Integrand size = 22, antiderivative size = 139 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d x^4+\frac {1}{5} b^2 (b B d+3 A c d+A b e) x^5+\frac {1}{6} b \left (3 A c^2 d+b^2 B e+3 b c (B d+A e)\right ) x^6+\frac {1}{7} c \left (A c^2 d+3 b^2 B e+3 b c (B d+A e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \]
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Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {785} \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d x^4+\frac {1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac {1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac {1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9 \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (A b^3 d x^3+b^2 (b B d+3 A c d+A b e) x^4+b \left (3 A c^2 d+b^2 B e+3 b c (B d+A e)\right ) x^5+c \left (A c^2 d+3 b^2 B e+3 b c (B d+A 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 \\ & = \frac {1}{4} A b^3 d x^4+\frac {1}{5} b^2 (b B d+3 A c d+A b e) x^5+\frac {1}{6} b \left (3 A c^2 d+b^2 B e+3 b c (B d+A e)\right ) x^6+\frac {1}{7} c \left (A c^2 d+3 b^2 B e+3 b c (B d+A 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 = 0.03 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d x^4+\frac {1}{5} b^2 (b B d+3 A c d+A b e) x^5+\frac {1}{6} b \left (3 b B c d+3 A c^2 d+b^2 B e+3 A b c e\right ) x^6+\frac {1}{7} c \left (3 b B c d+A c^2 d+3 b^2 B e+3 A b c e\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \]
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Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {B \,c^{3} e \,x^{9}}{9}+\frac {\left (\left (A e +B d \right ) c^{3}+3 B e b \,c^{2}\right ) x^{8}}{8}+\frac {\left (A d \,c^{3}+3 \left (A e +B d \right ) b \,c^{2}+3 B e \,b^{2} c \right ) x^{7}}{7}+\frac {\left (3 A d b \,c^{2}+3 b^{2} c \left (A e +B d \right )+b^{3} B e \right ) x^{6}}{6}+\frac {\left (3 A d \,b^{2} c +b^{3} \left (A e +B d \right )\right ) x^{5}}{5}+\frac {A \,b^{3} d \,x^{4}}{4}\) | \(138\) |
norman | \(\frac {B \,c^{3} e \,x^{9}}{9}+\left (\frac {1}{8} A \,c^{3} e +\frac {3}{8} B e b \,c^{2}+\frac {1}{8} B \,c^{3} d \right ) x^{8}+\left (\frac {3}{7} A b \,c^{2} e +\frac {1}{7} A d \,c^{3}+\frac {3}{7} B e \,b^{2} c +\frac {3}{7} B b \,c^{2} d \right ) x^{7}+\left (\frac {1}{2} A \,b^{2} c e +\frac {1}{2} A d b \,c^{2}+\frac {1}{6} b^{3} B e +\frac {1}{2} B \,b^{2} c d \right ) x^{6}+\left (\frac {1}{5} A \,b^{3} e +\frac {3}{5} A d \,b^{2} c +\frac {1}{5} B \,b^{3} d \right ) x^{5}+\frac {A \,b^{3} d \,x^{4}}{4}\) | \(148\) |
gosper | \(\frac {x^{4} \left (280 B \,c^{3} e \,x^{5}+315 x^{4} A \,c^{3} e +945 x^{4} B e b \,c^{2}+315 x^{4} B \,c^{3} d +1080 x^{3} A b \,c^{2} e +360 x^{3} A d \,c^{3}+1080 x^{3} B e \,b^{2} c +1080 x^{3} B b \,c^{2} d +1260 x^{2} A \,b^{2} c e +1260 x^{2} A d b \,c^{2}+420 x^{2} b^{3} B e +1260 x^{2} B \,b^{2} c d +504 x A \,b^{3} e +1512 x A d \,b^{2} c +504 x B \,b^{3} d +630 A d \,b^{3}\right )}{2520}\) | \(166\) |
risch | \(\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {1}{8} x^{8} A \,c^{3} e +\frac {3}{8} x^{8} B e b \,c^{2}+\frac {1}{8} x^{8} B \,c^{3} d +\frac {3}{7} x^{7} A b \,c^{2} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {3}{7} x^{7} B e \,b^{2} c +\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{4} A \,b^{3} d \,x^{4}\) | \(170\) |
parallelrisch | \(\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {1}{8} x^{8} A \,c^{3} e +\frac {3}{8} x^{8} B e b \,c^{2}+\frac {1}{8} x^{8} B \,c^{3} d +\frac {3}{7} x^{7} A b \,c^{2} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {3}{7} x^{7} B e \,b^{2} c +\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{4} A \,b^{3} d \,x^{4}\) | \(170\) |
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Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + A b c^{2}\right )} d + {\left (B b^{3} + 3 \, A b^{2} c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (A b^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d\right )} x^{5} \]
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Time = 0.03 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.27 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {A b^{3} d x^{4}}{4} + \frac {B c^{3} e x^{9}}{9} + x^{8} \left (\frac {A c^{3} e}{8} + \frac {3 B b c^{2} e}{8} + \frac {B c^{3} d}{8}\right ) + x^{7} \cdot \left (\frac {3 A b c^{2} e}{7} + \frac {A c^{3} d}{7} + \frac {3 B b^{2} c e}{7} + \frac {3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac {A b^{2} c e}{2} + \frac {A b c^{2} d}{2} + \frac {B b^{3} e}{6} + \frac {B b^{2} c d}{2}\right ) + x^{5} \left (\frac {A b^{3} e}{5} + \frac {3 A b^{2} c d}{5} + \frac {B b^{3} d}{5}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + A b c^{2}\right )} d + {\left (B b^{3} + 3 \, A b^{2} c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (A b^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d\right )} x^{5} \]
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Time = 0.25 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.22 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, B c^{3} d x^{8} + \frac {3}{8} \, B b c^{2} e x^{8} + \frac {1}{8} \, A c^{3} e x^{8} + \frac {3}{7} \, B b c^{2} d x^{7} + \frac {1}{7} \, A c^{3} d x^{7} + \frac {3}{7} \, B b^{2} c e x^{7} + \frac {3}{7} \, A b c^{2} e x^{7} + \frac {1}{2} \, B b^{2} c d x^{6} + \frac {1}{2} \, A b c^{2} d x^{6} + \frac {1}{6} \, B b^{3} e x^{6} + \frac {1}{2} \, A b^{2} c e x^{6} + \frac {1}{5} \, B b^{3} d x^{5} + \frac {3}{5} \, A b^{2} c d x^{5} + \frac {1}{5} \, A b^{3} e x^{5} + \frac {1}{4} \, A b^{3} d x^{4} \]
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Time = 10.70 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.06 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,A\,b^2\,c\,d}{5}\right )+x^8\,\left (\frac {A\,c^3\,e}{8}+\frac {B\,c^3\,d}{8}+\frac {3\,B\,b\,c^2\,e}{8}\right )+x^6\,\left (\frac {B\,b^3\,e}{6}+\frac {A\,b\,c^2\,d}{2}+\frac {A\,b^2\,c\,e}{2}+\frac {B\,b^2\,c\,d}{2}\right )+x^7\,\left (\frac {A\,c^3\,d}{7}+\frac {3\,A\,b\,c^2\,e}{7}+\frac {3\,B\,b\,c^2\,d}{7}+\frac {3\,B\,b^2\,c\,e}{7}\right )+\frac {A\,b^3\,d\,x^4}{4}+\frac {B\,c^3\,e\,x^9}{9} \]
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