Integrand size = 29, antiderivative size = 92 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=16 a^2 c^2 x+\frac {32}{3} a c^3 x^3+8 a c^2 d x^4+\frac {8}{5} c \left (2 c^3+a d^2\right ) x^5+\frac {16}{3} c^3 d x^6+\frac {24}{7} c^2 d^2 x^7+c d^3 x^8+\frac {d^4 x^9}{9} \]
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Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2086} \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=16 a^2 c^2 x+\frac {8}{5} c x^5 \left (a d^2+2 c^3\right )+\frac {32}{3} a c^3 x^3+8 a c^2 d x^4+\frac {16}{3} c^3 d x^6+\frac {24}{7} c^2 d^2 x^7+c d^3 x^8+\frac {d^4 x^9}{9} \]
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Rule 2086
Rubi steps \begin{align*} \text {integral}& = \int \left (16 a^2 c^2+32 a c^3 x^2+32 a c^2 d x^3+8 c \left (2 c^3+a d^2\right ) x^4+32 c^3 d x^5+24 c^2 d^2 x^6+8 c d^3 x^7+d^4 x^8\right ) \, dx \\ & = 16 a^2 c^2 x+\frac {32}{3} a c^3 x^3+8 a c^2 d x^4+\frac {8}{5} c \left (2 c^3+a d^2\right ) x^5+\frac {16}{3} c^3 d x^6+\frac {24}{7} c^2 d^2 x^7+c d^3 x^8+\frac {d^4 x^9}{9} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=16 a^2 c^2 x+\frac {32}{3} a c^3 x^3+8 a c^2 d x^4+\frac {8}{5} c \left (2 c^3+a d^2\right ) x^5+\frac {16}{3} c^3 d x^6+\frac {24}{7} c^2 d^2 x^7+c d^3 x^8+\frac {d^4 x^9}{9} \]
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Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90
method | result | size |
norman | \(\frac {d^{4} x^{9}}{9}+c \,d^{3} x^{8}+\frac {24 c^{2} d^{2} x^{7}}{7}+\frac {16 c^{3} d \,x^{6}}{3}+\left (\frac {8}{5} d^{2} a c +\frac {16}{5} c^{4}\right ) x^{5}+8 a \,c^{2} d \,x^{4}+\frac {32 a \,c^{3} x^{3}}{3}+16 a^{2} c^{2} x\) | \(83\) |
gosper | \(\frac {1}{9} d^{4} x^{9}+c \,d^{3} x^{8}+\frac {24}{7} c^{2} d^{2} x^{7}+\frac {16}{3} c^{3} d \,x^{6}+\frac {8}{5} x^{5} d^{2} a c +\frac {16}{5} x^{5} c^{4}+8 a \,c^{2} d \,x^{4}+\frac {32}{3} a \,c^{3} x^{3}+16 a^{2} c^{2} x\) | \(84\) |
default | \(\frac {d^{4} x^{9}}{9}+c \,d^{3} x^{8}+\frac {24 c^{2} d^{2} x^{7}}{7}+\frac {16 c^{3} d \,x^{6}}{3}+\frac {\left (8 d^{2} a c +16 c^{4}\right ) x^{5}}{5}+8 a \,c^{2} d \,x^{4}+\frac {32 a \,c^{3} x^{3}}{3}+16 a^{2} c^{2} x\) | \(84\) |
risch | \(\frac {1}{9} d^{4} x^{9}+c \,d^{3} x^{8}+\frac {24}{7} c^{2} d^{2} x^{7}+\frac {16}{3} c^{3} d \,x^{6}+\frac {8}{5} x^{5} d^{2} a c +\frac {16}{5} x^{5} c^{4}+8 a \,c^{2} d \,x^{4}+\frac {32}{3} a \,c^{3} x^{3}+16 a^{2} c^{2} x\) | \(84\) |
parallelrisch | \(\frac {1}{9} d^{4} x^{9}+c \,d^{3} x^{8}+\frac {24}{7} c^{2} d^{2} x^{7}+\frac {16}{3} c^{3} d \,x^{6}+\frac {8}{5} x^{5} d^{2} a c +\frac {16}{5} x^{5} c^{4}+8 a \,c^{2} d \,x^{4}+\frac {32}{3} a \,c^{3} x^{3}+16 a^{2} c^{2} x\) | \(84\) |
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Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=\frac {1}{9} \, d^{4} x^{9} + c d^{3} x^{8} + \frac {24}{7} \, c^{2} d^{2} x^{7} + \frac {16}{3} \, c^{3} d x^{6} + 8 \, a c^{2} d x^{4} + \frac {32}{3} \, a c^{3} x^{3} + \frac {8}{5} \, {\left (2 \, c^{4} + a c d^{2}\right )} x^{5} + 16 \, a^{2} c^{2} x \]
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Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.03 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=16 a^{2} c^{2} x + \frac {32 a c^{3} x^{3}}{3} + 8 a c^{2} d x^{4} + \frac {16 c^{3} d x^{6}}{3} + \frac {24 c^{2} d^{2} x^{7}}{7} + c d^{3} x^{8} + \frac {d^{4} x^{9}}{9} + x^{5} \cdot \left (\frac {8 a c d^{2}}{5} + \frac {16 c^{4}}{5}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=\frac {1}{9} \, d^{4} x^{9} + c d^{3} x^{8} + \frac {16}{7} \, c^{2} d^{2} x^{7} + \frac {16}{5} \, c^{4} x^{5} + 16 \, a^{2} c^{2} x + \frac {8}{15} \, {\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a c + \frac {8}{21} \, {\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\right )} c^{2} \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=\frac {1}{9} \, d^{4} x^{9} + c d^{3} x^{8} + \frac {24}{7} \, c^{2} d^{2} x^{7} + \frac {16}{3} \, c^{3} d x^{6} + \frac {16}{5} \, c^{4} x^{5} + \frac {8}{5} \, a c d^{2} x^{5} + 8 \, a c^{2} d x^{4} + \frac {32}{3} \, a c^{3} x^{3} + 16 \, a^{2} c^{2} x \]
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Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx=x^5\,\left (\frac {16\,c^4}{5}+\frac {8\,a\,c\,d^2}{5}\right )+\frac {d^4\,x^9}{9}+16\,a^2\,c^2\,x+\frac {32\,a\,c^3\,x^3}{3}+\frac {16\,c^3\,d\,x^6}{3}+c\,d^3\,x^8+\frac {24\,c^2\,d^2\,x^7}{7}+8\,a\,c^2\,d\,x^4 \]
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