Integrand size = 22, antiderivative size = 60 \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=-7 a x-\frac {b x^2}{2}+\frac {8 a^4}{b (a-b x)^2}-\frac {32 a^3}{b (a-b x)}-\frac {24 a^2 \log (a-b x)}{b} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45} \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=\frac {8 a^4}{b (a-b x)^2}-\frac {32 a^3}{b (a-b x)}-\frac {24 a^2 \log (a-b x)}{b}-7 a x-\frac {b x^2}{2} \]
[In]
[Out]
Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^4}{(a-b x)^3} \, dx \\ & = \int \left (-7 a-b x+\frac {16 a^4}{(a-b x)^3}-\frac {32 a^3}{(a-b x)^2}+\frac {24 a^2}{a-b x}\right ) \, dx \\ & = -7 a x-\frac {b x^2}{2}+\frac {8 a^4}{b (a-b x)^2}-\frac {32 a^3}{b (a-b x)}-\frac {24 a^2 \log (a-b x)}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=-7 a x-\frac {b x^2}{2}+\frac {8 a^4}{b (-a+b x)^2}+\frac {32 a^3}{b (-a+b x)}-\frac {24 a^2 \log (a-b x)}{b} \]
[In]
[Out]
Time = 2.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-\frac {b \,x^{2}}{2}-7 a x +\frac {32 a^{3} x -\frac {24 a^{4}}{b}}{\left (-b x +a \right )^{2}}-\frac {24 a^{2} \ln \left (-b x +a \right )}{b}\) | \(51\) |
default | \(-7 a x -\frac {b \,x^{2}}{2}+\frac {8 a^{4}}{b \left (-b x +a \right )^{2}}-\frac {32 a^{3}}{b \left (-b x +a \right )}-\frac {24 a^{2} \ln \left (-b x +a \right )}{b}\) | \(59\) |
norman | \(\frac {-23 a^{5} x -\frac {b^{5} x^{6}}{2}-7 a \,b^{4} x^{5}+46 a^{3} b^{2} x^{3}-\frac {25 a^{6}}{b}+\frac {83 a^{4} b \,x^{2}}{2}}{\left (-b^{2} x^{2}+a^{2}\right )^{2}}-\frac {24 a^{2} \ln \left (-b x +a \right )}{b}\) | \(84\) |
parallelrisch | \(-\frac {b^{5} x^{4}+48 \ln \left (b x -a \right ) x^{2} a^{2} b^{3}+12 a \,b^{4} x^{3}-96 \ln \left (b x -a \right ) x \,a^{3} b^{2}+48 \ln \left (b x -a \right ) a^{4} b -104 a^{3} b^{2} x +75 a^{4} b}{2 b^{2} \left (b x -a \right )^{2}}\) | \(97\) |
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=-\frac {b^{4} x^{4} + 12 \, a b^{3} x^{3} - 27 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x + 48 \, a^{4} + 48 \, {\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )} \log \left (b x - a\right )}{2 \, {\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=- \frac {24 a^{2} \log {\left (- a + b x \right )}}{b} - 7 a x - \frac {b x^{2}}{2} - \frac {24 a^{4} - 32 a^{3} b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=-\frac {1}{2} \, b x^{2} - 7 \, a x - \frac {24 \, a^{2} \log \left (b x - a\right )}{b} + \frac {8 \, {\left (4 \, a^{3} b x - 3 \, a^{4}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=-\frac {24 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} + \frac {8 \, {\left (4 \, a^{3} b x - 3 \, a^{4}\right )}}{{\left (b x - a\right )}^{2} b} - \frac {b^{7} x^{2} + 14 \, a b^{6} x}{2 \, b^{6}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx=\frac {32\,a^3\,x-\frac {24\,a^4}{b}}{a^2-2\,a\,b\,x+b^2\,x^2}-7\,a\,x-\frac {b\,x^2}{2}-\frac {24\,a^2\,\ln \left (b\,x-a\right )}{b} \]
[In]
[Out]