Integrand size = 20, antiderivative size = 156 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^5}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^6}{3 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^7}{7 e^5}-\frac {c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {(d+e x)^7 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]
[In]
[Out]
Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^6}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^7}{e^4}+\frac {c^2 (d+e x)^8}{e^4}\right ) \, dx \\ & = \frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^5}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^6}{3 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^7}{7 e^5}-\frac {c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^2 d^4 x+a d^3 (b d+2 a e) x^2+\frac {1}{3} d^2 \left (b^2 d^2+8 a b d e+2 a \left (c d^2+3 a e^2\right )\right ) x^3+\frac {1}{2} d \left (b c d^3+2 b^2 d^2 e+4 a c d^2 e+6 a b d e^2+2 a^2 e^3\right ) x^4+\frac {1}{5} \left (c^2 d^4+4 c d^2 e (2 b d+3 a e)+e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) x^5+\frac {1}{3} e \left (2 c^2 d^3+b e^2 (2 b d+a e)+2 c d e (3 b d+2 a e)\right ) x^6+\frac {1}{7} e^2 \left (6 c^2 d^2+b^2 e^2+2 c e (4 b d+a e)\right ) x^7+\frac {1}{4} c e^3 (2 c d+b e) x^8+\frac {1}{9} c^2 e^4 x^9 \]
[In]
[Out]
Time = 2.83 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {\left (2 e^{4} b c +4 c^{2} d \,e^{3}\right ) x^{8}}{8}+\frac {\left (6 c^{2} d^{2} e^{2}+8 d \,e^{3} b c +e^{4} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (4 d^{3} e \,c^{2}+12 b c \,d^{2} e^{2}+4 d \,e^{3} \left (2 a c +b^{2}\right )+2 a b \,e^{4}\right ) x^{6}}{6}+\frac {\left (c^{2} d^{4}+8 d^{3} e b c +6 d^{2} e^{2} \left (2 a c +b^{2}\right )+8 a b d \,e^{3}+a^{2} e^{4}\right ) x^{5}}{5}+\frac {\left (2 d^{4} b c +4 d^{3} e \left (2 a c +b^{2}\right )+12 d^{2} e^{2} a b +4 a^{2} d \,e^{3}\right ) x^{4}}{4}+\frac {\left (d^{4} \left (2 a c +b^{2}\right )+8 a b \,d^{3} e +6 a^{2} d^{2} e^{2}\right ) x^{3}}{3}+\frac {\left (4 a^{2} d^{3} e +2 a b \,d^{4}\right ) x^{2}}{2}+a^{2} d^{4} x\) | \(283\) |
norman | \(\frac {e^{4} c^{2} x^{9}}{9}+\left (\frac {1}{4} e^{4} b c +\frac {1}{2} c^{2} d \,e^{3}\right ) x^{8}+\left (\frac {2}{7} a c \,e^{4}+\frac {1}{7} b^{2} e^{4}+\frac {8}{7} d \,e^{3} b c +\frac {6}{7} c^{2} d^{2} e^{2}\right ) x^{7}+\left (\frac {1}{3} a b \,e^{4}+\frac {4}{3} a c d \,e^{3}+\frac {2}{3} b^{2} d \,e^{3}+2 b c \,d^{2} e^{2}+\frac {2}{3} d^{3} e \,c^{2}\right ) x^{6}+\left (\frac {1}{5} a^{2} e^{4}+\frac {8}{5} a b d \,e^{3}+\frac {12}{5} a c \,d^{2} e^{2}+\frac {6}{5} b^{2} d^{2} e^{2}+\frac {8}{5} d^{3} e b c +\frac {1}{5} c^{2} d^{4}\right ) x^{5}+\left (a^{2} d \,e^{3}+3 d^{2} e^{2} a b +2 a c \,d^{3} e +b^{2} d^{3} e +\frac {1}{2} d^{4} b c \right ) x^{4}+\left (2 a^{2} d^{2} e^{2}+\frac {8}{3} a b \,d^{3} e +\frac {2}{3} a c \,d^{4}+\frac {1}{3} b^{2} d^{4}\right ) x^{3}+\left (2 a^{2} d^{3} e +a b \,d^{4}\right ) x^{2}+a^{2} d^{4} x\) | \(292\) |
gosper | \(2 x^{4} a c \,d^{3} e +\frac {8}{3} x^{3} a b \,d^{3} e +\frac {8}{7} x^{7} d \,e^{3} b c +\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {8}{5} x^{5} d^{3} e b c +3 x^{4} d^{2} e^{2} a b +\frac {4}{3} x^{6} a c d \,e^{3}+2 x^{6} b c \,d^{2} e^{2}+\frac {8}{5} x^{5} a b d \,e^{3}+x^{4} a^{2} d \,e^{3}+\frac {1}{7} b^{2} e^{4} x^{7}+a b \,d^{4} x^{2}+x^{4} b^{2} d^{3} e +\frac {2}{3} x^{6} b^{2} d \,e^{3}+\frac {1}{5} c^{2} d^{4} x^{5}+2 x^{3} a^{2} d^{2} e^{2}+\frac {2}{3} x^{3} a c \,d^{4}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {6}{5} x^{5} b^{2} d^{2} e^{2}+\frac {6}{7} x^{7} c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {1}{3} x^{6} a b \,e^{4}+a^{2} d^{4} x +\frac {1}{2} x^{4} d^{4} b c +\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{3} d^{4} b^{2} x^{3}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+2 a^{2} d^{3} e \,x^{2}+\frac {1}{4} x^{8} e^{4} b c +\frac {1}{5} x^{5} a^{2} e^{4}\) | \(341\) |
risch | \(2 x^{4} a c \,d^{3} e +\frac {8}{3} x^{3} a b \,d^{3} e +\frac {8}{7} x^{7} d \,e^{3} b c +\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {8}{5} x^{5} d^{3} e b c +3 x^{4} d^{2} e^{2} a b +\frac {4}{3} x^{6} a c d \,e^{3}+2 x^{6} b c \,d^{2} e^{2}+\frac {8}{5} x^{5} a b d \,e^{3}+x^{4} a^{2} d \,e^{3}+\frac {1}{7} b^{2} e^{4} x^{7}+a b \,d^{4} x^{2}+x^{4} b^{2} d^{3} e +\frac {2}{3} x^{6} b^{2} d \,e^{3}+\frac {1}{5} c^{2} d^{4} x^{5}+2 x^{3} a^{2} d^{2} e^{2}+\frac {2}{3} x^{3} a c \,d^{4}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {6}{5} x^{5} b^{2} d^{2} e^{2}+\frac {6}{7} x^{7} c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {1}{3} x^{6} a b \,e^{4}+a^{2} d^{4} x +\frac {1}{2} x^{4} d^{4} b c +\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{3} d^{4} b^{2} x^{3}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+2 a^{2} d^{3} e \,x^{2}+\frac {1}{4} x^{8} e^{4} b c +\frac {1}{5} x^{5} a^{2} e^{4}\) | \(341\) |
parallelrisch | \(2 x^{4} a c \,d^{3} e +\frac {8}{3} x^{3} a b \,d^{3} e +\frac {8}{7} x^{7} d \,e^{3} b c +\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {8}{5} x^{5} d^{3} e b c +3 x^{4} d^{2} e^{2} a b +\frac {4}{3} x^{6} a c d \,e^{3}+2 x^{6} b c \,d^{2} e^{2}+\frac {8}{5} x^{5} a b d \,e^{3}+x^{4} a^{2} d \,e^{3}+\frac {1}{7} b^{2} e^{4} x^{7}+a b \,d^{4} x^{2}+x^{4} b^{2} d^{3} e +\frac {2}{3} x^{6} b^{2} d \,e^{3}+\frac {1}{5} c^{2} d^{4} x^{5}+2 x^{3} a^{2} d^{2} e^{2}+\frac {2}{3} x^{3} a c \,d^{4}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {6}{5} x^{5} b^{2} d^{2} e^{2}+\frac {6}{7} x^{7} c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {1}{3} x^{6} a b \,e^{4}+a^{2} d^{4} x +\frac {1}{2} x^{4} d^{4} b c +\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{3} d^{4} b^{2} x^{3}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+2 a^{2} d^{3} e \,x^{2}+\frac {1}{4} x^{8} e^{4} b c +\frac {1}{5} x^{5} a^{2} e^{4}\) | \(341\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.78 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e + 6 \, b c d^{2} e^{2} + a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 8 \, a b d e^{3} + a^{2} e^{4} + 6 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 6 \, a b d^{2} e^{2} + 2 \, a^{2} d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (146) = 292\).
Time = 0.04 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.16 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^{2} d^{4} x + \frac {c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac {b c e^{4}}{4} + \frac {c^{2} d e^{3}}{2}\right ) + x^{7} \cdot \left (\frac {2 a c e^{4}}{7} + \frac {b^{2} e^{4}}{7} + \frac {8 b c d e^{3}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {a b e^{4}}{3} + \frac {4 a c d e^{3}}{3} + \frac {2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {8 a b d e^{3}}{5} + \frac {12 a c d^{2} e^{2}}{5} + \frac {6 b^{2} d^{2} e^{2}}{5} + \frac {8 b c d^{3} e}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + 2 a c d^{3} e + b^{2} d^{3} e + \frac {b c d^{4}}{2}\right ) + x^{3} \cdot \left (2 a^{2} d^{2} e^{2} + \frac {8 a b d^{3} e}{3} + \frac {2 a c d^{4}}{3} + \frac {b^{2} d^{4}}{3}\right ) + x^{2} \cdot \left (2 a^{2} d^{3} e + a b d^{4}\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.78 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e + 6 \, b c d^{2} e^{2} + a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 8 \, a b d e^{3} + a^{2} e^{4} + 6 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 6 \, a b d^{2} e^{2} + 2 \, a^{2} d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (146) = 292\).
Time = 0.26 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.18 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{2} d e^{3} x^{8} + \frac {1}{4} \, b c e^{4} x^{8} + \frac {6}{7} \, c^{2} d^{2} e^{2} x^{7} + \frac {8}{7} \, b c d e^{3} x^{7} + \frac {1}{7} \, b^{2} e^{4} x^{7} + \frac {2}{7} \, a c e^{4} x^{7} + \frac {2}{3} \, c^{2} d^{3} e x^{6} + 2 \, b c d^{2} e^{2} x^{6} + \frac {2}{3} \, b^{2} d e^{3} x^{6} + \frac {4}{3} \, a c d e^{3} x^{6} + \frac {1}{3} \, a b e^{4} x^{6} + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {8}{5} \, b c d^{3} e x^{5} + \frac {6}{5} \, b^{2} d^{2} e^{2} x^{5} + \frac {12}{5} \, a c d^{2} e^{2} x^{5} + \frac {8}{5} \, a b d e^{3} x^{5} + \frac {1}{5} \, a^{2} e^{4} x^{5} + \frac {1}{2} \, b c d^{4} x^{4} + b^{2} d^{3} e x^{4} + 2 \, a c d^{3} e x^{4} + 3 \, a b d^{2} e^{2} x^{4} + a^{2} d e^{3} x^{4} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {2}{3} \, a c d^{4} x^{3} + \frac {8}{3} \, a b d^{3} e x^{3} + 2 \, a^{2} d^{2} e^{2} x^{3} + a b d^{4} x^{2} + 2 \, a^{2} d^{3} e x^{2} + a^{2} d^{4} x \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=x^4\,\left (a^2\,d\,e^3+3\,a\,b\,d^2\,e^2+2\,c\,a\,d^3\,e+b^2\,d^3\,e+\frac {c\,b\,d^4}{2}\right )+x^6\,\left (\frac {2\,b^2\,d\,e^3}{3}+2\,b\,c\,d^2\,e^2+\frac {a\,b\,e^4}{3}+\frac {2\,c^2\,d^3\,e}{3}+\frac {4\,a\,c\,d\,e^3}{3}\right )+x^5\,\left (\frac {a^2\,e^4}{5}+\frac {8\,a\,b\,d\,e^3}{5}+\frac {12\,a\,c\,d^2\,e^2}{5}+\frac {6\,b^2\,d^2\,e^2}{5}+\frac {8\,b\,c\,d^3\,e}{5}+\frac {c^2\,d^4}{5}\right )+x^3\,\left (2\,a^2\,d^2\,e^2+\frac {8\,a\,b\,d^3\,e}{3}+\frac {2\,c\,a\,d^4}{3}+\frac {b^2\,d^4}{3}\right )+x^7\,\left (\frac {b^2\,e^4}{7}+\frac {8\,b\,c\,d\,e^3}{7}+\frac {6\,c^2\,d^2\,e^2}{7}+\frac {2\,a\,c\,e^4}{7}\right )+a^2\,d^4\,x+\frac {c^2\,e^4\,x^9}{9}+a\,d^3\,x^2\,\left (2\,a\,e+b\,d\right )+\frac {c\,e^3\,x^8\,\left (b\,e+2\,c\,d\right )}{4} \]
[In]
[Out]