Integrand size = 19, antiderivative size = 263 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=a^4 \text {b1} x+\frac {1}{2} a^3 (8 b \text {b1}+a \text {c1}) x^2+\frac {4}{3} a^2 \left (6 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+a \left (8 b^3 \text {b1}+6 a b \text {b1} c+6 a b^2 \text {c1}+a^2 c \text {c1}\right ) x^4+\frac {2}{5} \left (8 b^4 \text {b1}+24 a b^2 \text {b1} c+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+12 a^2 b c \text {c1}\right ) x^5+\frac {1}{3} \left (16 b^3 \text {b1} c+12 a b \text {b1} c^2+8 b^4 \text {c1}+24 a b^2 c \text {c1}+3 a^2 c^2 \text {c1}\right ) x^6+\frac {4}{7} c \left (6 b^2 \text {b1} c+a \text {b1} c^2+8 b^3 \text {c1}+6 a b c \text {c1}\right ) x^7+\frac {1}{2} c^2 \left (2 b \text {b1} c+6 b^2 \text {c1}+a c \text {c1}\right ) x^8+\frac {1}{9} c^3 (\text {b1} c+8 b \text {c1}) x^9+\frac {1}{10} c^4 \text {c1} x^{10} \]
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Time = 0.27 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {645} \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=a^4 \text {b1} x+\frac {1}{2} a^3 x^2 (a \text {c1}+8 b \text {b1})+\frac {4}{3} a^2 x^3 \left (2 a b \text {c1}+a \text {b1} c+6 b^2 \text {b1}\right )+a x^4 \left (a^2 c \text {c1}+6 a b^2 \text {c1}+6 a b \text {b1} c+8 b^3 \text {b1}\right )+\frac {1}{3} x^6 \left (3 a^2 c^2 \text {c1}+24 a b^2 c \text {c1}+12 a b \text {b1} c^2+8 b^4 \text {c1}+16 b^3 \text {b1} c\right )+\frac {2}{5} x^5 \left (12 a^2 b c \text {c1}+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+24 a b^2 \text {b1} c+8 b^4 \text {b1}\right )+\frac {1}{2} c^2 x^8 \left (a c \text {c1}+6 b^2 \text {c1}+2 b \text {b1} c\right )+\frac {4}{7} c x^7 \left (6 a b c \text {c1}+a \text {b1} c^2+8 b^3 \text {c1}+6 b^2 \text {b1} c\right )+\frac {1}{9} c^3 x^9 (8 b \text {c1}+\text {b1} c)+\frac {1}{10} c^4 \text {c1} x^{10} \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 \text {b1}+a^3 (8 b \text {b1}+a \text {c1}) x+4 a^2 \left (6 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^2+4 a \left (8 b^3 \text {b1}+6 a b \text {b1} c+6 a b^2 \text {c1}+a^2 c \text {c1}\right ) x^3+2 \left (8 b^4 \text {b1}+24 a b^2 \text {b1} c+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+12 a^2 b c \text {c1}\right ) x^4+2 \left (16 b^3 \text {b1} c+12 a b \text {b1} c^2+8 b^4 \text {c1}+24 a b^2 c \text {c1}+3 a^2 c^2 \text {c1}\right ) x^5+4 c \left (6 b^2 \text {b1} c+a \text {b1} c^2+8 b^3 \text {c1}+6 a b c \text {c1}\right ) x^6+4 c^2 \left (2 b \text {b1} c+6 b^2 \text {c1}+a c \text {c1}\right ) x^7+c^3 (\text {b1} c+8 b \text {c1}) x^8+c^4 \text {c1} x^9\right ) \, dx \\ & = a^4 \text {b1} x+\frac {1}{2} a^3 (8 b \text {b1}+a \text {c1}) x^2+\frac {4}{3} a^2 \left (6 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+a \left (8 b^3 \text {b1}+6 a b \text {b1} c+6 a b^2 \text {c1}+a^2 c \text {c1}\right ) x^4+\frac {2}{5} \left (8 b^4 \text {b1}+24 a b^2 \text {b1} c+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+12 a^2 b c \text {c1}\right ) x^5+\frac {1}{3} \left (16 b^3 \text {b1} c+12 a b \text {b1} c^2+8 b^4 \text {c1}+24 a b^2 c \text {c1}+3 a^2 c^2 \text {c1}\right ) x^6+\frac {4}{7} c \left (6 b^2 \text {b1} c+a \text {b1} c^2+8 b^3 \text {c1}+6 a b c \text {c1}\right ) x^7+\frac {1}{2} c^2 \left (2 b \text {b1} c+6 b^2 \text {c1}+a c \text {c1}\right ) x^8+\frac {1}{9} c^3 (\text {b1} c+8 b \text {c1}) x^9+\frac {1}{10} c^4 \text {c1} x^{10} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=a^4 \text {b1} x+\frac {1}{2} a^3 (8 b \text {b1}+a \text {c1}) x^2+\frac {4}{3} a^2 \left (6 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+a \left (8 b^3 \text {b1}+6 a b \text {b1} c+6 a b^2 \text {c1}+a^2 c \text {c1}\right ) x^4+\frac {2}{5} \left (8 b^4 \text {b1}+24 a b^2 \text {b1} c+3 a^2 \text {b1} c^2+16 a b^3 \text {c1}+12 a^2 b c \text {c1}\right ) x^5+\frac {1}{3} \left (16 b^3 \text {b1} c+12 a b \text {b1} c^2+8 b^4 \text {c1}+24 a b^2 c \text {c1}+3 a^2 c^2 \text {c1}\right ) x^6+\frac {4}{7} c \left (6 b^2 \text {b1} c+a \text {b1} c^2+8 b^3 \text {c1}+6 a b c \text {c1}\right ) x^7+\frac {1}{2} c^2 \left (2 b \text {b1} c+6 b^2 \text {c1}+a c \text {c1}\right ) x^8+\frac {1}{9} c^3 (\text {b1} c+8 b \text {c1}) x^9+\frac {1}{10} c^4 \text {c1} x^{10} \]
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Time = 0.35 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {c^{4} \operatorname {c1} \,x^{10}}{10}+\left (\frac {8}{9} \operatorname {c1} b \,c^{3}+\frac {1}{9} \operatorname {b1} \,c^{4}\right ) x^{9}+\left (\frac {1}{2} a \,c^{3} \operatorname {c1} +3 b^{2} c^{2} \operatorname {c1} +\operatorname {b1} b \,c^{3}\right ) x^{8}+\left (\frac {24}{7} a b \,c^{2} \operatorname {c1} +\frac {4}{7} a \operatorname {b1} \,c^{3}+\frac {32}{7} b^{3} c \operatorname {c1} +\frac {24}{7} b^{2} \operatorname {b1} \,c^{2}\right ) x^{7}+\left (a^{2} c^{2} \operatorname {c1} +8 a \,b^{2} c \operatorname {c1} +4 a b \operatorname {b1} \,c^{2}+\frac {8}{3} b^{4} \operatorname {c1} +\frac {16}{3} b^{3} \operatorname {b1} c \right ) x^{6}+\left (\frac {24}{5} a^{2} b c \operatorname {c1} +\frac {6}{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} a \,b^{3} \operatorname {c1} +\frac {48}{5} a \,b^{2} \operatorname {b1} c +\frac {16}{5} b^{4} \operatorname {b1} \right ) x^{5}+\left (a^{3} c \operatorname {c1} +6 a^{2} b^{2} \operatorname {c1} +6 a^{2} b \operatorname {b1} c +8 a \,b^{3} \operatorname {b1} \right ) x^{4}+\left (\frac {8}{3} \operatorname {c1} \,a^{3} b +\frac {4}{3} a^{3} \operatorname {b1} c +8 a^{2} b^{2} \operatorname {b1} \right ) x^{3}+\left (\frac {1}{2} \operatorname {c1} \,a^{4}+4 \operatorname {b1} \,a^{3} b \right ) x^{2}+a^{4} \operatorname {b1} x\) | \(264\) |
gosper | \(a^{4} \operatorname {b1} x +\frac {8}{9} x^{9} \operatorname {c1} b \,c^{3}+\frac {1}{2} x^{2} \operatorname {c1} \,a^{4}+\frac {1}{9} x^{9} \operatorname {b1} \,c^{4}+\frac {8}{3} x^{6} b^{4} \operatorname {c1} +\frac {16}{5} x^{5} b^{4} \operatorname {b1} +\frac {1}{2} x^{8} a \,c^{3} \operatorname {c1} +3 x^{8} b^{2} c^{2} \operatorname {c1} +\frac {1}{10} c^{4} \operatorname {c1} \,x^{10}+8 a \,b^{3} \operatorname {b1} \,x^{4}+a^{3} c \operatorname {c1} \,x^{4}+6 a^{2} b^{2} \operatorname {c1} \,x^{4}+\frac {4}{3} x^{3} a^{3} \operatorname {b1} c +8 x^{3} a^{2} b^{2} \operatorname {b1} +4 x^{2} \operatorname {b1} \,a^{3} b +\frac {32}{7} x^{7} b^{3} c \operatorname {c1} +\frac {24}{7} x^{7} b^{2} \operatorname {b1} \,c^{2}+x^{6} a^{2} c^{2} \operatorname {c1} +\frac {16}{3} x^{6} b^{3} \operatorname {b1} c +\frac {6}{5} x^{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} x^{5} a \,b^{3} \operatorname {c1} +\frac {8}{3} x^{3} \operatorname {c1} \,a^{3} b +x^{8} \operatorname {b1} b \,c^{3}+\frac {4}{7} x^{7} a \operatorname {b1} \,c^{3}+\frac {24}{7} x^{7} a b \,c^{2} \operatorname {c1} +8 x^{6} a \,b^{2} c \operatorname {c1} +4 x^{6} a b \operatorname {b1} \,c^{2}+\frac {24}{5} x^{5} a^{2} b c \operatorname {c1} +\frac {48}{5} x^{5} a \,b^{2} \operatorname {b1} c +6 a^{2} b \operatorname {b1} c \,x^{4}\) | \(308\) |
risch | \(a^{4} \operatorname {b1} x +\frac {8}{9} x^{9} \operatorname {c1} b \,c^{3}+\frac {1}{2} x^{2} \operatorname {c1} \,a^{4}+\frac {1}{9} x^{9} \operatorname {b1} \,c^{4}+\frac {8}{3} x^{6} b^{4} \operatorname {c1} +\frac {16}{5} x^{5} b^{4} \operatorname {b1} +\frac {1}{2} x^{8} a \,c^{3} \operatorname {c1} +3 x^{8} b^{2} c^{2} \operatorname {c1} +\frac {1}{10} c^{4} \operatorname {c1} \,x^{10}+8 a \,b^{3} \operatorname {b1} \,x^{4}+a^{3} c \operatorname {c1} \,x^{4}+6 a^{2} b^{2} \operatorname {c1} \,x^{4}+\frac {4}{3} x^{3} a^{3} \operatorname {b1} c +8 x^{3} a^{2} b^{2} \operatorname {b1} +4 x^{2} \operatorname {b1} \,a^{3} b +\frac {32}{7} x^{7} b^{3} c \operatorname {c1} +\frac {24}{7} x^{7} b^{2} \operatorname {b1} \,c^{2}+x^{6} a^{2} c^{2} \operatorname {c1} +\frac {16}{3} x^{6} b^{3} \operatorname {b1} c +\frac {6}{5} x^{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} x^{5} a \,b^{3} \operatorname {c1} +\frac {8}{3} x^{3} \operatorname {c1} \,a^{3} b +x^{8} \operatorname {b1} b \,c^{3}+\frac {4}{7} x^{7} a \operatorname {b1} \,c^{3}+\frac {24}{7} x^{7} a b \,c^{2} \operatorname {c1} +8 x^{6} a \,b^{2} c \operatorname {c1} +4 x^{6} a b \operatorname {b1} \,c^{2}+\frac {24}{5} x^{5} a^{2} b c \operatorname {c1} +\frac {48}{5} x^{5} a \,b^{2} \operatorname {b1} c +6 a^{2} b \operatorname {b1} c \,x^{4}\) | \(308\) |
parallelrisch | \(a^{4} \operatorname {b1} x +\frac {8}{9} x^{9} \operatorname {c1} b \,c^{3}+\frac {1}{2} x^{2} \operatorname {c1} \,a^{4}+\frac {1}{9} x^{9} \operatorname {b1} \,c^{4}+\frac {8}{3} x^{6} b^{4} \operatorname {c1} +\frac {16}{5} x^{5} b^{4} \operatorname {b1} +\frac {1}{2} x^{8} a \,c^{3} \operatorname {c1} +3 x^{8} b^{2} c^{2} \operatorname {c1} +\frac {1}{10} c^{4} \operatorname {c1} \,x^{10}+8 a \,b^{3} \operatorname {b1} \,x^{4}+a^{3} c \operatorname {c1} \,x^{4}+6 a^{2} b^{2} \operatorname {c1} \,x^{4}+\frac {4}{3} x^{3} a^{3} \operatorname {b1} c +8 x^{3} a^{2} b^{2} \operatorname {b1} +4 x^{2} \operatorname {b1} \,a^{3} b +\frac {32}{7} x^{7} b^{3} c \operatorname {c1} +\frac {24}{7} x^{7} b^{2} \operatorname {b1} \,c^{2}+x^{6} a^{2} c^{2} \operatorname {c1} +\frac {16}{3} x^{6} b^{3} \operatorname {b1} c +\frac {6}{5} x^{5} a^{2} \operatorname {b1} \,c^{2}+\frac {32}{5} x^{5} a \,b^{3} \operatorname {c1} +\frac {8}{3} x^{3} \operatorname {c1} \,a^{3} b +x^{8} \operatorname {b1} b \,c^{3}+\frac {4}{7} x^{7} a \operatorname {b1} \,c^{3}+\frac {24}{7} x^{7} a b \,c^{2} \operatorname {c1} +8 x^{6} a \,b^{2} c \operatorname {c1} +4 x^{6} a b \operatorname {b1} \,c^{2}+\frac {24}{5} x^{5} a^{2} b c \operatorname {c1} +\frac {48}{5} x^{5} a \,b^{2} \operatorname {b1} c +6 a^{2} b \operatorname {b1} c \,x^{4}\) | \(308\) |
default | \(\frac {c^{4} \operatorname {c1} \,x^{10}}{10}+\frac {\left (8 \operatorname {c1} b \,c^{3}+\operatorname {b1} \,c^{4}\right ) x^{9}}{9}+\frac {\left (8 \operatorname {b1} b \,c^{3}+\operatorname {c1} \left (2 \left (2 a c +4 b^{2}\right ) c^{2}+16 b^{2} c^{2}\right )\right ) x^{8}}{8}+\frac {\left (\operatorname {b1} \left (2 \left (2 a c +4 b^{2}\right ) c^{2}+16 b^{2} c^{2}\right )+\operatorname {c1} \left (8 a b \,c^{2}+8 \left (2 a c +4 b^{2}\right ) b c \right )\right ) x^{7}}{7}+\frac {\left (\operatorname {b1} \left (8 a b \,c^{2}+8 \left (2 a c +4 b^{2}\right ) b c \right )+\operatorname {c1} \left (2 a^{2} c^{2}+32 a \,b^{2} c +\left (2 a c +4 b^{2}\right )^{2}\right )\right ) x^{6}}{6}+\frac {\left (\operatorname {b1} \left (2 a^{2} c^{2}+32 a \,b^{2} c +\left (2 a c +4 b^{2}\right )^{2}\right )+\operatorname {c1} \left (8 a^{2} b c +8 a b \left (2 a c +4 b^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (\operatorname {b1} \left (8 a^{2} b c +8 a b \left (2 a c +4 b^{2}\right )\right )+\operatorname {c1} \left (2 a^{2} \left (2 a c +4 b^{2}\right )+16 a^{2} b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\operatorname {b1} \left (2 a^{2} \left (2 a c +4 b^{2}\right )+16 a^{2} b^{2}\right )+8 \operatorname {c1} \,a^{3} b \right ) x^{3}}{3}+\frac {\left (\operatorname {c1} \,a^{4}+8 \operatorname {b1} \,a^{3} b \right ) x^{2}}{2}+a^{4} \operatorname {b1} x\) | \(363\) |
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Time = 0.25 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.04 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} c_{1} x^{10} + \frac {1}{9} \, {\left (b_{1} c^{4} + 8 \, b c^{3} c_{1}\right )} x^{9} + \frac {1}{2} \, {\left (2 \, b b_{1} c^{3} + {\left (6 \, b^{2} c^{2} + a c^{3}\right )} c_{1}\right )} x^{8} + \frac {4}{7} \, {\left (6 \, b^{2} b_{1} c^{2} + a b_{1} c^{3} + 2 \, {\left (4 \, b^{3} c + 3 \, a b c^{2}\right )} c_{1}\right )} x^{7} + \frac {1}{3} \, {\left (16 \, b^{3} b_{1} c + 12 \, a b b_{1} c^{2} + {\left (8 \, b^{4} + 24 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} c_{1}\right )} x^{6} + a^{4} b_{1} x + \frac {2}{5} \, {\left (8 \, b^{4} b_{1} + 24 \, a b^{2} b_{1} c + 3 \, a^{2} b_{1} c^{2} + 4 \, {\left (4 \, a b^{3} + 3 \, a^{2} b c\right )} c_{1}\right )} x^{5} + {\left (8 \, a b^{3} b_{1} + 6 \, a^{2} b b_{1} c + {\left (6 \, a^{2} b^{2} + a^{3} c\right )} c_{1}\right )} x^{4} + \frac {4}{3} \, {\left (6 \, a^{2} b^{2} b_{1} + a^{3} b_{1} c + 2 \, a^{3} b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (8 \, a^{3} b b_{1} + a^{4} c_{1}\right )} x^{2} \]
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Time = 0.04 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.19 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=a^{4} b_{1} x + \frac {c^{4} c_{1} x^{10}}{10} + x^{9} \cdot \left (\frac {8 b c^{3} c_{1}}{9} + \frac {b_{1} c^{4}}{9}\right ) + x^{8} \left (\frac {a c^{3} c_{1}}{2} + 3 b^{2} c^{2} c_{1} + b b_{1} c^{3}\right ) + x^{7} \cdot \left (\frac {24 a b c^{2} c_{1}}{7} + \frac {4 a b_{1} c^{3}}{7} + \frac {32 b^{3} c c_{1}}{7} + \frac {24 b^{2} b_{1} c^{2}}{7}\right ) + x^{6} \left (a^{2} c^{2} c_{1} + 8 a b^{2} c c_{1} + 4 a b b_{1} c^{2} + \frac {8 b^{4} c_{1}}{3} + \frac {16 b^{3} b_{1} c}{3}\right ) + x^{5} \cdot \left (\frac {24 a^{2} b c c_{1}}{5} + \frac {6 a^{2} b_{1} c^{2}}{5} + \frac {32 a b^{3} c_{1}}{5} + \frac {48 a b^{2} b_{1} c}{5} + \frac {16 b^{4} b_{1}}{5}\right ) + x^{4} \left (a^{3} c c_{1} + 6 a^{2} b^{2} c_{1} + 6 a^{2} b b_{1} c + 8 a b^{3} b_{1}\right ) + x^{3} \cdot \left (\frac {8 a^{3} b c_{1}}{3} + \frac {4 a^{3} b_{1} c}{3} + 8 a^{2} b^{2} b_{1}\right ) + x^{2} \left (\frac {a^{4} c_{1}}{2} + 4 a^{3} b b_{1}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.04 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} c_{1} x^{10} + \frac {1}{9} \, {\left (b_{1} c^{4} + 8 \, b c^{3} c_{1}\right )} x^{9} + \frac {1}{2} \, {\left (2 \, b b_{1} c^{3} + {\left (6 \, b^{2} c^{2} + a c^{3}\right )} c_{1}\right )} x^{8} + \frac {4}{7} \, {\left (6 \, b^{2} b_{1} c^{2} + a b_{1} c^{3} + 2 \, {\left (4 \, b^{3} c + 3 \, a b c^{2}\right )} c_{1}\right )} x^{7} + \frac {1}{3} \, {\left (16 \, b^{3} b_{1} c + 12 \, a b b_{1} c^{2} + {\left (8 \, b^{4} + 24 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} c_{1}\right )} x^{6} + a^{4} b_{1} x + \frac {2}{5} \, {\left (8 \, b^{4} b_{1} + 24 \, a b^{2} b_{1} c + 3 \, a^{2} b_{1} c^{2} + 4 \, {\left (4 \, a b^{3} + 3 \, a^{2} b c\right )} c_{1}\right )} x^{5} + {\left (8 \, a b^{3} b_{1} + 6 \, a^{2} b b_{1} c + {\left (6 \, a^{2} b^{2} + a^{3} c\right )} c_{1}\right )} x^{4} + \frac {4}{3} \, {\left (6 \, a^{2} b^{2} b_{1} + a^{3} b_{1} c + 2 \, a^{3} b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (8 \, a^{3} b b_{1} + a^{4} c_{1}\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.17 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=\frac {1}{10} \, c^{4} c_{1} x^{10} + \frac {1}{9} \, b_{1} c^{4} x^{9} + \frac {8}{9} \, b c^{3} c_{1} x^{9} + b b_{1} c^{3} x^{8} + 3 \, b^{2} c^{2} c_{1} x^{8} + \frac {1}{2} \, a c^{3} c_{1} x^{8} + \frac {24}{7} \, b^{2} b_{1} c^{2} x^{7} + \frac {4}{7} \, a b_{1} c^{3} x^{7} + \frac {32}{7} \, b^{3} c c_{1} x^{7} + \frac {24}{7} \, a b c^{2} c_{1} x^{7} + \frac {16}{3} \, b^{3} b_{1} c x^{6} + 4 \, a b b_{1} c^{2} x^{6} + \frac {8}{3} \, b^{4} c_{1} x^{6} + 8 \, a b^{2} c c_{1} x^{6} + a^{2} c^{2} c_{1} x^{6} + \frac {16}{5} \, b^{4} b_{1} x^{5} + \frac {48}{5} \, a b^{2} b_{1} c x^{5} + \frac {6}{5} \, a^{2} b_{1} c^{2} x^{5} + \frac {32}{5} \, a b^{3} c_{1} x^{5} + \frac {24}{5} \, a^{2} b c c_{1} x^{5} + 8 \, a b^{3} b_{1} x^{4} + 6 \, a^{2} b b_{1} c x^{4} + 6 \, a^{2} b^{2} c_{1} x^{4} + a^{3} c c_{1} x^{4} + 8 \, a^{2} b^{2} b_{1} x^{3} + \frac {4}{3} \, a^{3} b_{1} c x^{3} + \frac {8}{3} \, a^{3} b c_{1} x^{3} + 4 \, a^{3} b b_{1} x^{2} + \frac {1}{2} \, a^{4} c_{1} x^{2} + a^{4} b_{1} x \]
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Time = 0.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^4 \, dx=x^9\,\left (\frac {b_{1}\,c^4}{9}+\frac {8\,b\,c_{1}\,c^3}{9}\right )+x^3\,\left (\frac {8\,c_{1}\,a^3\,b}{3}+\frac {4\,b_{1}\,c\,a^3}{3}+8\,b_{1}\,a^2\,b^2\right )+x^8\,\left (3\,c_{1}\,b^2\,c^2+b_{1}\,b\,c^3+\frac {a\,c_{1}\,c^3}{2}\right )+x^5\,\left (\frac {24\,c_{1}\,a^2\,b\,c}{5}+\frac {6\,b_{1}\,a^2\,c^2}{5}+\frac {32\,c_{1}\,a\,b^3}{5}+\frac {48\,b_{1}\,a\,b^2\,c}{5}+\frac {16\,b_{1}\,b^4}{5}\right )+x^6\,\left (c_{1}\,a^2\,c^2+8\,c_{1}\,a\,b^2\,c+4\,b_{1}\,a\,b\,c^2+\frac {8\,c_{1}\,b^4}{3}+\frac {16\,b_{1}\,b^3\,c}{3}\right )+x^4\,\left (c\,c_{1}\,a^3+6\,c_{1}\,a^2\,b^2+6\,b_{1}\,c\,a^2\,b+8\,b_{1}\,a\,b^3\right )+x^7\,\left (\frac {32\,c_{1}\,b^3\,c}{7}+\frac {24\,b_{1}\,b^2\,c^2}{7}+\frac {24\,a\,c_{1}\,b\,c^2}{7}+\frac {4\,a\,b_{1}\,c^3}{7}\right )+x^2\,\left (\frac {c_{1}\,a^4}{2}+4\,b\,b_{1}\,a^3\right )+\frac {c^4\,c_{1}\,x^{10}}{10}+a^4\,b_{1}\,x \]
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