Integrand size = 20, antiderivative size = 163 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=-\frac {(b d-a e)^3 (B d-A e)}{8 e^5 (d+e x)^8}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{7 e^5 (d+e x)^7}-\frac {b (b d-a e) (2 b B d-A b e-a B e)}{2 e^5 (d+e x)^6}+\frac {b^2 (4 b B d-A b e-3 a B e)}{5 e^5 (d+e x)^5}-\frac {b^3 B}{4 e^5 (d+e x)^4} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 163, 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} \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=\frac {b^2 (-3 a B e-A b e+4 b B d)}{5 e^5 (d+e x)^5}-\frac {b (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (d+e x)^6}+\frac {(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (d+e x)^7}-\frac {(b d-a e)^3 (B d-A e)}{8 e^5 (d+e x)^8}-\frac {b^3 B}{4 e^5 (d+e x)^4} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^9}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^8}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^7}+\frac {b^2 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^6}+\frac {b^3 B}{e^4 (d+e x)^5}\right ) \, dx \\ & = -\frac {(b d-a e)^3 (B d-A e)}{8 e^5 (d+e x)^8}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{7 e^5 (d+e x)^7}-\frac {b (b d-a e) (2 b B d-A b e-a B e)}{2 e^5 (d+e x)^6}+\frac {b^2 (4 b B d-A b e-3 a B e)}{5 e^5 (d+e x)^5}-\frac {b^3 B}{4 e^5 (d+e x)^4} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=-\frac {5 a^3 e^3 (7 A e+B (d+8 e x))+5 a^2 b e^2 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a b^2 e \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )}{280 e^5 (d+e x)^8} \]
[In]
[Out]
Time = 0.70 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.61
method | result | size |
risch | \(\frac {-\frac {b^{3} B \,x^{4}}{4 e}-\frac {b^{2} \left (A b e +3 B a e +B b d \right ) x^{3}}{5 e^{2}}-\frac {b \left (5 A a b \,e^{2}+A \,b^{2} d e +5 B \,a^{2} e^{2}+3 B a b d e +b^{2} B \,d^{2}\right ) x^{2}}{10 e^{3}}-\frac {\left (15 A \,a^{2} b \,e^{3}+5 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +5 B \,a^{3} e^{3}+5 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}\right ) x}{35 e^{4}}-\frac {35 a^{3} A \,e^{4}+15 A \,a^{2} b d \,e^{3}+5 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +5 B \,a^{3} d \,e^{3}+5 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{280 e^{5}}}{\left (e x +d \right )^{8}}\) | \(262\) |
default | \(-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{8 e^{5} \left (e x +d \right )^{8}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{2 e^{5} \left (e x +d \right )^{6}}-\frac {b^{3} B}{4 e^{5} \left (e x +d \right )^{4}}\) | \(281\) |
norman | \(\frac {-\frac {b^{3} B \,x^{4}}{4 e}-\frac {\left (A \,b^{3} e^{4}+3 B a \,b^{2} e^{4}+b^{3} B d \,e^{3}\right ) x^{3}}{5 e^{5}}-\frac {\left (5 A a \,b^{2} e^{5}+A \,b^{3} d \,e^{4}+5 B \,a^{2} b \,e^{5}+3 B a \,b^{2} d \,e^{4}+b^{3} B \,d^{2} e^{3}\right ) x^{2}}{10 e^{6}}-\frac {\left (15 a^{2} b A \,e^{6}+5 A a \,b^{2} d \,e^{5}+A \,b^{3} d^{2} e^{4}+5 a^{3} B \,e^{6}+5 B \,a^{2} b d \,e^{5}+3 B a \,b^{2} d^{2} e^{4}+B \,b^{3} d^{3} e^{3}\right ) x}{35 e^{7}}-\frac {35 a^{3} A \,e^{7}+15 A \,a^{2} b d \,e^{6}+5 A a \,b^{2} d^{2} e^{5}+A \,b^{3} d^{3} e^{4}+5 B \,a^{3} d \,e^{6}+5 B \,a^{2} b \,d^{2} e^{5}+3 B a \,b^{2} d^{3} e^{4}+B \,b^{3} d^{4} e^{3}}{280 e^{8}}}{\left (e x +d \right )^{8}}\) | \(298\) |
gosper | \(-\frac {70 B \,x^{4} b^{3} e^{4}+56 A \,x^{3} b^{3} e^{4}+168 B \,x^{3} a \,b^{2} e^{4}+56 B \,x^{3} b^{3} d \,e^{3}+140 A \,x^{2} a \,b^{2} e^{4}+28 A \,x^{2} b^{3} d \,e^{3}+140 B \,x^{2} a^{2} b \,e^{4}+84 B \,x^{2} a \,b^{2} d \,e^{3}+28 B \,x^{2} b^{3} d^{2} e^{2}+120 A x \,a^{2} b \,e^{4}+40 A x a \,b^{2} d \,e^{3}+8 A x \,b^{3} d^{2} e^{2}+40 B x \,a^{3} e^{4}+40 B x \,a^{2} b d \,e^{3}+24 B x a \,b^{2} d^{2} e^{2}+8 B x \,b^{3} d^{3} e +35 a^{3} A \,e^{4}+15 A \,a^{2} b d \,e^{3}+5 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +5 B \,a^{3} d \,e^{3}+5 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{280 e^{5} \left (e x +d \right )^{8}}\) | \(299\) |
parallelrisch | \(-\frac {70 b^{3} B \,x^{4} e^{7}+56 A \,b^{3} e^{7} x^{3}+168 B a \,b^{2} e^{7} x^{3}+56 B \,b^{3} d \,e^{6} x^{3}+140 A a \,b^{2} e^{7} x^{2}+28 A \,b^{3} d \,e^{6} x^{2}+140 B \,a^{2} b \,e^{7} x^{2}+84 B a \,b^{2} d \,e^{6} x^{2}+28 B \,b^{3} d^{2} e^{5} x^{2}+120 A \,a^{2} b \,e^{7} x +40 A a \,b^{2} d \,e^{6} x +8 A \,b^{3} d^{2} e^{5} x +40 B \,a^{3} e^{7} x +40 B \,a^{2} b d \,e^{6} x +24 B a \,b^{2} d^{2} e^{5} x +8 B \,b^{3} d^{3} e^{4} x +35 a^{3} A \,e^{7}+15 A \,a^{2} b d \,e^{6}+5 A a \,b^{2} d^{2} e^{5}+A \,b^{3} d^{3} e^{4}+5 B \,a^{3} d \,e^{6}+5 B \,a^{2} b \,d^{2} e^{5}+3 B a \,b^{2} d^{3} e^{4}+B \,b^{3} d^{4} e^{3}}{280 e^{8} \left (e x +d \right )^{8}}\) | \(308\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (153) = 306\).
Time = 0.23 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=-\frac {70 \, B b^{3} e^{4} x^{4} + B b^{3} d^{4} + 35 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 56 \, {\left (B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 28 \, {\left (B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 5 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 8 \, {\left (B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (153) = 306\).
Time = 0.22 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=-\frac {70 \, B b^{3} e^{4} x^{4} + B b^{3} d^{4} + 35 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 56 \, {\left (B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 28 \, {\left (B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 5 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 8 \, {\left (B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 5 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=-\frac {70 \, B b^{3} e^{4} x^{4} + 56 \, B b^{3} d e^{3} x^{3} + 168 \, B a b^{2} e^{4} x^{3} + 56 \, A b^{3} e^{4} x^{3} + 28 \, B b^{3} d^{2} e^{2} x^{2} + 84 \, B a b^{2} d e^{3} x^{2} + 28 \, A b^{3} d e^{3} x^{2} + 140 \, B a^{2} b e^{4} x^{2} + 140 \, A a b^{2} e^{4} x^{2} + 8 \, B b^{3} d^{3} e x + 24 \, B a b^{2} d^{2} e^{2} x + 8 \, A b^{3} d^{2} e^{2} x + 40 \, B a^{2} b d e^{3} x + 40 \, A a b^{2} d e^{3} x + 40 \, B a^{3} e^{4} x + 120 \, A a^{2} b e^{4} x + B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 5 \, B a^{2} b d^{2} e^{2} + 5 \, A a b^{2} d^{2} e^{2} + 5 \, B a^{3} d e^{3} + 15 \, A a^{2} b d e^{3} + 35 \, A a^{3} e^{4}}{280 \, {\left (e x + d\right )}^{8} e^{5}} \]
[In]
[Out]
Time = 1.46 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.08 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx=-\frac {\frac {5\,B\,a^3\,d\,e^3+35\,A\,a^3\,e^4+5\,B\,a^2\,b\,d^2\,e^2+15\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+5\,A\,a\,b^2\,d^2\,e^2+B\,b^3\,d^4+A\,b^3\,d^3\,e}{280\,e^5}+\frac {x\,\left (5\,B\,a^3\,e^3+5\,B\,a^2\,b\,d\,e^2+15\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+5\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{35\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+B\,b\,d\right )}{5\,e^2}+\frac {b\,x^2\,\left (5\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+5\,A\,a\,b\,e^2+B\,b^2\,d^2+A\,b^2\,d\,e\right )}{10\,e^3}+\frac {B\,b^3\,x^4}{4\,e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
[In]
[Out]