Integrand size = 11, antiderivative size = 64 \[ \int x^3 (a+b x)^5 \, dx=-\frac {a^3 (a+b x)^6}{6 b^4}+\frac {3 a^2 (a+b x)^7}{7 b^4}-\frac {3 a (a+b x)^8}{8 b^4}+\frac {(a+b x)^9}{9 b^4} \]
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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} \[ \int x^3 (a+b x)^5 \, dx=-\frac {a^3 (a+b x)^6}{6 b^4}+\frac {3 a^2 (a+b x)^7}{7 b^4}+\frac {(a+b x)^9}{9 b^4}-\frac {3 a (a+b x)^8}{8 b^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 (a+b x)^5}{b^3}+\frac {3 a^2 (a+b x)^6}{b^3}-\frac {3 a (a+b x)^7}{b^3}+\frac {(a+b x)^8}{b^3}\right ) \, dx \\ & = -\frac {a^3 (a+b x)^6}{6 b^4}+\frac {3 a^2 (a+b x)^7}{7 b^4}-\frac {3 a (a+b x)^8}{8 b^4}+\frac {(a+b x)^9}{9 b^4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int x^3 (a+b x)^5 \, dx=\frac {a^5 x^4}{4}+a^4 b x^5+\frac {5}{3} a^3 b^2 x^6+\frac {10}{7} a^2 b^3 x^7+\frac {5}{8} a b^4 x^8+\frac {b^5 x^9}{9} \]
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Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\) | \(57\) |
default | \(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\) | \(57\) |
norman | \(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\) | \(57\) |
risch | \(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\) | \(57\) |
parallelrisch | \(\frac {1}{9} b^{5} x^{9}+\frac {5}{8} a \,b^{4} x^{8}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{3} a^{3} b^{2} x^{6}+a^{4} b \,x^{5}+\frac {1}{4} a^{5} x^{4}\) | \(57\) |
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none
Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int x^3 (a+b x)^5 \, dx=\frac {1}{9} \, b^{5} x^{9} + \frac {5}{8} \, a b^{4} x^{8} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac {1}{4} \, a^{5} x^{4} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int x^3 (a+b x)^5 \, dx=\frac {a^{5} x^{4}}{4} + a^{4} b x^{5} + \frac {5 a^{3} b^{2} x^{6}}{3} + \frac {10 a^{2} b^{3} x^{7}}{7} + \frac {5 a b^{4} x^{8}}{8} + \frac {b^{5} x^{9}}{9} \]
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none
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int x^3 (a+b x)^5 \, dx=\frac {1}{9} \, b^{5} x^{9} + \frac {5}{8} \, a b^{4} x^{8} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac {1}{4} \, a^{5} x^{4} \]
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none
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int x^3 (a+b x)^5 \, dx=\frac {1}{9} \, b^{5} x^{9} + \frac {5}{8} \, a b^{4} x^{8} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac {1}{4} \, a^{5} x^{4} \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int x^3 (a+b x)^5 \, dx=\frac {a^5\,x^4}{4}+a^4\,b\,x^5+\frac {5\,a^3\,b^2\,x^6}{3}+\frac {10\,a^2\,b^3\,x^7}{7}+\frac {5\,a\,b^4\,x^8}{8}+\frac {b^5\,x^9}{9} \]
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