Integrand size = 18, antiderivative size = 77 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=-\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}+\frac {2 b B d-A b e-a B e}{4 e^3 (d+e x)^4}-\frac {b B}{3 e^3 (d+e x)^3} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=\frac {-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac {b B}{3 e^3 (d+e x)^3} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^6}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)^5}+\frac {b B}{e^2 (d+e x)^4}\right ) \, dx \\ & = -\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}+\frac {2 b B d-A b e-a B e}{4 e^3 (d+e x)^4}-\frac {b B}{3 e^3 (d+e x)^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=-\frac {3 a e (4 A e+B (d+5 e x))+b \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{60 e^3 (d+e x)^5} \]
[In]
[Out]
Time = 2.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(-\frac {20 b B \,x^{2} e^{2}+15 A x b \,e^{2}+15 B x a \,e^{2}+10 B x b d e +12 A a \,e^{2}+3 A b d e +3 B a d e +2 b B \,d^{2}}{60 e^{3} \left (e x +d \right )^{5}}\) | \(73\) |
risch | \(\frac {-\frac {b B \,x^{2}}{3 e}-\frac {\left (3 A b e +3 B a e +2 B b d \right ) x}{12 e^{2}}-\frac {12 A a \,e^{2}+3 A b d e +3 B a d e +2 b B \,d^{2}}{60 e^{3}}}{\left (e x +d \right )^{5}}\) | \(74\) |
default | \(-\frac {A a \,e^{2}-A b d e -B a d e +b B \,d^{2}}{5 e^{3} \left (e x +d \right )^{5}}-\frac {b B}{3 e^{3} \left (e x +d \right )^{3}}-\frac {A b e +B a e -2 B b d}{4 e^{3} \left (e x +d \right )^{4}}\) | \(79\) |
parallelrisch | \(-\frac {20 b B \,x^{2} e^{4}+15 A b \,e^{4} x +15 B a \,e^{4} x +10 B b d \,e^{3} x +12 A a \,e^{4}+3 A b d \,e^{3}+3 B a d \,e^{3}+2 b B \,d^{2} e^{2}}{60 e^{5} \left (e x +d \right )^{5}}\) | \(82\) |
norman | \(\frac {-\frac {b B \,x^{2}}{3 e}-\frac {\left (3 A b \,e^{3}+3 B a \,e^{3}+2 b B d \,e^{2}\right ) x}{12 e^{4}}-\frac {12 A a \,e^{4}+3 A b d \,e^{3}+3 B a d \,e^{3}+2 b B \,d^{2} e^{2}}{60 e^{5}}}{\left (e x +d \right )^{5}}\) | \(88\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \, {\left (B a + A b\right )} d e + 5 \, {\left (2 \, B b d e + 3 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
[In]
[Out]
Time = 1.85 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=\frac {- 12 A a e^{2} - 3 A b d e - 3 B a d e - 2 B b d^{2} - 20 B b e^{2} x^{2} + x \left (- 15 A b e^{2} - 15 B a e^{2} - 10 B b d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \, {\left (B a + A b\right )} d e + 5 \, {\left (2 \, B b d e + 3 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b e^{2} x^{2} + 10 \, B b d e x + 15 \, B a e^{2} x + 15 \, A b e^{2} x + 2 \, B b d^{2} + 3 \, B a d e + 3 \, A b d e + 12 \, A a e^{2}}{60 \, {\left (e x + d\right )}^{5} e^{3}} \]
[In]
[Out]
Time = 1.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx=-\frac {\frac {12\,A\,a\,e^2+2\,B\,b\,d^2+3\,A\,b\,d\,e+3\,B\,a\,d\,e}{60\,e^3}+\frac {x\,\left (3\,A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{12\,e^2}+\frac {B\,b\,x^2}{3\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]
[In]
[Out]