Integrand size = 11, antiderivative size = 64 \[ \int x^3 (a+b x)^7 \, dx=-\frac {a^3 (a+b x)^8}{8 b^4}+\frac {a^2 (a+b x)^9}{3 b^4}-\frac {3 a (a+b x)^{10}}{10 b^4}+\frac {(a+b x)^{11}}{11 b^4} \]
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Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^3 (a+b x)^7 \, dx=-\frac {a^3 (a+b x)^8}{8 b^4}+\frac {a^2 (a+b x)^9}{3 b^4}+\frac {(a+b x)^{11}}{11 b^4}-\frac {3 a (a+b x)^{10}}{10 b^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 (a+b x)^7}{b^3}+\frac {3 a^2 (a+b x)^8}{b^3}-\frac {3 a (a+b x)^9}{b^3}+\frac {(a+b x)^{10}}{b^3}\right ) \, dx \\ & = -\frac {a^3 (a+b x)^8}{8 b^4}+\frac {a^2 (a+b x)^9}{3 b^4}-\frac {3 a (a+b x)^{10}}{10 b^4}+\frac {(a+b x)^{11}}{11 b^4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.45 \[ \int x^3 (a+b x)^7 \, dx=\frac {a^7 x^4}{4}+\frac {7}{5} a^6 b x^5+\frac {7}{2} a^5 b^2 x^6+5 a^4 b^3 x^7+\frac {35}{8} a^3 b^4 x^8+\frac {7}{3} a^2 b^5 x^9+\frac {7}{10} a b^6 x^{10}+\frac {b^7 x^{11}}{11} \]
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Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.25
method | result | size |
gosper | \(\frac {1}{11} b^{7} x^{11}+\frac {7}{10} a \,b^{6} x^{10}+\frac {7}{3} a^{2} b^{5} x^{9}+\frac {35}{8} a^{3} b^{4} x^{8}+5 a^{4} b^{3} x^{7}+\frac {7}{2} a^{5} b^{2} x^{6}+\frac {7}{5} a^{6} b \,x^{5}+\frac {1}{4} a^{7} x^{4}\) | \(80\) |
default | \(\frac {1}{11} b^{7} x^{11}+\frac {7}{10} a \,b^{6} x^{10}+\frac {7}{3} a^{2} b^{5} x^{9}+\frac {35}{8} a^{3} b^{4} x^{8}+5 a^{4} b^{3} x^{7}+\frac {7}{2} a^{5} b^{2} x^{6}+\frac {7}{5} a^{6} b \,x^{5}+\frac {1}{4} a^{7} x^{4}\) | \(80\) |
norman | \(\frac {1}{11} b^{7} x^{11}+\frac {7}{10} a \,b^{6} x^{10}+\frac {7}{3} a^{2} b^{5} x^{9}+\frac {35}{8} a^{3} b^{4} x^{8}+5 a^{4} b^{3} x^{7}+\frac {7}{2} a^{5} b^{2} x^{6}+\frac {7}{5} a^{6} b \,x^{5}+\frac {1}{4} a^{7} x^{4}\) | \(80\) |
risch | \(\frac {1}{11} b^{7} x^{11}+\frac {7}{10} a \,b^{6} x^{10}+\frac {7}{3} a^{2} b^{5} x^{9}+\frac {35}{8} a^{3} b^{4} x^{8}+5 a^{4} b^{3} x^{7}+\frac {7}{2} a^{5} b^{2} x^{6}+\frac {7}{5} a^{6} b \,x^{5}+\frac {1}{4} a^{7} x^{4}\) | \(80\) |
parallelrisch | \(\frac {1}{11} b^{7} x^{11}+\frac {7}{10} a \,b^{6} x^{10}+\frac {7}{3} a^{2} b^{5} x^{9}+\frac {35}{8} a^{3} b^{4} x^{8}+5 a^{4} b^{3} x^{7}+\frac {7}{2} a^{5} b^{2} x^{6}+\frac {7}{5} a^{6} b \,x^{5}+\frac {1}{4} a^{7} x^{4}\) | \(80\) |
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Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.23 \[ \int x^3 (a+b x)^7 \, dx=\frac {1}{11} \, b^{7} x^{11} + \frac {7}{10} \, a b^{6} x^{10} + \frac {7}{3} \, a^{2} b^{5} x^{9} + \frac {35}{8} \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{7} + \frac {7}{2} \, a^{5} b^{2} x^{6} + \frac {7}{5} \, a^{6} b x^{5} + \frac {1}{4} \, a^{7} x^{4} \]
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Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.44 \[ \int x^3 (a+b x)^7 \, dx=\frac {a^{7} x^{4}}{4} + \frac {7 a^{6} b x^{5}}{5} + \frac {7 a^{5} b^{2} x^{6}}{2} + 5 a^{4} b^{3} x^{7} + \frac {35 a^{3} b^{4} x^{8}}{8} + \frac {7 a^{2} b^{5} x^{9}}{3} + \frac {7 a b^{6} x^{10}}{10} + \frac {b^{7} x^{11}}{11} \]
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Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.23 \[ \int x^3 (a+b x)^7 \, dx=\frac {1}{11} \, b^{7} x^{11} + \frac {7}{10} \, a b^{6} x^{10} + \frac {7}{3} \, a^{2} b^{5} x^{9} + \frac {35}{8} \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{7} + \frac {7}{2} \, a^{5} b^{2} x^{6} + \frac {7}{5} \, a^{6} b x^{5} + \frac {1}{4} \, a^{7} x^{4} \]
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.23 \[ \int x^3 (a+b x)^7 \, dx=\frac {1}{11} \, b^{7} x^{11} + \frac {7}{10} \, a b^{6} x^{10} + \frac {7}{3} \, a^{2} b^{5} x^{9} + \frac {35}{8} \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{7} + \frac {7}{2} \, a^{5} b^{2} x^{6} + \frac {7}{5} \, a^{6} b x^{5} + \frac {1}{4} \, a^{7} x^{4} \]
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Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.23 \[ \int x^3 (a+b x)^7 \, dx=\frac {a^7\,x^4}{4}+\frac {7\,a^6\,b\,x^5}{5}+\frac {7\,a^5\,b^2\,x^6}{2}+5\,a^4\,b^3\,x^7+\frac {35\,a^3\,b^4\,x^8}{8}+\frac {7\,a^2\,b^5\,x^9}{3}+\frac {7\,a\,b^6\,x^{10}}{10}+\frac {b^7\,x^{11}}{11} \]
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