Integrand size = 23, antiderivative size = 86 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=a A d x+\frac {1}{2} a (B d+A e) x^2+\frac {1}{3} (A c d+a C d+a B e) x^3+\frac {1}{4} (B c d+A c e+a C e) x^4+\frac {1}{5} c (C d+B e) x^5+\frac {1}{6} c C e x^6 \]
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Time = 0.07 (sec) , antiderivative size = 86, 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 = {1642} \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{4} x^4 (a C e+A c e+B c d)+\frac {1}{3} x^3 (a B e+a C d+A c d)+\frac {1}{2} a x^2 (A e+B d)+a A d x+\frac {1}{5} c x^5 (B e+C d)+\frac {1}{6} c C e x^6 \]
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Rule 1642
Rubi steps \begin{align*} \text {integral}& = \int \left (a A d+a (B d+A e) x+(A c d+a C d+a B e) x^2+(B c d+A c e+a C e) x^3+c (C d+B e) x^4+c C e x^5\right ) \, dx \\ & = a A d x+\frac {1}{2} a (B d+A e) x^2+\frac {1}{3} (A c d+a C d+a B e) x^3+\frac {1}{4} (B c d+A c e+a C e) x^4+\frac {1}{5} c (C d+B e) x^5+\frac {1}{6} c C e x^6 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=a A d x+\frac {1}{2} a (B d+A e) x^2+\frac {1}{3} (A c d+a C d+a B e) x^3+\frac {1}{4} (B c d+A c e+a C e) x^4+\frac {1}{5} c (C d+B e) x^5+\frac {1}{6} c C e x^6 \]
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Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {c C e \,x^{6}}{6}+\frac {\left (B c e +c d C \right ) x^{5}}{5}+\frac {\left (A c e +B c d +C a e \right ) x^{4}}{4}+\frac {\left (A c d +B a e +C a d \right ) x^{3}}{3}+\frac {\left (a A e +B a d \right ) x^{2}}{2}+a A d x\) | \(79\) |
norman | \(\frac {c C e \,x^{6}}{6}+\left (\frac {1}{5} B c e +\frac {1}{5} c d C \right ) x^{5}+\left (\frac {1}{4} A c e +\frac {1}{4} B c d +\frac {1}{4} C a e \right ) x^{4}+\left (\frac {1}{3} A c d +\frac {1}{3} B a e +\frac {1}{3} C a d \right ) x^{3}+\left (\frac {1}{2} a A e +\frac {1}{2} B a d \right ) x^{2}+a A d x\) | \(85\) |
gosper | \(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} B c e \,x^{5}+\frac {1}{5} x^{5} c d C +\frac {1}{4} x^{4} A c e +\frac {1}{4} x^{4} B c d +\frac {1}{4} x^{4} C a e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} a A e +\frac {1}{2} x^{2} B a d +a A d x\) | \(95\) |
risch | \(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} B c e \,x^{5}+\frac {1}{5} x^{5} c d C +\frac {1}{4} x^{4} A c e +\frac {1}{4} x^{4} B c d +\frac {1}{4} x^{4} C a e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} a A e +\frac {1}{2} x^{2} B a d +a A d x\) | \(95\) |
parallelrisch | \(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} B c e \,x^{5}+\frac {1}{5} x^{5} c d C +\frac {1}{4} x^{4} A c e +\frac {1}{4} x^{4} B c d +\frac {1}{4} x^{4} C a e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} a A e +\frac {1}{2} x^{2} B a d +a A d x\) | \(95\) |
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Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, {\left (C c d + B c e\right )} x^{5} + \frac {1}{4} \, {\left (B c d + {\left (C a + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left (B a e + {\left (C a + A c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (B a d + A a e\right )} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=A a d x + \frac {C c e x^{6}}{6} + x^{5} \left (\frac {B c e}{5} + \frac {C c d}{5}\right ) + x^{4} \left (\frac {A c e}{4} + \frac {B c d}{4} + \frac {C a e}{4}\right ) + x^{3} \left (\frac {A c d}{3} + \frac {B a e}{3} + \frac {C a d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {B a d}{2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, {\left (C c d + B c e\right )} x^{5} + \frac {1}{4} \, {\left (B c d + {\left (C a + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left (B a e + {\left (C a + A c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (B a d + A a e\right )} x^{2} \]
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Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, C c d x^{5} + \frac {1}{5} \, B c e x^{5} + \frac {1}{4} \, B c d x^{4} + \frac {1}{4} \, C a e x^{4} + \frac {1}{4} \, A c e x^{4} + \frac {1}{3} \, C a d x^{3} + \frac {1}{3} \, A c d x^{3} + \frac {1}{3} \, B a e x^{3} + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A a e x^{2} + A a d x \]
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Time = 12.59 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {C\,c\,e\,x^6}{6}+\frac {c\,\left (B\,e+C\,d\right )\,x^5}{5}+\left (\frac {A\,c\,e}{4}+\frac {B\,c\,d}{4}+\frac {C\,a\,e}{4}\right )\,x^4+\left (\frac {A\,c\,d}{3}+\frac {B\,a\,e}{3}+\frac {C\,a\,d}{3}\right )\,x^3+\frac {a\,\left (A\,e+B\,d\right )\,x^2}{2}+A\,a\,d\,x \]
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