Integrand size = 24, antiderivative size = 101 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^5}{640 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^7}{896 c^4}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^9}{384 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4} \]
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Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac {3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac {d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3 (b d+2 c d x)^4}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^6}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^8}{64 c^3 d^4}+\frac {(b d+2 c d x)^{10}}{64 c^3 d^6}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^5}{640 c^4}+\frac {3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^7}{896 c^4}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^9}{384 c^4}+\frac {d^4 (b+2 c x)^{11}}{1408 c^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(101)=202\).
Time = 0.04 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.56 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=d^4 \left (a^3 b^4 x+\frac {1}{2} a^2 b^3 \left (3 b^2+8 a c\right ) x^2+a b^2 \left (b^4+9 a b^2 c+8 a^2 c^2\right ) x^3+\frac {1}{4} b \left (b^6+30 a b^4 c+96 a^2 b^2 c^2+32 a^3 c^3\right ) x^4+\frac {1}{5} c \left (11 b^6+123 a b^4 c+168 a^2 b^2 c^2+16 a^3 c^3\right ) x^5+\frac {1}{2} b c^2 \left (17 b^4+88 a b^2 c+48 a^2 c^2\right ) x^6+\frac {3}{7} c^3 \left (43 b^4+104 a b^2 c+16 a^2 c^2\right ) x^7+24 b c^4 \left (b^2+a c\right ) x^8+\frac {8}{3} c^5 \left (7 b^2+2 a c\right ) x^9+8 b c^6 x^{10}+\frac {16 c^7 x^{11}}{11}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(93)=186\).
Time = 2.78 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84
method | result | size |
gosper | \(\frac {x \left (6720 c^{7} x^{10}+36960 b \,c^{6} x^{9}+24640 x^{8} a \,c^{6}+86240 x^{8} b^{2} c^{5}+110880 a b \,c^{5} x^{7}+110880 b^{3} c^{4} x^{7}+31680 x^{6} a^{2} c^{5}+205920 x^{6} a \,b^{2} c^{4}+85140 x^{6} b^{4} c^{3}+110880 a^{2} b \,c^{4} x^{5}+203280 x^{5} a \,b^{3} c^{3}+39270 x^{5} b^{5} c^{2}+14784 x^{4} a^{3} c^{4}+155232 a^{2} b^{2} c^{3} x^{4}+113652 c^{2} x^{4} a \,b^{4}+10164 x^{4} b^{6} c +36960 x^{3} a^{3} c^{3} b +110880 x^{3} a^{2} c^{2} b^{3}+34650 x^{3} a \,b^{5} c +1155 x^{3} b^{7}+36960 a^{3} b^{2} c^{2} x^{2}+41580 a^{2} b^{4} c \,x^{2}+4620 a \,b^{6} x^{2}+18480 x \,a^{3} b^{3} c +6930 a^{2} x \,b^{5}+4620 a^{3} b^{4}\right ) d^{4}}{4620}\) | \(287\) |
norman | \(\left (\frac {16}{3} a \,c^{6} d^{4}+\frac {56}{3} c^{5} b^{2} d^{4}\right ) x^{9}+\left (4 b^{3} d^{4} c \,a^{3}+\frac {3}{2} b^{5} d^{4} a^{2}\right ) x^{2}+\left (\frac {48}{7} a^{2} c^{5} d^{4}+\frac {312}{7} a \,b^{2} c^{4} d^{4}+\frac {129}{7} b^{4} d^{4} c^{3}\right ) x^{7}+\left (24 c^{4} d^{4} a^{2} b +44 a \,b^{3} c^{3} d^{4}+\frac {17}{2} b^{5} d^{4} c^{2}\right ) x^{6}+\left (\frac {16}{5} c^{4} d^{4} a^{3}+\frac {168}{5} b^{2} c^{3} d^{4} a^{2}+\frac {123}{5} a \,b^{4} c^{2} d^{4}+\frac {11}{5} b^{6} c \,d^{4}\right ) x^{5}+\left (8 b \,c^{3} d^{4} a^{3}+24 b^{3} d^{4} c^{2} a^{2}+\frac {15}{2} a \,b^{5} c \,d^{4}+\frac {1}{4} b^{7} d^{4}\right ) x^{4}+\left (24 a b \,c^{5} d^{4}+24 b^{3} d^{4} c^{4}\right ) x^{8}+\left (8 b^{2} d^{4} c^{2} a^{3}+9 a^{2} b^{4} c \,d^{4}+a \,b^{6} d^{4}\right ) x^{3}+b^{4} d^{4} a^{3} x +\frac {16 c^{7} d^{4} x^{11}}{11}+8 b \,c^{6} d^{4} x^{10}\) | \(333\) |
risch | \(24 d^{4} x^{4} a^{2} c^{2} b^{3}+\frac {15}{2} d^{4} x^{4} a \,b^{5} c +4 d^{4} x^{2} a^{3} b^{3} c +24 d^{4} a b \,c^{5} x^{8}+8 d^{4} a^{3} b^{2} c^{2} x^{3}+9 d^{4} a^{2} b^{4} c \,x^{3}+\frac {168}{5} d^{4} a^{2} b^{2} c^{3} x^{5}+\frac {312}{7} d^{4} x^{7} a \,b^{2} c^{4}+24 d^{4} x^{6} a^{2} b \,c^{4}+44 d^{4} x^{6} a \,b^{3} c^{3}+\frac {1}{4} d^{4} x^{4} b^{7}+8 b \,c^{6} d^{4} x^{10}+b^{4} d^{4} a^{3} x +\frac {16}{5} d^{4} x^{5} a^{3} c^{4}+\frac {11}{5} d^{4} x^{5} b^{6} c +24 d^{4} b^{3} c^{4} x^{8}+\frac {56}{3} d^{4} x^{9} b^{2} c^{5}+\frac {48}{7} d^{4} x^{7} a^{2} c^{5}+\frac {129}{7} d^{4} x^{7} b^{4} c^{3}+\frac {16}{3} d^{4} x^{9} a \,c^{6}+\frac {16}{11} c^{7} d^{4} x^{11}+d^{4} x^{3} a \,b^{6}+\frac {3}{2} d^{4} x^{2} a^{2} b^{5}+\frac {17}{2} d^{4} x^{6} b^{5} c^{2}+\frac {123}{5} d^{4} a \,b^{4} c^{2} x^{5}+8 d^{4} x^{4} a^{3} c^{3} b\) | \(362\) |
parallelrisch | \(24 d^{4} x^{4} a^{2} c^{2} b^{3}+\frac {15}{2} d^{4} x^{4} a \,b^{5} c +4 d^{4} x^{2} a^{3} b^{3} c +24 d^{4} a b \,c^{5} x^{8}+8 d^{4} a^{3} b^{2} c^{2} x^{3}+9 d^{4} a^{2} b^{4} c \,x^{3}+\frac {168}{5} d^{4} a^{2} b^{2} c^{3} x^{5}+\frac {312}{7} d^{4} x^{7} a \,b^{2} c^{4}+24 d^{4} x^{6} a^{2} b \,c^{4}+44 d^{4} x^{6} a \,b^{3} c^{3}+\frac {1}{4} d^{4} x^{4} b^{7}+8 b \,c^{6} d^{4} x^{10}+b^{4} d^{4} a^{3} x +\frac {16}{5} d^{4} x^{5} a^{3} c^{4}+\frac {11}{5} d^{4} x^{5} b^{6} c +24 d^{4} b^{3} c^{4} x^{8}+\frac {56}{3} d^{4} x^{9} b^{2} c^{5}+\frac {48}{7} d^{4} x^{7} a^{2} c^{5}+\frac {129}{7} d^{4} x^{7} b^{4} c^{3}+\frac {16}{3} d^{4} x^{9} a \,c^{6}+\frac {16}{11} c^{7} d^{4} x^{11}+d^{4} x^{3} a \,b^{6}+\frac {3}{2} d^{4} x^{2} a^{2} b^{5}+\frac {17}{2} d^{4} x^{6} b^{5} c^{2}+\frac {123}{5} d^{4} a \,b^{4} c^{2} x^{5}+8 d^{4} x^{4} a^{3} c^{3} b\) | \(362\) |
default | \(\frac {16 c^{7} d^{4} x^{11}}{11}+8 b \,c^{6} d^{4} x^{10}+\frac {\left (120 c^{5} b^{2} d^{4}+16 c^{4} d^{4} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{9}}{9}+\frac {\left (80 b^{3} d^{4} c^{4}+32 b \,c^{3} d^{4} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+16 c^{4} d^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (25 b^{4} d^{4} c^{3}+24 b^{2} d^{4} c^{2} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+32 b \,c^{3} d^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+16 c^{4} d^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{7}}{7}+\frac {\left (3 b^{5} d^{4} c^{2}+8 b^{3} d^{4} c \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+24 b^{2} d^{4} c^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+32 b \,c^{3} d^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+48 c^{4} d^{4} a^{2} b \right ) x^{6}}{6}+\frac {\left (b^{4} d^{4} \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+8 b^{3} d^{4} c \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+24 b^{2} d^{4} c^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+96 b^{2} c^{3} d^{4} a^{2}+16 c^{4} d^{4} a^{3}\right ) x^{5}}{5}+\frac {\left (b^{4} d^{4} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+8 b^{3} d^{4} c \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+72 b^{3} d^{4} c^{2} a^{2}+32 b \,c^{3} d^{4} a^{3}\right ) x^{4}}{4}+\frac {\left (b^{4} d^{4} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+24 a^{2} b^{4} c \,d^{4}+24 b^{2} d^{4} c^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (8 b^{3} d^{4} c \,a^{3}+3 b^{5} d^{4} a^{2}\right ) x^{2}}{2}+b^{4} d^{4} a^{3} x\) | \(672\) |
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Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (93) = 186\).
Time = 0.25 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.87 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {8}{3} \, {\left (7 \, b^{2} c^{5} + 2 \, a c^{6}\right )} d^{4} x^{9} + 24 \, {\left (b^{3} c^{4} + a b c^{5}\right )} d^{4} x^{8} + a^{3} b^{4} d^{4} x + \frac {3}{7} \, {\left (43 \, b^{4} c^{3} + 104 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{7} + \frac {1}{2} \, {\left (17 \, b^{5} c^{2} + 88 \, a b^{3} c^{3} + 48 \, a^{2} b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (11 \, b^{6} c + 123 \, a b^{4} c^{2} + 168 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (b^{7} + 30 \, a b^{5} c + 96 \, a^{2} b^{3} c^{2} + 32 \, a^{3} b c^{3}\right )} d^{4} x^{4} + {\left (a b^{6} + 9 \, a^{2} b^{4} c + 8 \, a^{3} b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{5} + 8 \, a^{3} b^{3} c\right )} d^{4} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (97) = 194\).
Time = 0.05 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.67 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^{3} b^{4} d^{4} x + 8 b c^{6} d^{4} x^{10} + \frac {16 c^{7} d^{4} x^{11}}{11} + x^{9} \cdot \left (\frac {16 a c^{6} d^{4}}{3} + \frac {56 b^{2} c^{5} d^{4}}{3}\right ) + x^{8} \cdot \left (24 a b c^{5} d^{4} + 24 b^{3} c^{4} d^{4}\right ) + x^{7} \cdot \left (\frac {48 a^{2} c^{5} d^{4}}{7} + \frac {312 a b^{2} c^{4} d^{4}}{7} + \frac {129 b^{4} c^{3} d^{4}}{7}\right ) + x^{6} \cdot \left (24 a^{2} b c^{4} d^{4} + 44 a b^{3} c^{3} d^{4} + \frac {17 b^{5} c^{2} d^{4}}{2}\right ) + x^{5} \cdot \left (\frac {16 a^{3} c^{4} d^{4}}{5} + \frac {168 a^{2} b^{2} c^{3} d^{4}}{5} + \frac {123 a b^{4} c^{2} d^{4}}{5} + \frac {11 b^{6} c d^{4}}{5}\right ) + x^{4} \cdot \left (8 a^{3} b c^{3} d^{4} + 24 a^{2} b^{3} c^{2} d^{4} + \frac {15 a b^{5} c d^{4}}{2} + \frac {b^{7} d^{4}}{4}\right ) + x^{3} \cdot \left (8 a^{3} b^{2} c^{2} d^{4} + 9 a^{2} b^{4} c d^{4} + a b^{6} d^{4}\right ) + x^{2} \cdot \left (4 a^{3} b^{3} c d^{4} + \frac {3 a^{2} b^{5} d^{4}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (93) = 186\).
Time = 0.19 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.87 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {8}{3} \, {\left (7 \, b^{2} c^{5} + 2 \, a c^{6}\right )} d^{4} x^{9} + 24 \, {\left (b^{3} c^{4} + a b c^{5}\right )} d^{4} x^{8} + a^{3} b^{4} d^{4} x + \frac {3}{7} \, {\left (43 \, b^{4} c^{3} + 104 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{7} + \frac {1}{2} \, {\left (17 \, b^{5} c^{2} + 88 \, a b^{3} c^{3} + 48 \, a^{2} b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (11 \, b^{6} c + 123 \, a b^{4} c^{2} + 168 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (b^{7} + 30 \, a b^{5} c + 96 \, a^{2} b^{3} c^{2} + 32 \, a^{3} b c^{3}\right )} d^{4} x^{4} + {\left (a b^{6} + 9 \, a^{2} b^{4} c + 8 \, a^{3} b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b^{5} + 8 \, a^{3} b^{3} c\right )} d^{4} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (93) = 186\).
Time = 0.28 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.57 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac {56}{3} \, b^{2} c^{5} d^{4} x^{9} + \frac {16}{3} \, a c^{6} d^{4} x^{9} + 24 \, b^{3} c^{4} d^{4} x^{8} + 24 \, a b c^{5} d^{4} x^{8} + \frac {129}{7} \, b^{4} c^{3} d^{4} x^{7} + \frac {312}{7} \, a b^{2} c^{4} d^{4} x^{7} + \frac {48}{7} \, a^{2} c^{5} d^{4} x^{7} + \frac {17}{2} \, b^{5} c^{2} d^{4} x^{6} + 44 \, a b^{3} c^{3} d^{4} x^{6} + 24 \, a^{2} b c^{4} d^{4} x^{6} + \frac {11}{5} \, b^{6} c d^{4} x^{5} + \frac {123}{5} \, a b^{4} c^{2} d^{4} x^{5} + \frac {168}{5} \, a^{2} b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a^{3} c^{4} d^{4} x^{5} + \frac {1}{4} \, b^{7} d^{4} x^{4} + \frac {15}{2} \, a b^{5} c d^{4} x^{4} + 24 \, a^{2} b^{3} c^{2} d^{4} x^{4} + 8 \, a^{3} b c^{3} d^{4} x^{4} + a b^{6} d^{4} x^{3} + 9 \, a^{2} b^{4} c d^{4} x^{3} + 8 \, a^{3} b^{2} c^{2} d^{4} x^{3} + \frac {3}{2} \, a^{2} b^{5} d^{4} x^{2} + 4 \, a^{3} b^{3} c d^{4} x^{2} + a^{3} b^{4} d^{4} x \]
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Time = 0.15 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.71 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {16\,c^7\,d^4\,x^{11}}{11}+\frac {3\,c^3\,d^4\,x^7\,\left (16\,a^2\,c^2+104\,a\,b^2\,c+43\,b^4\right )}{7}+a^3\,b^4\,d^4\,x+8\,b\,c^6\,d^4\,x^{10}+\frac {c\,d^4\,x^5\,\left (16\,a^3\,c^3+168\,a^2\,b^2\,c^2+123\,a\,b^4\,c+11\,b^6\right )}{5}+\frac {8\,c^5\,d^4\,x^9\,\left (7\,b^2+2\,a\,c\right )}{3}+\frac {b\,d^4\,x^4\,\left (32\,a^3\,c^3+96\,a^2\,b^2\,c^2+30\,a\,b^4\,c+b^6\right )}{4}+a\,b^2\,d^4\,x^3\,\left (8\,a^2\,c^2+9\,a\,b^2\,c+b^4\right )+\frac {a^2\,b^3\,d^4\,x^2\,\left (3\,b^2+8\,a\,c\right )}{2}+24\,b\,c^4\,d^4\,x^8\,\left (b^2+a\,c\right )+\frac {b\,c^2\,d^4\,x^6\,\left (48\,a^2\,c^2+88\,a\,b^2\,c+17\,b^4\right )}{2} \]
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