Integrand size = 11, antiderivative size = 105 \[ \int \frac {x^7}{(a+b x)^4} \, dx=-\frac {20 a^3 x}{b^7}+\frac {5 a^2 x^2}{b^6}-\frac {4 a x^3}{3 b^5}+\frac {x^4}{4 b^4}+\frac {a^7}{3 b^8 (a+b x)^3}-\frac {7 a^6}{2 b^8 (a+b x)^2}+\frac {21 a^5}{b^8 (a+b x)}+\frac {35 a^4 \log (a+b x)}{b^8} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 105, 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} \[ \int \frac {x^7}{(a+b x)^4} \, dx=\frac {a^7}{3 b^8 (a+b x)^3}-\frac {7 a^6}{2 b^8 (a+b x)^2}+\frac {21 a^5}{b^8 (a+b x)}+\frac {35 a^4 \log (a+b x)}{b^8}-\frac {20 a^3 x}{b^7}+\frac {5 a^2 x^2}{b^6}-\frac {4 a x^3}{3 b^5}+\frac {x^4}{4 b^4} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {20 a^3}{b^7}+\frac {10 a^2 x}{b^6}-\frac {4 a x^2}{b^5}+\frac {x^3}{b^4}-\frac {a^7}{b^7 (a+b x)^4}+\frac {7 a^6}{b^7 (a+b x)^3}-\frac {21 a^5}{b^7 (a+b x)^2}+\frac {35 a^4}{b^7 (a+b x)}\right ) \, dx \\ & = -\frac {20 a^3 x}{b^7}+\frac {5 a^2 x^2}{b^6}-\frac {4 a x^3}{3 b^5}+\frac {x^4}{4 b^4}+\frac {a^7}{3 b^8 (a+b x)^3}-\frac {7 a^6}{2 b^8 (a+b x)^2}+\frac {21 a^5}{b^8 (a+b x)}+\frac {35 a^4 \log (a+b x)}{b^8} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{(a+b x)^4} \, dx=\frac {-240 a^3 b x+60 a^2 b^2 x^2-16 a b^3 x^3+3 b^4 x^4+\frac {4 a^7}{(a+b x)^3}-\frac {42 a^6}{(a+b x)^2}+\frac {252 a^5}{a+b x}+420 a^4 \log (a+b x)}{12 b^8} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {x^{4}}{4 b^{4}}-\frac {4 a \,x^{3}}{3 b^{5}}+\frac {5 a^{2} x^{2}}{b^{6}}-\frac {20 a^{3} x}{b^{7}}+\frac {21 a^{5} b \,x^{2}+\frac {77 a^{6} x}{2}+\frac {107 a^{7}}{6 b}}{b^{7} \left (b x +a \right )^{3}}+\frac {35 a^{4} \ln \left (b x +a \right )}{b^{8}}\) | \(88\) |
norman | \(\frac {\frac {x^{7}}{4 b}-\frac {7 a \,x^{6}}{12 b^{2}}+\frac {7 a^{2} x^{5}}{4 b^{3}}-\frac {35 a^{3} x^{4}}{4 b^{4}}+\frac {385 a^{7}}{6 b^{8}}+\frac {105 a^{5} x^{2}}{b^{6}}+\frac {315 a^{6} x}{2 b^{7}}}{\left (b x +a \right )^{3}}+\frac {35 a^{4} \ln \left (b x +a \right )}{b^{8}}\) | \(92\) |
default | \(-\frac {-\frac {1}{4} b^{3} x^{4}+\frac {4}{3} a \,b^{2} x^{3}-5 a^{2} b \,x^{2}+20 a^{3} x}{b^{7}}+\frac {35 a^{4} \ln \left (b x +a \right )}{b^{8}}+\frac {a^{7}}{3 b^{8} \left (b x +a \right )^{3}}-\frac {7 a^{6}}{2 b^{8} \left (b x +a \right )^{2}}+\frac {21 a^{5}}{b^{8} \left (b x +a \right )}\) | \(99\) |
parallelrisch | \(\frac {3 b^{7} x^{7}-7 a \,b^{6} x^{6}+21 a^{2} b^{5} x^{5}+420 \ln \left (b x +a \right ) x^{3} a^{4} b^{3}-105 a^{3} b^{4} x^{4}+1260 \ln \left (b x +a \right ) x^{2} a^{5} b^{2}+1260 \ln \left (b x +a \right ) x \,a^{6} b +1260 a^{5} b^{2} x^{2}+420 \ln \left (b x +a \right ) a^{7}+1890 a^{6} b x +770 a^{7}}{12 b^{8} \left (b x +a \right )^{3}}\) | \(134\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.44 \[ \int \frac {x^7}{(a+b x)^4} \, dx=\frac {3 \, b^{7} x^{7} - 7 \, a b^{6} x^{6} + 21 \, a^{2} b^{5} x^{5} - 105 \, a^{3} b^{4} x^{4} - 556 \, a^{4} b^{3} x^{3} - 408 \, a^{5} b^{2} x^{2} + 222 \, a^{6} b x + 214 \, a^{7} + 420 \, {\left (a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} + 3 \, a^{6} b x + a^{7}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.13 \[ \int \frac {x^7}{(a+b x)^4} \, dx=\frac {35 a^{4} \log {\left (a + b x \right )}}{b^{8}} - \frac {20 a^{3} x}{b^{7}} + \frac {5 a^{2} x^{2}}{b^{6}} - \frac {4 a x^{3}}{3 b^{5}} + \frac {107 a^{7} + 231 a^{6} b x + 126 a^{5} b^{2} x^{2}}{6 a^{3} b^{8} + 18 a^{2} b^{9} x + 18 a b^{10} x^{2} + 6 b^{11} x^{3}} + \frac {x^{4}}{4 b^{4}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09 \[ \int \frac {x^7}{(a+b x)^4} \, dx=\frac {126 \, a^{5} b^{2} x^{2} + 231 \, a^{6} b x + 107 \, a^{7}}{6 \, {\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} + \frac {35 \, a^{4} \log \left (b x + a\right )}{b^{8}} + \frac {3 \, b^{3} x^{4} - 16 \, a b^{2} x^{3} + 60 \, a^{2} b x^{2} - 240 \, a^{3} x}{12 \, b^{7}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{(a+b x)^4} \, dx=\frac {35 \, a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac {126 \, a^{5} b^{2} x^{2} + 231 \, a^{6} b x + 107 \, a^{7}}{6 \, {\left (b x + a\right )}^{3} b^{8}} + \frac {3 \, b^{12} x^{4} - 16 \, a b^{11} x^{3} + 60 \, a^{2} b^{10} x^{2} - 240 \, a^{3} b^{9} x}{12 \, b^{16}} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{(a+b x)^4} \, dx=\frac {\frac {{\left (a+b\,x\right )}^4}{4}-\frac {7\,a\,{\left (a+b\,x\right )}^3}{3}+\frac {21\,a^2\,{\left (a+b\,x\right )}^2}{2}+\frac {21\,a^5}{a+b\,x}-\frac {7\,a^6}{2\,{\left (a+b\,x\right )}^2}+\frac {a^7}{3\,{\left (a+b\,x\right )}^3}+35\,a^4\,\ln \left (a+b\,x\right )-35\,a^3\,b\,x}{b^8} \]
[In]
[Out]