Integrand size = 17, antiderivative size = 44 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=a \text {b1} x+\frac {1}{2} (2 b \text {b1}+a \text {c1}) x^2+\frac {1}{3} (\text {b1} c+2 b \text {c1}) x^3+\frac {1}{4} c \text {c1} x^4 \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {645} \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=\frac {1}{2} x^2 (a \text {c1}+2 b \text {b1})+a \text {b1} x+\frac {1}{3} x^3 (2 b \text {c1}+\text {b1} c)+\frac {1}{4} c \text {c1} x^4 \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (a \text {b1}+(2 b \text {b1}+a \text {c1}) x+(\text {b1} c+2 b \text {c1}) x^2+c \text {c1} x^3\right ) \, dx \\ & = a \text {b1} x+\frac {1}{2} (2 b \text {b1}+a \text {c1}) x^2+\frac {1}{3} (\text {b1} c+2 b \text {c1}) x^3+\frac {1}{4} c \text {c1} x^4 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=\frac {1}{12} x (6 a (2 \text {b1}+\text {c1} x)+x (4 b (3 \text {b1}+2 \text {c1} x)+c x (4 \text {b1}+3 \text {c1} x))) \]
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Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {c \operatorname {c1} \,x^{4}}{4}+\left (\frac {2 b \operatorname {c1}}{3}+\frac {\operatorname {b1} c}{3}\right ) x^{3}+\left (\frac {a \operatorname {c1}}{2}+b \operatorname {b1} \right ) x^{2}+a \operatorname {b1} x\) | \(38\) |
default | \(a \operatorname {b1} x +\frac {\left (a \operatorname {c1} +2 b \operatorname {b1} \right ) x^{2}}{2}+\frac {\left (2 b \operatorname {c1} +\operatorname {b1} c \right ) x^{3}}{3}+\frac {c \operatorname {c1} \,x^{4}}{4}\) | \(39\) |
gosper | \(\frac {1}{4} c \operatorname {c1} \,x^{4}+\frac {2}{3} x^{3} b \operatorname {c1} +\frac {1}{3} x^{3} \operatorname {b1} c +\frac {1}{2} x^{2} a \operatorname {c1} +x^{2} b \operatorname {b1} +a \operatorname {b1} x\) | \(40\) |
risch | \(\frac {1}{4} c \operatorname {c1} \,x^{4}+\frac {2}{3} x^{3} b \operatorname {c1} +\frac {1}{3} x^{3} \operatorname {b1} c +\frac {1}{2} x^{2} a \operatorname {c1} +x^{2} b \operatorname {b1} +a \operatorname {b1} x\) | \(40\) |
parallelrisch | \(\frac {1}{4} c \operatorname {c1} \,x^{4}+\frac {2}{3} x^{3} b \operatorname {c1} +\frac {1}{3} x^{3} \operatorname {b1} c +\frac {1}{2} x^{2} a \operatorname {c1} +x^{2} b \operatorname {b1} +a \operatorname {b1} x\) | \(40\) |
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Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=\frac {1}{4} \, c c_{1} x^{4} + \frac {1}{3} \, {\left (b_{1} c + 2 \, b c_{1}\right )} x^{3} + a b_{1} x + \frac {1}{2} \, {\left (2 \, b b_{1} + a c_{1}\right )} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=a b_{1} x + \frac {c c_{1} x^{4}}{4} + x^{3} \cdot \left (\frac {2 b c_{1}}{3} + \frac {b_{1} c}{3}\right ) + x^{2} \left (\frac {a c_{1}}{2} + b b_{1}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=\frac {1}{4} \, c c_{1} x^{4} + \frac {1}{3} \, {\left (b_{1} c + 2 \, b c_{1}\right )} x^{3} + a b_{1} x + \frac {1}{2} \, {\left (2 \, b b_{1} + a c_{1}\right )} x^{2} \]
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Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=\frac {1}{4} \, c c_{1} x^{4} + \frac {1}{3} \, b_{1} c x^{3} + \frac {2}{3} \, b c_{1} x^{3} + b b_{1} x^{2} + \frac {1}{2} \, a c_{1} x^{2} + a b_{1} x \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx=\frac {c\,c_{1}\,x^4}{4}+\left (\frac {2\,b\,c_{1}}{3}+\frac {b_{1}\,c}{3}\right )\,x^3+\left (\frac {a\,c_{1}}{2}+b\,b_{1}\right )\,x^2+a\,b_{1}\,x \]
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