Integrand size = 11, antiderivative size = 81 \[ \int x^4 (a+b x)^7 \, dx=\frac {a^4 (a+b x)^8}{8 b^5}-\frac {4 a^3 (a+b x)^9}{9 b^5}+\frac {3 a^2 (a+b x)^{10}}{5 b^5}-\frac {4 a (a+b x)^{11}}{11 b^5}+\frac {(a+b x)^{12}}{12 b^5} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 81, 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} \[ \int x^4 (a+b x)^7 \, dx=\frac {a^4 (a+b x)^8}{8 b^5}-\frac {4 a^3 (a+b x)^9}{9 b^5}+\frac {3 a^2 (a+b x)^{10}}{5 b^5}+\frac {(a+b x)^{12}}{12 b^5}-\frac {4 a (a+b x)^{11}}{11 b^5} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4 (a+b x)^7}{b^4}-\frac {4 a^3 (a+b x)^8}{b^4}+\frac {6 a^2 (a+b x)^9}{b^4}-\frac {4 a (a+b x)^{10}}{b^4}+\frac {(a+b x)^{11}}{b^4}\right ) \, dx \\ & = \frac {a^4 (a+b x)^8}{8 b^5}-\frac {4 a^3 (a+b x)^9}{9 b^5}+\frac {3 a^2 (a+b x)^{10}}{5 b^5}-\frac {4 a (a+b x)^{11}}{11 b^5}+\frac {(a+b x)^{12}}{12 b^5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.15 \[ \int x^4 (a+b x)^7 \, dx=\frac {a^7 x^5}{5}+\frac {7}{6} a^6 b x^6+3 a^5 b^2 x^7+\frac {35}{8} a^4 b^3 x^8+\frac {35}{9} a^3 b^4 x^9+\frac {21}{10} a^2 b^5 x^{10}+\frac {7}{11} a b^6 x^{11}+\frac {b^7 x^{12}}{12} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99
method | result | size |
gosper | \(\frac {1}{12} b^{7} x^{12}+\frac {7}{11} a \,b^{6} x^{11}+\frac {21}{10} a^{2} b^{5} x^{10}+\frac {35}{9} a^{3} b^{4} x^{9}+\frac {35}{8} a^{4} b^{3} x^{8}+3 a^{5} b^{2} x^{7}+\frac {7}{6} a^{6} b \,x^{6}+\frac {1}{5} a^{7} x^{5}\) | \(80\) |
default | \(\frac {1}{12} b^{7} x^{12}+\frac {7}{11} a \,b^{6} x^{11}+\frac {21}{10} a^{2} b^{5} x^{10}+\frac {35}{9} a^{3} b^{4} x^{9}+\frac {35}{8} a^{4} b^{3} x^{8}+3 a^{5} b^{2} x^{7}+\frac {7}{6} a^{6} b \,x^{6}+\frac {1}{5} a^{7} x^{5}\) | \(80\) |
norman | \(\frac {1}{12} b^{7} x^{12}+\frac {7}{11} a \,b^{6} x^{11}+\frac {21}{10} a^{2} b^{5} x^{10}+\frac {35}{9} a^{3} b^{4} x^{9}+\frac {35}{8} a^{4} b^{3} x^{8}+3 a^{5} b^{2} x^{7}+\frac {7}{6} a^{6} b \,x^{6}+\frac {1}{5} a^{7} x^{5}\) | \(80\) |
risch | \(\frac {1}{12} b^{7} x^{12}+\frac {7}{11} a \,b^{6} x^{11}+\frac {21}{10} a^{2} b^{5} x^{10}+\frac {35}{9} a^{3} b^{4} x^{9}+\frac {35}{8} a^{4} b^{3} x^{8}+3 a^{5} b^{2} x^{7}+\frac {7}{6} a^{6} b \,x^{6}+\frac {1}{5} a^{7} x^{5}\) | \(80\) |
parallelrisch | \(\frac {1}{12} b^{7} x^{12}+\frac {7}{11} a \,b^{6} x^{11}+\frac {21}{10} a^{2} b^{5} x^{10}+\frac {35}{9} a^{3} b^{4} x^{9}+\frac {35}{8} a^{4} b^{3} x^{8}+3 a^{5} b^{2} x^{7}+\frac {7}{6} a^{6} b \,x^{6}+\frac {1}{5} a^{7} x^{5}\) | \(80\) |
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int x^4 (a+b x)^7 \, dx=\frac {1}{12} \, b^{7} x^{12} + \frac {7}{11} \, a b^{6} x^{11} + \frac {21}{10} \, a^{2} b^{5} x^{10} + \frac {35}{9} \, a^{3} b^{4} x^{9} + \frac {35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac {7}{6} \, a^{6} b x^{6} + \frac {1}{5} \, a^{7} x^{5} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int x^4 (a+b x)^7 \, dx=\frac {a^{7} x^{5}}{5} + \frac {7 a^{6} b x^{6}}{6} + 3 a^{5} b^{2} x^{7} + \frac {35 a^{4} b^{3} x^{8}}{8} + \frac {35 a^{3} b^{4} x^{9}}{9} + \frac {21 a^{2} b^{5} x^{10}}{10} + \frac {7 a b^{6} x^{11}}{11} + \frac {b^{7} x^{12}}{12} \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int x^4 (a+b x)^7 \, dx=\frac {1}{12} \, b^{7} x^{12} + \frac {7}{11} \, a b^{6} x^{11} + \frac {21}{10} \, a^{2} b^{5} x^{10} + \frac {35}{9} \, a^{3} b^{4} x^{9} + \frac {35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac {7}{6} \, a^{6} b x^{6} + \frac {1}{5} \, a^{7} x^{5} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int x^4 (a+b x)^7 \, dx=\frac {1}{12} \, b^{7} x^{12} + \frac {7}{11} \, a b^{6} x^{11} + \frac {21}{10} \, a^{2} b^{5} x^{10} + \frac {35}{9} \, a^{3} b^{4} x^{9} + \frac {35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac {7}{6} \, a^{6} b x^{6} + \frac {1}{5} \, a^{7} x^{5} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int x^4 (a+b x)^7 \, dx=\frac {a^7\,x^5}{5}+\frac {7\,a^6\,b\,x^6}{6}+3\,a^5\,b^2\,x^7+\frac {35\,a^4\,b^3\,x^8}{8}+\frac {35\,a^3\,b^4\,x^9}{9}+\frac {21\,a^2\,b^5\,x^{10}}{10}+\frac {7\,a\,b^6\,x^{11}}{11}+\frac {b^7\,x^{12}}{12} \]
[In]
[Out]