c d^{2} + 5 \, {\left (c d x^{2} + b d x\right )}^{4} b^{2} d + 60 \, {\left (c d x^{2} + b d x\right )}^{4} a c d + 16 \, {\left (c d x^{2} + b d x\right )}^{5} c}{20 \, d^{2}} \]
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Time = 9.99 (sec) , antiderivative size = 229, normalized size of antiderivative = 4.16 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {d^3\,x^4\,\left (8\,a^3\,c^3+54\,a^2\,b^2\,c^2+24\,a\,b^4\,c+b^6\right )}{4}+\frac {4\,c^6\,d^3\,x^{10}}{5}+\frac {c^2\,d^3\,x^6\,\left (8\,a^2\,c^2+38\,a\,b^2\,c+11\,b^4\right )}{2}+a^3\,b^3\,d^3\,x+4\,b\,c^5\,d^3\,x^9+\frac {3\,c^4\,d^3\,x^8\,\left (11\,b^2+4\,a\,c\right )}{4}+\frac {3\,b\,c\,d^3\,x^5\,\left (20\,a^2\,c^2+25\,a\,b^2\,c+3\,b^4\right )}{5}+a\,b\,d^3\,x^3\,\left (4\,a^2\,c^2+7\,a\,b^2\,c+b^4\right )+\frac {3\,a^2\,b^2\,d^3\,x^2\,\left (b^2+2\,a\,c\right )}{2}+3\,b\,c^3\,d^3\,x^7\,\left (3\,b^2+4\,a\,c\right ) \]
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