Integrand size = 23, antiderivative size = 69 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a A x+\frac {1}{2} a B x^2+\frac {1}{3} (A b+a C) x^3+\frac {1}{4} b B x^4+\frac {1}{5} (A c+b C) x^5+\frac {1}{6} B c x^6+\frac {1}{7} c C x^7 \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1671} \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{2} a B x^2+\frac {1}{5} x^5 (A c+b C)+\frac {1}{4} b B x^4+\frac {1}{6} B c x^6+\frac {1}{7} c C x^7 \]
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Rule 1671
Rubi steps \begin{align*} \text {integral}& = \int \left (a A+a B x+(A b+a C) x^2+b B x^3+(A c+b C) x^4+B c x^5+c C x^6\right ) \, dx \\ & = a A x+\frac {1}{2} a B x^2+\frac {1}{3} (A b+a C) x^3+\frac {1}{4} b B x^4+\frac {1}{5} (A c+b C) x^5+\frac {1}{6} B c x^6+\frac {1}{7} c C x^7 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a A x+\frac {1}{2} a B x^2+\frac {1}{3} (A b+a C) x^3+\frac {1}{4} b B x^4+\frac {1}{5} (A c+b C) x^5+\frac {1}{6} B c x^6+\frac {1}{7} c C x^7 \]
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Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84
method | result | size |
default | \(a A x +\frac {B a \,x^{2}}{2}+\frac {\left (A b +C a \right ) x^{3}}{3}+\frac {b B \,x^{4}}{4}+\frac {\left (A c +C b \right ) x^{5}}{5}+\frac {B c \,x^{6}}{6}+\frac {c C \,x^{7}}{7}\) | \(58\) |
norman | \(\frac {c C \,x^{7}}{7}+\frac {B c \,x^{6}}{6}+\left (\frac {A c}{5}+\frac {C b}{5}\right ) x^{5}+\frac {b B \,x^{4}}{4}+\left (\frac {A b}{3}+\frac {C a}{3}\right ) x^{3}+\frac {B a \,x^{2}}{2}+a A x\) | \(60\) |
gosper | \(\frac {1}{7} c C \,x^{7}+\frac {1}{6} B c \,x^{6}+\frac {1}{5} x^{5} A c +\frac {1}{5} x^{5} C b +\frac {1}{4} b B \,x^{4}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} C a +\frac {1}{2} B a \,x^{2}+a A x\) | \(62\) |
risch | \(\frac {1}{7} c C \,x^{7}+\frac {1}{6} B c \,x^{6}+\frac {1}{5} x^{5} A c +\frac {1}{5} x^{5} C b +\frac {1}{4} b B \,x^{4}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} C a +\frac {1}{2} B a \,x^{2}+a A x\) | \(62\) |
parallelrisch | \(\frac {1}{7} c C \,x^{7}+\frac {1}{6} B c \,x^{6}+\frac {1}{5} x^{5} A c +\frac {1}{5} x^{5} C b +\frac {1}{4} b B \,x^{4}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} C a +\frac {1}{2} B a \,x^{2}+a A x\) | \(62\) |
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Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, C c x^{7} + \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, B b x^{4} + \frac {1}{5} \, {\left (C b + A c\right )} x^{5} + \frac {1}{2} \, B a x^{2} + \frac {1}{3} \, {\left (C a + A b\right )} x^{3} + A a x \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=A a x + \frac {B a x^{2}}{2} + \frac {B b x^{4}}{4} + \frac {B c x^{6}}{6} + \frac {C c x^{7}}{7} + x^{5} \left (\frac {A c}{5} + \frac {C b}{5}\right ) + x^{3} \left (\frac {A b}{3} + \frac {C a}{3}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, C c x^{7} + \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, B b x^{4} + \frac {1}{5} \, {\left (C b + A c\right )} x^{5} + \frac {1}{2} \, B a x^{2} + \frac {1}{3} \, {\left (C a + A b\right )} x^{3} + A a x \]
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Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, C c x^{7} + \frac {1}{6} \, B c x^{6} + \frac {1}{5} \, C b x^{5} + \frac {1}{5} \, A c x^{5} + \frac {1}{4} \, B b x^{4} + \frac {1}{3} \, C a x^{3} + \frac {1}{3} \, A b x^{3} + \frac {1}{2} \, B a x^{2} + A a x \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {C\,c\,x^7}{7}+\frac {B\,c\,x^6}{6}+\left (\frac {A\,c}{5}+\frac {C\,b}{5}\right )\,x^5+\frac {B\,b\,x^4}{4}+\left (\frac {A\,b}{3}+\frac {C\,a}{3}\right )\,x^3+\frac {B\,a\,x^2}{2}+A\,a\,x \]
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