Integrand size = 20, antiderivative size = 87 \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9 \]
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Time = 0.03 (sec) , antiderivative size = 87, 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 = {76} \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9 \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 c^4 x^3-3 a^4 b c^4 x^4+2 a^3 b^2 c^4 x^5+2 a^2 b^3 c^4 x^6-3 a b^4 c^4 x^7+b^5 c^4 x^8\right ) \, dx \\ & = \frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9 \]
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Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(\frac {x^{4} \left (280 b^{5} x^{5}-945 a \,b^{4} x^{4}+720 a^{2} b^{3} x^{3}+840 a^{3} b^{2} x^{2}-1512 a^{4} b x +630 a^{5}\right ) c^{4}}{2520}\) | \(61\) |
default | \(\frac {1}{4} a^{5} c^{4} x^{4}-\frac {3}{5} a^{4} b \,c^{4} x^{5}+\frac {1}{3} a^{3} b^{2} c^{4} x^{6}+\frac {2}{7} a^{2} b^{3} c^{4} x^{7}-\frac {3}{8} a \,b^{4} c^{4} x^{8}+\frac {1}{9} b^{5} c^{4} x^{9}\) | \(76\) |
norman | \(\frac {1}{4} a^{5} c^{4} x^{4}-\frac {3}{5} a^{4} b \,c^{4} x^{5}+\frac {1}{3} a^{3} b^{2} c^{4} x^{6}+\frac {2}{7} a^{2} b^{3} c^{4} x^{7}-\frac {3}{8} a \,b^{4} c^{4} x^{8}+\frac {1}{9} b^{5} c^{4} x^{9}\) | \(76\) |
risch | \(\frac {1}{4} a^{5} c^{4} x^{4}-\frac {3}{5} a^{4} b \,c^{4} x^{5}+\frac {1}{3} a^{3} b^{2} c^{4} x^{6}+\frac {2}{7} a^{2} b^{3} c^{4} x^{7}-\frac {3}{8} a \,b^{4} c^{4} x^{8}+\frac {1}{9} b^{5} c^{4} x^{9}\) | \(76\) |
parallelrisch | \(\frac {1}{4} a^{5} c^{4} x^{4}-\frac {3}{5} a^{4} b \,c^{4} x^{5}+\frac {1}{3} a^{3} b^{2} c^{4} x^{6}+\frac {2}{7} a^{2} b^{3} c^{4} x^{7}-\frac {3}{8} a \,b^{4} c^{4} x^{8}+\frac {1}{9} b^{5} c^{4} x^{9}\) | \(76\) |
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none
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{9} \, b^{5} c^{4} x^{9} - \frac {3}{8} \, a b^{4} c^{4} x^{8} + \frac {2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac {1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac {3}{5} \, a^{4} b c^{4} x^{5} + \frac {1}{4} \, a^{5} c^{4} x^{4} \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {a^{5} c^{4} x^{4}}{4} - \frac {3 a^{4} b c^{4} x^{5}}{5} + \frac {a^{3} b^{2} c^{4} x^{6}}{3} + \frac {2 a^{2} b^{3} c^{4} x^{7}}{7} - \frac {3 a b^{4} c^{4} x^{8}}{8} + \frac {b^{5} c^{4} x^{9}}{9} \]
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Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{9} \, b^{5} c^{4} x^{9} - \frac {3}{8} \, a b^{4} c^{4} x^{8} + \frac {2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac {1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac {3}{5} \, a^{4} b c^{4} x^{5} + \frac {1}{4} \, a^{5} c^{4} x^{4} \]
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {1}{9} \, b^{5} c^{4} x^{9} - \frac {3}{8} \, a b^{4} c^{4} x^{8} + \frac {2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac {1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac {3}{5} \, a^{4} b c^{4} x^{5} + \frac {1}{4} \, a^{5} c^{4} x^{4} \]
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Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^3 (a+b x) (a c-b c x)^4 \, dx=\frac {a^5\,c^4\,x^4}{4}-\frac {3\,a^4\,b\,c^4\,x^5}{5}+\frac {a^3\,b^2\,c^4\,x^6}{3}+\frac {2\,a^2\,b^3\,c^4\,x^7}{7}-\frac {3\,a\,b^4\,c^4\,x^8}{8}+\frac {b^5\,c^4\,x^9}{9} \]
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