Integrand size = 18, antiderivative size = 46 \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=a A x+\frac {1}{2} A b x^2+\frac {1}{3} (A c+a C) x^3+\frac {1}{4} b C x^4+\frac {1}{5} c C x^5 \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1671} \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=\frac {1}{3} x^3 (a C+A c)+a A x+\frac {1}{2} A b x^2+\frac {1}{4} b C x^4+\frac {1}{5} c C x^5 \]
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Rule 1671
Rubi steps \begin{align*} \text {integral}& = \int \left (a A+A b x+(A c+a C) x^2+b C x^3+c C x^4\right ) \, dx \\ & = a A x+\frac {1}{2} A b x^2+\frac {1}{3} (A c+a C) x^3+\frac {1}{4} b C x^4+\frac {1}{5} c C x^5 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=a A x+\frac {1}{2} A b x^2+\frac {1}{3} (A c+a C) x^3+\frac {1}{4} b C x^4+\frac {1}{5} c C x^5 \]
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Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85
method | result | size |
default | \(a A x +\frac {A b \,x^{2}}{2}+\frac {\left (A c +C a \right ) x^{3}}{3}+\frac {b C \,x^{4}}{4}+\frac {c C \,x^{5}}{5}\) | \(39\) |
norman | \(\frac {c C \,x^{5}}{5}+\frac {b C \,x^{4}}{4}+\left (\frac {A c}{3}+\frac {C a}{3}\right ) x^{3}+\frac {A b \,x^{2}}{2}+a A x\) | \(40\) |
gosper | \(\frac {1}{5} c C \,x^{5}+\frac {1}{4} b C \,x^{4}+\frac {1}{3} A c \,x^{3}+\frac {1}{3} x^{3} C a +\frac {1}{2} A b \,x^{2}+a A x\) | \(41\) |
risch | \(\frac {1}{5} c C \,x^{5}+\frac {1}{4} b C \,x^{4}+\frac {1}{3} A c \,x^{3}+\frac {1}{3} x^{3} C a +\frac {1}{2} A b \,x^{2}+a A x\) | \(41\) |
parallelrisch | \(\frac {1}{5} c C \,x^{5}+\frac {1}{4} b C \,x^{4}+\frac {1}{3} A c \,x^{3}+\frac {1}{3} x^{3} C a +\frac {1}{2} A b \,x^{2}+a A x\) | \(41\) |
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Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=\frac {1}{5} \, C c x^{5} + \frac {1}{4} \, C b x^{4} + \frac {1}{2} \, A b x^{2} + \frac {1}{3} \, {\left (C a + A c\right )} x^{3} + A a x \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=A a x + \frac {A b x^{2}}{2} + \frac {C b x^{4}}{4} + \frac {C c x^{5}}{5} + x^{3} \left (\frac {A c}{3} + \frac {C a}{3}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=\frac {1}{5} \, C c x^{5} + \frac {1}{4} \, C b x^{4} + \frac {1}{2} \, A b x^{2} + \frac {1}{3} \, {\left (C a + A c\right )} x^{3} + A a x \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=\frac {1}{5} \, C c x^{5} + \frac {1}{4} \, C b x^{4} + \frac {1}{3} \, C a x^{3} + \frac {1}{3} \, A c x^{3} + \frac {1}{2} \, A b x^{2} + A a x \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \left (a+b x+c x^2\right ) \left (A+C x^2\right ) \, dx=\frac {C\,c\,x^5}{5}+\frac {C\,b\,x^4}{4}+\left (\frac {A\,c}{3}+\frac {C\,a}{3}\right )\,x^3+\frac {A\,b\,x^2}{2}+A\,a\,x \]
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