Integrand size = 15, antiderivative size = 65 \[ \int (a+b x)^2 (c+d x)^3 \, dx=\frac {(b c-a d)^2 (c+d x)^4}{4 d^3}-\frac {2 b (b c-a d) (c+d x)^5}{5 d^3}+\frac {b^2 (c+d x)^6}{6 d^3} \]
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Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^2 (c+d x)^3 \, dx=-\frac {2 b (c+d x)^5 (b c-a d)}{5 d^3}+\frac {(c+d x)^4 (b c-a d)^2}{4 d^3}+\frac {b^2 (c+d x)^6}{6 d^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^2 (c+d x)^3}{d^2}-\frac {2 b (b c-a d) (c+d x)^4}{d^2}+\frac {b^2 (c+d x)^5}{d^2}\right ) \, dx \\ & = \frac {(b c-a d)^2 (c+d x)^4}{4 d^3}-\frac {2 b (b c-a d) (c+d x)^5}{5 d^3}+\frac {b^2 (c+d x)^6}{6 d^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.88 \[ \int (a+b x)^2 (c+d x)^3 \, dx=a^2 c^3 x+\frac {1}{2} a c^2 (2 b c+3 a d) x^2+\frac {1}{3} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^3+\frac {1}{4} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^4+\frac {1}{5} b d^2 (3 b c+2 a d) x^5+\frac {1}{6} b^2 d^3 x^6 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(120\) vs. \(2(59)=118\).
Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.86
method | result | size |
norman | \(\frac {b^{2} d^{3} x^{6}}{6}+\left (\frac {2}{5} a b \,d^{3}+\frac {3}{5} b^{2} c \,d^{2}\right ) x^{5}+\left (\frac {1}{4} a^{2} d^{3}+\frac {3}{2} a b c \,d^{2}+\frac {3}{4} b^{2} c^{2} d \right ) x^{4}+\left (a^{2} c \,d^{2}+2 a b \,c^{2} d +\frac {1}{3} b^{2} c^{3}\right ) x^{3}+\left (\frac {3}{2} a^{2} c^{2} d +a \,c^{3} b \right ) x^{2}+a^{2} c^{3} x\) | \(121\) |
default | \(\frac {b^{2} d^{3} x^{6}}{6}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{4}}{4}+\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{2} c^{2} d +2 a \,c^{3} b \right ) x^{2}}{2}+a^{2} c^{3} x\) | \(125\) |
gosper | \(\frac {1}{6} b^{2} d^{3} x^{6}+\frac {2}{5} x^{5} a b \,d^{3}+\frac {3}{5} x^{5} b^{2} c \,d^{2}+\frac {1}{4} x^{4} a^{2} d^{3}+\frac {3}{2} x^{4} a b c \,d^{2}+\frac {3}{4} x^{4} b^{2} c^{2} d +x^{3} a^{2} c \,d^{2}+2 x^{3} a b \,c^{2} d +\frac {1}{3} x^{3} b^{2} c^{3}+\frac {3}{2} x^{2} a^{2} c^{2} d +x^{2} a \,c^{3} b +a^{2} c^{3} x\) | \(131\) |
risch | \(\frac {1}{6} b^{2} d^{3} x^{6}+\frac {2}{5} x^{5} a b \,d^{3}+\frac {3}{5} x^{5} b^{2} c \,d^{2}+\frac {1}{4} x^{4} a^{2} d^{3}+\frac {3}{2} x^{4} a b c \,d^{2}+\frac {3}{4} x^{4} b^{2} c^{2} d +x^{3} a^{2} c \,d^{2}+2 x^{3} a b \,c^{2} d +\frac {1}{3} x^{3} b^{2} c^{3}+\frac {3}{2} x^{2} a^{2} c^{2} d +x^{2} a \,c^{3} b +a^{2} c^{3} x\) | \(131\) |
parallelrisch | \(\frac {1}{6} b^{2} d^{3} x^{6}+\frac {2}{5} x^{5} a b \,d^{3}+\frac {3}{5} x^{5} b^{2} c \,d^{2}+\frac {1}{4} x^{4} a^{2} d^{3}+\frac {3}{2} x^{4} a b c \,d^{2}+\frac {3}{4} x^{4} b^{2} c^{2} d +x^{3} a^{2} c \,d^{2}+2 x^{3} a b \,c^{2} d +\frac {1}{3} x^{3} b^{2} c^{3}+\frac {3}{2} x^{2} a^{2} c^{2} d +x^{2} a \,c^{3} b +a^{2} c^{3} x\) | \(131\) |
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (59) = 118\).
Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.91 \[ \int (a+b x)^2 (c+d x)^3 \, dx=\frac {1}{6} \, b^{2} d^{3} x^{6} + a^{2} c^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (56) = 112\).
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.05 \[ \int (a+b x)^2 (c+d x)^3 \, dx=a^{2} c^{3} x + \frac {b^{2} d^{3} x^{6}}{6} + x^{5} \cdot \left (\frac {2 a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + \frac {3 a b c d^{2}}{2} + \frac {3 b^{2} c^{2} d}{4}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + \frac {b^{2} c^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2} d}{2} + a b c^{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (59) = 118\).
Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.91 \[ \int (a+b x)^2 (c+d x)^3 \, dx=\frac {1}{6} \, b^{2} d^{3} x^{6} + a^{2} c^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (59) = 118\).
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.00 \[ \int (a+b x)^2 (c+d x)^3 \, dx=\frac {1}{6} \, b^{2} d^{3} x^{6} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {2}{5} \, a b d^{3} x^{5} + \frac {3}{4} \, b^{2} c^{2} d x^{4} + \frac {3}{2} \, a b c d^{2} x^{4} + \frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{2} d x^{3} + a^{2} c d^{2} x^{3} + a b c^{3} x^{2} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x \]
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Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.77 \[ \int (a+b x)^2 (c+d x)^3 \, dx=x^3\,\left (a^2\,c\,d^2+2\,a\,b\,c^2\,d+\frac {b^2\,c^3}{3}\right )+x^4\,\left (\frac {a^2\,d^3}{4}+\frac {3\,a\,b\,c\,d^2}{2}+\frac {3\,b^2\,c^2\,d}{4}\right )+a^2\,c^3\,x+\frac {b^2\,d^3\,x^6}{6}+\frac {a\,c^2\,x^2\,\left (3\,a\,d+2\,b\,c\right )}{2}+\frac {b\,d^2\,x^5\,\left (2\,a\,d+3\,b\,c\right )}{5} \]
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