Integrand size = 23, antiderivative size = 310 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 A d x+\frac {1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac {1}{3} a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^4+\frac {1}{5} \left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^5+\frac {1}{6} \left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^6+\frac {1}{7} c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \]
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Time = 0.41 (sec) , antiderivative size = 310, 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 = {785} \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 A d x+\frac {1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac {1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac {1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac {1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac {1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac {1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9 \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A d+a^2 (3 A b d+a B d+a A e) x+a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^2+\left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^3+\left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^4+\left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^5+c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^6+c^2 (B c d+3 b B e+A c e) x^7+B c^3 e x^8\right ) \, dx \\ & = a^3 A d x+\frac {1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac {1}{3} a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^4+\frac {1}{5} \left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^5+\frac {1}{6} \left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^6+\frac {1}{7} c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 A d x+\frac {1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac {1}{3} a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^4+\frac {1}{5} \left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^5+\frac {1}{6} \left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^6+\frac {1}{7} c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \]
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Time = 0.34 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.09
method | result | size |
norman | \(\frac {B \,c^{3} e \,x^{9}}{9}+\left (\frac {1}{8} A \,c^{3} e +\frac {3}{8} B e b \,c^{2}+\frac {1}{8} B \,c^{3} d \right ) x^{8}+\left (\frac {3}{7} A b \,c^{2} e +\frac {1}{7} A d \,c^{3}+\frac {3}{7} B a \,c^{2} e +\frac {3}{7} B e \,b^{2} c +\frac {3}{7} B b \,c^{2} d \right ) x^{7}+\left (\frac {1}{2} A a \,c^{2} e +\frac {1}{2} A \,b^{2} c e +\frac {1}{2} A d b \,c^{2}+B a b c e +\frac {1}{2} B a \,c^{2} d +\frac {1}{6} b^{3} B e +\frac {1}{2} B \,b^{2} c d \right ) x^{6}+\left (\frac {6}{5} A a b c e +\frac {3}{5} A a \,c^{2} d +\frac {1}{5} A \,b^{3} e +\frac {3}{5} A d \,b^{2} c +\frac {3}{5} B \,a^{2} c e +\frac {3}{5} B e \,b^{2} a +\frac {6}{5} B a b c d +\frac {1}{5} B \,b^{3} d \right ) x^{5}+\left (\frac {3}{4} A \,a^{2} c e +\frac {3}{4} A a \,b^{2} e +\frac {3}{2} A a b c d +\frac {1}{4} A d \,b^{3}+\frac {3}{4} B e b \,a^{2}+\frac {3}{4} B \,a^{2} c d +\frac {3}{4} B a \,b^{2} d \right ) x^{4}+\left (A \,a^{2} b e +A \,a^{2} c d +d A \,b^{2} a +\frac {1}{3} B e \,a^{3}+B \,a^{2} b d \right ) x^{3}+\left (\frac {1}{2} A \,a^{3} e +\frac {3}{2} d A b \,a^{2}+\frac {1}{2} B \,a^{3} d \right ) x^{2}+a^{3} A d x\) | \(339\) |
default | \(\frac {B \,c^{3} e \,x^{9}}{9}+\frac {\left (\left (A e +B d \right ) c^{3}+3 B e b \,c^{2}\right ) x^{8}}{8}+\frac {\left (A d \,c^{3}+3 \left (A e +B d \right ) b \,c^{2}+B e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 A d b \,c^{2}+\left (A e +B d \right ) \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d A \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+\left (A e +B d \right ) \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B e \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )\right ) x^{5}}{5}+\frac {\left (d A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+\left (A e +B d \right ) \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 B e b \,a^{2}\right ) x^{4}}{4}+\frac {\left (d A \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 \left (A e +B d \right ) b \,a^{2}+B e \,a^{3}\right ) x^{3}}{3}+\frac {\left (3 d A b \,a^{2}+\left (A e +B d \right ) a^{3}\right ) x^{2}}{2}+a^{3} A d x\) | \(375\) |
gosper | \(\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{4} A \,b^{3} d \,x^{4}+\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {3}{8} x^{8} B e b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2} e +\frac {3}{7} x^{7} B e \,b^{2} c +a^{3} A d x +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{2} x^{2} B \,a^{3} d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {3}{2} x^{4} A a b c d +\frac {6}{5} x^{5} B a b c d +x^{6} B a b c e +\frac {6}{5} x^{5} A a b c e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {3}{5} x^{5} B e \,b^{2} a +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {3}{4} x^{4} B e b \,a^{2}+\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} d A \,b^{2} a +x^{3} B \,a^{2} b d +\frac {3}{2} x^{2} d A b \,a^{2}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} d A a \,c^{2}+\frac {3}{5} x^{5} B e c \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} d A c \,a^{2}+\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}\) | \(418\) |
risch | \(\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{4} A \,b^{3} d \,x^{4}+\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {3}{8} x^{8} B e b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2} e +\frac {3}{7} x^{7} B e \,b^{2} c +a^{3} A d x +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{2} x^{2} B \,a^{3} d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {3}{2} x^{4} A a b c d +\frac {6}{5} x^{5} B a b c d +x^{6} B a b c e +\frac {6}{5} x^{5} A a b c e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {3}{5} x^{5} B e \,b^{2} a +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {3}{4} x^{4} B e b \,a^{2}+\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} d A \,b^{2} a +x^{3} B \,a^{2} b d +\frac {3}{2} x^{2} d A b \,a^{2}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} d A a \,c^{2}+\frac {3}{5} x^{5} B e c \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} d A c \,a^{2}+\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}\) | \(418\) |
parallelrisch | \(\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{4} A \,b^{3} d \,x^{4}+\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {3}{8} x^{8} B e b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2} e +\frac {3}{7} x^{7} B e \,b^{2} c +a^{3} A d x +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{2} x^{2} B \,a^{3} d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {3}{2} x^{4} A a b c d +\frac {6}{5} x^{5} B a b c d +x^{6} B a b c e +\frac {6}{5} x^{5} A a b c e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {3}{5} x^{5} B e \,b^{2} a +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {3}{4} x^{4} B e b \,a^{2}+\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} d A \,b^{2} a +x^{3} B \,a^{2} b d +\frac {3}{2} x^{2} d A b \,a^{2}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} d A a \,c^{2}+\frac {3}{5} x^{5} B e c \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} d A c \,a^{2}+\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}\) | \(418\) |
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Time = 0.31 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e\right )} x^{6} + A a^{3} d x + \frac {1}{5} \, {\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]
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Time = 0.04 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.40 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=A a^{3} d x + \frac {B c^{3} e x^{9}}{9} + x^{8} \left (\frac {A c^{3} e}{8} + \frac {3 B b c^{2} e}{8} + \frac {B c^{3} d}{8}\right ) + x^{7} \cdot \left (\frac {3 A b c^{2} e}{7} + \frac {A c^{3} d}{7} + \frac {3 B a c^{2} e}{7} + \frac {3 B b^{2} c e}{7} + \frac {3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac {A a c^{2} e}{2} + \frac {A b^{2} c e}{2} + \frac {A b c^{2} d}{2} + B a b c e + \frac {B a c^{2} d}{2} + \frac {B b^{3} e}{6} + \frac {B b^{2} c d}{2}\right ) + x^{5} \cdot \left (\frac {6 A a b c e}{5} + \frac {3 A a c^{2} d}{5} + \frac {A b^{3} e}{5} + \frac {3 A b^{2} c d}{5} + \frac {3 B a^{2} c e}{5} + \frac {3 B a b^{2} e}{5} + \frac {6 B a b c d}{5} + \frac {B b^{3} d}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} c e}{4} + \frac {3 A a b^{2} e}{4} + \frac {3 A a b c d}{2} + \frac {A b^{3} d}{4} + \frac {3 B a^{2} b e}{4} + \frac {3 B a^{2} c d}{4} + \frac {3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a^{2} c d + A a b^{2} d + \frac {B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac {A a^{3} e}{2} + \frac {3 A a^{2} b d}{2} + \frac {B a^{3} d}{2}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e\right )} x^{6} + A a^{3} d x + \frac {1}{5} \, {\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]
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Time = 0.27 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.35 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, B c^{3} d x^{8} + \frac {3}{8} \, B b c^{2} e x^{8} + \frac {1}{8} \, A c^{3} e x^{8} + \frac {3}{7} \, B b c^{2} d x^{7} + \frac {1}{7} \, A c^{3} d x^{7} + \frac {3}{7} \, B b^{2} c e x^{7} + \frac {3}{7} \, B a c^{2} e x^{7} + \frac {3}{7} \, A b c^{2} e x^{7} + \frac {1}{2} \, B b^{2} c d x^{6} + \frac {1}{2} \, B a c^{2} d x^{6} + \frac {1}{2} \, A b c^{2} d x^{6} + \frac {1}{6} \, B b^{3} e x^{6} + B a b c e x^{6} + \frac {1}{2} \, A b^{2} c e x^{6} + \frac {1}{2} \, A a c^{2} e x^{6} + \frac {1}{5} \, B b^{3} d x^{5} + \frac {6}{5} \, B a b c d x^{5} + \frac {3}{5} \, A b^{2} c d x^{5} + \frac {3}{5} \, A a c^{2} d x^{5} + \frac {3}{5} \, B a b^{2} e x^{5} + \frac {1}{5} \, A b^{3} e x^{5} + \frac {3}{5} \, B a^{2} c e x^{5} + \frac {6}{5} \, A a b c e x^{5} + \frac {3}{4} \, B a b^{2} d x^{4} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {3}{4} \, B a^{2} c d x^{4} + \frac {3}{2} \, A a b c d x^{4} + \frac {3}{4} \, B a^{2} b e x^{4} + \frac {3}{4} \, A a b^{2} e x^{4} + \frac {3}{4} \, A a^{2} c e x^{4} + B a^{2} b d x^{3} + A a b^{2} d x^{3} + A a^{2} c d x^{3} + \frac {1}{3} \, B a^{3} e x^{3} + A a^{2} b e x^{3} + \frac {1}{2} \, B a^{3} d x^{2} + \frac {3}{2} \, A a^{2} b d x^{2} + \frac {1}{2} \, A a^{3} e x^{2} + A a^{3} d x \]
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Time = 0.13 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.09 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=x^7\,\left (\frac {A\,c^3\,d}{7}+\frac {3\,A\,b\,c^2\,e}{7}+\frac {3\,B\,a\,c^2\,e}{7}+\frac {3\,B\,b\,c^2\,d}{7}+\frac {3\,B\,b^2\,c\,e}{7}\right )+x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,A\,a\,c^2\,d}{5}+\frac {3\,A\,b^2\,c\,d}{5}+\frac {3\,B\,a\,b^2\,e}{5}+\frac {3\,B\,a^2\,c\,e}{5}+\frac {6\,A\,a\,b\,c\,e}{5}+\frac {6\,B\,a\,b\,c\,d}{5}\right )+x^2\,\left (\frac {A\,a^3\,e}{2}+\frac {B\,a^3\,d}{2}+\frac {3\,A\,a^2\,b\,d}{2}\right )+x^8\,\left (\frac {A\,c^3\,e}{8}+\frac {B\,c^3\,d}{8}+\frac {3\,B\,b\,c^2\,e}{8}\right )+x^4\,\left (\frac {A\,b^3\,d}{4}+\frac {3\,A\,a\,b^2\,e}{4}+\frac {3\,B\,a\,b^2\,d}{4}+\frac {3\,A\,a^2\,c\,e}{4}+\frac {3\,B\,a^2\,b\,e}{4}+\frac {3\,B\,a^2\,c\,d}{4}+\frac {3\,A\,a\,b\,c\,d}{2}\right )+x^6\,\left (\frac {B\,b^3\,e}{6}+\frac {A\,a\,c^2\,e}{2}+\frac {A\,b\,c^2\,d}{2}+\frac {B\,a\,c^2\,d}{2}+\frac {A\,b^2\,c\,e}{2}+\frac {B\,b^2\,c\,d}{2}+B\,a\,b\,c\,e\right )+x^3\,\left (\frac {B\,a^3\,e}{3}+A\,a\,b^2\,d+A\,a^2\,b\,e+A\,a^2\,c\,d+B\,a^2\,b\,d\right )+A\,a^3\,d\,x+\frac {B\,c^3\,e\,x^9}{9} \]
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