Integrand size = 20, antiderivative size = 120 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=\frac {(b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^5}-\frac {(b d-a e) (3 b B d-2 A b e-a B e)}{4 e^4 (d+e x)^4}+\frac {b (3 b B d-A b e-2 a B e)}{3 e^4 (d+e x)^3}-\frac {b^2 B}{2 e^4 (d+e x)^2} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=\frac {b (-2 a B e-A b e+3 b B d)}{3 e^4 (d+e x)^3}-\frac {(b d-a e) (-a B e-2 A b e+3 b B d)}{4 e^4 (d+e x)^4}+\frac {(b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^5}-\frac {b^2 B}{2 e^4 (d+e x)^2} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^6}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^5}+\frac {b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)^4}+\frac {b^2 B}{e^3 (d+e x)^3}\right ) \, dx \\ & = \frac {(b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^5}-\frac {(b d-a e) (3 b B d-2 A b e-a B e)}{4 e^4 (d+e x)^4}+\frac {b (3 b B d-A b e-2 a B e)}{3 e^4 (d+e x)^3}-\frac {b^2 B}{2 e^4 (d+e x)^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=-\frac {3 a^2 e^2 (4 A e+B (d+5 e x))+2 a b e \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+b^2 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]
[In]
[Out]
Time = 0.68 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.31
method | result | size |
risch | \(\frac {-\frac {b^{2} B \,x^{3}}{2 e}-\frac {b \left (2 A b e +4 B a e +3 B b d \right ) x^{2}}{6 e^{2}}-\frac {\left (6 A a b \,e^{2}+2 A \,b^{2} d e +3 B \,a^{2} e^{2}+4 B a b d e +3 b^{2} B \,d^{2}\right ) x}{12 e^{3}}-\frac {12 a^{2} A \,e^{3}+6 A a b d \,e^{2}+2 A \,b^{2} d^{2} e +3 B \,a^{2} d \,e^{2}+4 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{60 e^{4}}}{\left (e x +d \right )^{5}}\) | \(157\) |
default | \(-\frac {a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}}{5 e^{4} \left (e x +d \right )^{5}}-\frac {b \left (A b e +2 B a e -3 B b d \right )}{3 e^{4} \left (e x +d \right )^{3}}-\frac {b^{2} B}{2 e^{4} \left (e x +d \right )^{2}}-\frac {2 A a b \,e^{2}-2 A \,b^{2} d e +B \,a^{2} e^{2}-4 B a b d e +3 b^{2} B \,d^{2}}{4 e^{4} \left (e x +d \right )^{4}}\) | \(166\) |
gosper | \(-\frac {30 b^{2} B \,x^{3} e^{3}+20 A \,x^{2} b^{2} e^{3}+40 B \,x^{2} a b \,e^{3}+30 B \,x^{2} b^{2} d \,e^{2}+30 A x a b \,e^{3}+10 A x \,b^{2} d \,e^{2}+15 B x \,a^{2} e^{3}+20 B x a b d \,e^{2}+15 B x \,b^{2} d^{2} e +12 a^{2} A \,e^{3}+6 A a b d \,e^{2}+2 A \,b^{2} d^{2} e +3 B \,a^{2} d \,e^{2}+4 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{60 e^{4} \left (e x +d \right )^{5}}\) | \(169\) |
norman | \(\frac {-\frac {b^{2} B \,x^{3}}{2 e}-\frac {\left (2 A \,b^{2} e^{2}+4 B a b \,e^{2}+3 b^{2} B d e \right ) x^{2}}{6 e^{3}}-\frac {\left (6 A a b \,e^{3}+2 A \,b^{2} d \,e^{2}+3 B \,a^{2} e^{3}+4 B a b d \,e^{2}+3 b^{2} B \,d^{2} e \right ) x}{12 e^{4}}-\frac {12 a^{2} A \,e^{4}+6 A a b d \,e^{3}+2 A \,b^{2} d^{2} e^{2}+3 B \,a^{2} d \,e^{3}+4 B a b \,d^{2} e^{2}+3 b^{2} B \,d^{3} e}{60 e^{5}}}{\left (e x +d \right )^{5}}\) | \(176\) |
parallelrisch | \(-\frac {30 b^{2} B \,x^{3} e^{4}+20 A \,b^{2} e^{4} x^{2}+40 B a b \,e^{4} x^{2}+30 B \,b^{2} d \,e^{3} x^{2}+30 A a b \,e^{4} x +10 A \,b^{2} d \,e^{3} x +15 B \,a^{2} e^{4} x +20 B a b d \,e^{3} x +15 B \,b^{2} d^{2} e^{2} x +12 a^{2} A \,e^{4}+6 A a b d \,e^{3}+2 A \,b^{2} d^{2} e^{2}+3 B \,a^{2} d \,e^{3}+4 B a b \,d^{2} e^{2}+3 b^{2} B \,d^{3} e}{60 e^{5} \left (e x +d \right )^{5}}\) | \(176\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=-\frac {30 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 12 \, A a^{2} e^{3} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 10 \, {\left (3 \, B b^{2} d e^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \, {\left (3 \, B b^{2} d^{2} e + 2 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (114) = 228\).
Time = 12.62 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=\frac {- 12 A a^{2} e^{3} - 6 A a b d e^{2} - 2 A b^{2} d^{2} e - 3 B a^{2} d e^{2} - 4 B a b d^{2} e - 3 B b^{2} d^{3} - 30 B b^{2} e^{3} x^{3} + x^{2} \left (- 20 A b^{2} e^{3} - 40 B a b e^{3} - 30 B b^{2} d e^{2}\right ) + x \left (- 30 A a b e^{3} - 10 A b^{2} d e^{2} - 15 B a^{2} e^{3} - 20 B a b d e^{2} - 15 B b^{2} d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=-\frac {30 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 12 \, A a^{2} e^{3} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 10 \, {\left (3 \, B b^{2} d e^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \, {\left (3 \, B b^{2} d^{2} e + 2 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=-\frac {30 \, B b^{2} e^{3} x^{3} + 30 \, B b^{2} d e^{2} x^{2} + 40 \, B a b e^{3} x^{2} + 20 \, A b^{2} e^{3} x^{2} + 15 \, B b^{2} d^{2} e x + 20 \, B a b d e^{2} x + 10 \, A b^{2} d e^{2} x + 15 \, B a^{2} e^{3} x + 30 \, A a b e^{3} x + 3 \, B b^{2} d^{3} + 4 \, B a b d^{2} e + 2 \, A b^{2} d^{2} e + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} + 12 \, A a^{2} e^{3}}{60 \, {\left (e x + d\right )}^{5} e^{4}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx=-\frac {\frac {3\,B\,a^2\,d\,e^2+12\,A\,a^2\,e^3+4\,B\,a\,b\,d^2\,e+6\,A\,a\,b\,d\,e^2+3\,B\,b^2\,d^3+2\,A\,b^2\,d^2\,e}{60\,e^4}+\frac {x\,\left (3\,B\,a^2\,e^2+4\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+3\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{12\,e^3}+\frac {b\,x^2\,\left (2\,A\,b\,e+4\,B\,a\,e+3\,B\,b\,d\right )}{6\,e^2}+\frac {B\,b^2\,x^3}{2\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]
[In]
[Out]