Integrand size = 19, antiderivative size = 96 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=a^2 \text {b1} x+\frac {1}{2} a (4 b \text {b1}+a \text {c1}) x^2+\frac {2}{3} \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+\frac {1}{2} \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^4+\frac {1}{5} c (\text {b1} c+4 b \text {c1}) x^5+\frac {1}{6} c^2 \text {c1} x^6 \]
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Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {645} \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=a^2 \text {b1} x+\frac {1}{2} x^4 \left (a c \text {c1}+2 b^2 \text {c1}+2 b \text {b1} c\right )+\frac {2}{3} x^3 \left (2 a b \text {c1}+a \text {b1} c+2 b^2 \text {b1}\right )+\frac {1}{2} a x^2 (a \text {c1}+4 b \text {b1})+\frac {1}{5} c x^5 (4 b \text {c1}+\text {b1} c)+\frac {1}{6} c^2 \text {c1} x^6 \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \text {b1}+a (4 b \text {b1}+a \text {c1}) x+2 \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^2+2 \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^3+c (\text {b1} c+4 b \text {c1}) x^4+c^2 \text {c1} x^5\right ) \, dx \\ & = a^2 \text {b1} x+\frac {1}{2} a (4 b \text {b1}+a \text {c1}) x^2+\frac {2}{3} \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+\frac {1}{2} \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^4+\frac {1}{5} c (\text {b1} c+4 b \text {c1}) x^5+\frac {1}{6} c^2 \text {c1} x^6 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=\frac {1}{30} x \left (15 a^2 (2 \text {b1}+\text {c1} x)+5 a x (4 b (3 \text {b1}+2 \text {c1} x)+c x (4 \text {b1}+3 \text {c1} x))+x^2 \left (10 b^2 (4 \text {b1}+3 \text {c1} x)+6 b c x (5 \text {b1}+4 \text {c1} x)+c^2 x^2 (6 \text {b1}+5 \text {c1} x)\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93
method | result | size |
norman | \(\frac {c^{2} \operatorname {c1} \,x^{6}}{6}+\left (\frac {4}{5} \operatorname {c1} b c +\frac {1}{5} \operatorname {b1} \,c^{2}\right ) x^{5}+\left (\frac {1}{2} a c \operatorname {c1} +b^{2} \operatorname {c1} +b \operatorname {b1} c \right ) x^{4}+\left (\frac {4}{3} a b \operatorname {c1} +\frac {2}{3} a \operatorname {b1} c +\frac {4}{3} b^{2} \operatorname {b1} \right ) x^{3}+\left (\frac {1}{2} \operatorname {c1} \,a^{2}+2 \operatorname {b1} a b \right ) x^{2}+a^{2} \operatorname {b1} x\) | \(89\) |
default | \(\frac {c^{2} \operatorname {c1} \,x^{6}}{6}+\frac {\left (4 \operatorname {c1} b c +\operatorname {b1} \,c^{2}\right ) x^{5}}{5}+\frac {\left (4 b \operatorname {b1} c +\operatorname {c1} \left (2 a c +4 b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\operatorname {b1} \left (2 a c +4 b^{2}\right )+4 a b \operatorname {c1} \right ) x^{3}}{3}+\frac {\left (\operatorname {c1} \,a^{2}+4 \operatorname {b1} a b \right ) x^{2}}{2}+a^{2} \operatorname {b1} x\) | \(95\) |
gosper | \(\frac {1}{6} c^{2} \operatorname {c1} \,x^{6}+\frac {4}{5} x^{5} \operatorname {c1} b c +\frac {1}{5} x^{5} \operatorname {b1} \,c^{2}+\frac {1}{2} x^{4} a c \operatorname {c1} +x^{4} b^{2} \operatorname {c1} +x^{4} b \operatorname {b1} c +\frac {4}{3} x^{3} a b \operatorname {c1} +\frac {2}{3} x^{3} a \operatorname {b1} c +\frac {4}{3} x^{3} b^{2} \operatorname {b1} +\frac {1}{2} x^{2} \operatorname {c1} \,a^{2}+2 x^{2} \operatorname {b1} a b +a^{2} \operatorname {b1} x\) | \(99\) |
risch | \(\frac {1}{6} c^{2} \operatorname {c1} \,x^{6}+\frac {4}{5} x^{5} \operatorname {c1} b c +\frac {1}{5} x^{5} \operatorname {b1} \,c^{2}+\frac {1}{2} x^{4} a c \operatorname {c1} +x^{4} b^{2} \operatorname {c1} +x^{4} b \operatorname {b1} c +\frac {4}{3} x^{3} a b \operatorname {c1} +\frac {2}{3} x^{3} a \operatorname {b1} c +\frac {4}{3} x^{3} b^{2} \operatorname {b1} +\frac {1}{2} x^{2} \operatorname {c1} \,a^{2}+2 x^{2} \operatorname {b1} a b +a^{2} \operatorname {b1} x\) | \(99\) |
parallelrisch | \(\frac {1}{6} c^{2} \operatorname {c1} \,x^{6}+\frac {4}{5} x^{5} \operatorname {c1} b c +\frac {1}{5} x^{5} \operatorname {b1} \,c^{2}+\frac {1}{2} x^{4} a c \operatorname {c1} +x^{4} b^{2} \operatorname {c1} +x^{4} b \operatorname {b1} c +\frac {4}{3} x^{3} a b \operatorname {c1} +\frac {2}{3} x^{3} a \operatorname {b1} c +\frac {4}{3} x^{3} b^{2} \operatorname {b1} +\frac {1}{2} x^{2} \operatorname {c1} \,a^{2}+2 x^{2} \operatorname {b1} a b +a^{2} \operatorname {b1} x\) | \(99\) |
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Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, {\left (b_{1} c^{2} + 4 \, b c c_{1}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, b b_{1} c + {\left (2 \, b^{2} + a c\right )} c_{1}\right )} x^{4} + a^{2} b_{1} x + \frac {2}{3} \, {\left (2 \, b^{2} b_{1} + a b_{1} c + 2 \, a b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b b_{1} + a^{2} c_{1}\right )} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=a^{2} b_{1} x + \frac {c^{2} c_{1} x^{6}}{6} + x^{5} \cdot \left (\frac {4 b c c_{1}}{5} + \frac {b_{1} c^{2}}{5}\right ) + x^{4} \left (\frac {a c c_{1}}{2} + b^{2} c_{1} + b b_{1} c\right ) + x^{3} \cdot \left (\frac {4 a b c_{1}}{3} + \frac {2 a b_{1} c}{3} + \frac {4 b^{2} b_{1}}{3}\right ) + x^{2} \left (\frac {a^{2} c_{1}}{2} + 2 a b b_{1}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, {\left (b_{1} c^{2} + 4 \, b c c_{1}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, b b_{1} c + {\left (2 \, b^{2} + a c\right )} c_{1}\right )} x^{4} + a^{2} b_{1} x + \frac {2}{3} \, {\left (2 \, b^{2} b_{1} + a b_{1} c + 2 \, a b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b b_{1} + a^{2} c_{1}\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.02 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, b_{1} c^{2} x^{5} + \frac {4}{5} \, b c c_{1} x^{5} + b b_{1} c x^{4} + b^{2} c_{1} x^{4} + \frac {1}{2} \, a c c_{1} x^{4} + \frac {4}{3} \, b^{2} b_{1} x^{3} + \frac {2}{3} \, a b_{1} c x^{3} + \frac {4}{3} \, a b c_{1} x^{3} + 2 \, a b b_{1} x^{2} + \frac {1}{2} \, a^{2} c_{1} x^{2} + a^{2} b_{1} x \]
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Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx=x^3\,\left (\frac {4\,b_{1}\,b^2}{3}+\frac {4\,a\,c_{1}\,b}{3}+\frac {2\,a\,b_{1}\,c}{3}\right )+x^4\,\left (c_{1}\,b^2+b_{1}\,c\,b+\frac {a\,c\,c_{1}}{2}\right )+x^2\,\left (\frac {c_{1}\,a^2}{2}+2\,b\,b_{1}\,a\right )+x^5\,\left (\frac {b_{1}\,c^2}{5}+\frac {4\,b\,c_{1}\,c}{5}\right )+\frac {c^2\,c_{1}\,x^6}{6}+a^2\,b_{1}\,x \]
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