Integrand size = 21, antiderivative size = 92 \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=-\frac {(b c-a d)^3 (c+d x)^4}{4 d^4}+\frac {3 b (b c-a d)^2 (c+d x)^5}{5 d^4}-\frac {b^2 (b c-a d) (c+d x)^6}{2 d^4}+\frac {b^3 (c+d x)^7}{7 d^4} \]
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Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {624, 45} \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=-\frac {b^2 (c+d x)^6 (b c-a d)}{2 d^4}+\frac {3 b (c+d x)^5 (b c-a d)^2}{5 d^4}-\frac {(c+d x)^4 (b c-a d)^3}{4 d^4}+\frac {b^3 (c+d x)^7}{7 d^4} \]
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Rule 45
Rule 624
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b c+b d x)^3 (a d+b d x)^3 \, dx}{b^3 d^3} \\ & = \frac {\int \left (-(b c-a d)^3 (b c+b d x)^3+3 (b c-a d)^2 (b c+b d x)^4-3 (b c-a d) (b c+b d x)^5+(b c+b d x)^6\right ) \, dx}{b^3 d^3} \\ & = -\frac {(b c-a d)^3 (c+d x)^4}{4 d^4}+\frac {3 b (b c-a d)^2 (c+d x)^5}{5 d^4}-\frac {b^2 (b c-a d) (c+d x)^6}{2 d^4}+\frac {b^3 (c+d x)^7}{7 d^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.75 \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=a^3 c^3 x+\frac {3}{2} a^2 c^2 (b c+a d) x^2+a c \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^3+\frac {1}{4} \left (b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right ) x^4+\frac {3}{5} b d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^5+\frac {1}{2} b^2 d^2 (b c+a d) x^6+\frac {1}{7} b^3 d^3 x^7 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(171\) vs. \(2(84)=168\).
Time = 2.42 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.87
method | result | size |
norman | \(\frac {b^{3} d^{3} x^{7}}{7}+\left (\frac {1}{2} a \,b^{2} d^{3}+\frac {1}{2} b^{3} c \,d^{2}\right ) x^{6}+\left (\frac {3}{5} a^{2} b \,d^{3}+\frac {9}{5} a \,b^{2} c \,d^{2}+\frac {3}{5} b^{3} c^{2} d \right ) x^{5}+\left (\frac {1}{4} a^{3} d^{3}+\frac {9}{4} a^{2} b c \,d^{2}+\frac {9}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{4}+\left (a^{3} c \,d^{2}+3 a^{2} b \,c^{2} d +b^{2} c^{3} a \right ) x^{3}+\left (\frac {3}{2} a^{3} c^{2} d +\frac {3}{2} a^{2} b \,c^{3}\right ) x^{2}+a^{3} c^{3} x\) | \(172\) |
risch | \(\frac {1}{7} b^{3} d^{3} x^{7}+\frac {1}{2} a \,b^{2} d^{3} x^{6}+\frac {1}{2} b^{3} c \,d^{2} x^{6}+\frac {3}{5} a^{2} b \,d^{3} x^{5}+\frac {9}{5} a \,b^{2} c \,d^{2} x^{5}+\frac {3}{5} b^{3} c^{2} d \,x^{5}+\frac {1}{4} a^{3} d^{3} x^{4}+\frac {9}{4} a^{2} b c \,d^{2} x^{4}+\frac {9}{4} a \,b^{2} c^{2} d \,x^{4}+\frac {1}{4} c^{3} b^{3} x^{4}+d^{2} a^{3} c \,x^{3}+3 a^{2} b \,c^{2} d \,x^{3}+a \,b^{2} c^{3} x^{3}+\frac {3}{2} a^{3} c^{2} d \,x^{2}+\frac {3}{2} a^{2} b \,c^{3} x^{2}+a^{3} c^{3} x\) | \(189\) |
parallelrisch | \(\frac {1}{7} b^{3} d^{3} x^{7}+\frac {1}{2} a \,b^{2} d^{3} x^{6}+\frac {1}{2} b^{3} c \,d^{2} x^{6}+\frac {3}{5} a^{2} b \,d^{3} x^{5}+\frac {9}{5} a \,b^{2} c \,d^{2} x^{5}+\frac {3}{5} b^{3} c^{2} d \,x^{5}+\frac {1}{4} a^{3} d^{3} x^{4}+\frac {9}{4} a^{2} b c \,d^{2} x^{4}+\frac {9}{4} a \,b^{2} c^{2} d \,x^{4}+\frac {1}{4} c^{3} b^{3} x^{4}+d^{2} a^{3} c \,x^{3}+3 a^{2} b \,c^{2} d \,x^{3}+a \,b^{2} c^{3} x^{3}+\frac {3}{2} a^{3} c^{2} d \,x^{2}+\frac {3}{2} a^{2} b \,c^{3} x^{2}+a^{3} c^{3} x\) | \(189\) |
gosper | \(\frac {x \left (20 b^{3} d^{3} x^{6}+70 a \,b^{2} d^{3} x^{5}+70 b^{3} c \,d^{2} x^{5}+84 a^{2} b \,d^{3} x^{4}+252 a \,b^{2} c \,d^{2} x^{4}+84 b^{3} c^{2} d \,x^{4}+35 x^{3} a^{3} d^{3}+315 x^{3} a^{2} b c \,d^{2}+315 x^{3} a \,b^{2} c^{2} d +35 x^{3} b^{3} c^{3}+140 a^{3} c \,d^{2} x^{2}+420 a^{2} b \,c^{2} d \,x^{2}+140 a \,b^{2} c^{3} x^{2}+210 a^{3} c^{2} d x +210 a^{2} b \,c^{3} x +140 c^{3} a^{3}\right )}{140}\) | \(190\) |
default | \(\frac {b^{3} d^{3} x^{7}}{7}+\frac {\left (a d +b c \right ) b^{2} d^{2} x^{6}}{2}+\frac {\left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +b d \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{5}}{5}+\frac {\left (4 a c b d \left (a d +b c \right )+\left (a d +b c \right ) \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{4}}{4}+\frac {\left (a c \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 \left (a d +b c \right )^{2} a c +a^{2} b \,c^{2} d \right ) x^{3}}{3}+\frac {3 a^{2} c^{2} \left (a d +b c \right ) x^{2}}{2}+a^{3} c^{3} x\) | \(194\) |
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Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.82 \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {1}{7} \, b^{3} d^{3} x^{7} + a^{3} c^{3} x + \frac {1}{2} \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} + {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (80) = 160\).
Time = 0.03 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.07 \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=a^{3} c^{3} x + \frac {b^{3} d^{3} x^{7}}{7} + x^{6} \left (\frac {a b^{2} d^{3}}{2} + \frac {b^{3} c d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b d^{3}}{5} + \frac {9 a b^{2} c d^{2}}{5} + \frac {3 b^{3} c^{2} d}{5}\right ) + x^{4} \left (\frac {a^{3} d^{3}}{4} + \frac {9 a^{2} b c d^{2}}{4} + \frac {9 a b^{2} c^{2} d}{4} + \frac {b^{3} c^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 3 a^{2} b c^{2} d + a b^{2} c^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} c^{2} d}{2} + \frac {3 a^{2} b c^{3}}{2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.52 \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {1}{7} \, b^{3} d^{3} x^{7} + \frac {1}{2} \, {\left (b c + a d\right )} b^{2} d^{2} x^{6} + \frac {3}{5} \, {\left (b c + a d\right )}^{2} b d x^{5} + a^{3} c^{3} x + \frac {1}{4} \, {\left (b c + a d\right )}^{3} x^{4} + \frac {1}{2} \, {\left (2 \, b d x^{3} + 3 \, {\left (b c + a d\right )} x^{2}\right )} a^{2} c^{2} + \frac {1}{10} \, {\left (6 \, b^{2} d^{2} x^{5} + 15 \, {\left (b c + a d\right )} b d x^{4} + 10 \, {\left (b c + a d\right )}^{2} x^{3}\right )} a c \]
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (84) = 168\).
Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.04 \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {1}{7} \, b^{3} d^{3} x^{7} + \frac {1}{2} \, b^{3} c d^{2} x^{6} + \frac {1}{2} \, a b^{2} d^{3} x^{6} + \frac {3}{5} \, b^{3} c^{2} d x^{5} + \frac {9}{5} \, a b^{2} c d^{2} x^{5} + \frac {3}{5} \, a^{2} b d^{3} x^{5} + \frac {1}{4} \, b^{3} c^{3} x^{4} + \frac {9}{4} \, a b^{2} c^{2} d x^{4} + \frac {9}{4} \, a^{2} b c d^{2} x^{4} + \frac {1}{4} \, a^{3} d^{3} x^{4} + a b^{2} c^{3} x^{3} + 3 \, a^{2} b c^{2} d x^{3} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{3} x^{2} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \]
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Time = 0.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65 \[ \int \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=x^4\,\left (\frac {a^3\,d^3}{4}+\frac {9\,a^2\,b\,c\,d^2}{4}+\frac {9\,a\,b^2\,c^2\,d}{4}+\frac {b^3\,c^3}{4}\right )+a^3\,c^3\,x+\frac {b^3\,d^3\,x^7}{7}+a\,c\,x^3\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )+\frac {3\,b\,d\,x^5\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{5}+\frac {3\,a^2\,c^2\,x^2\,\left (a\,d+b\,c\right )}{2}+\frac {b^2\,d^2\,x^6\,\left (a\,d+b\,c\right )}{2} \]
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