Integrand size = 20, antiderivative size = 163 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{6 e^5 (d+e x)^6}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{5 e^5 (d+e x)^5}+\frac {b^2 (4 b B d-A b e-3 a B e)}{4 e^5 (d+e x)^4}-\frac {b^3 B}{3 e^5 (d+e x)^3} \]
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Time = 0.10 (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} \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=\frac {b^2 (-3 a B e-A b e+4 b B d)}{4 e^5 (d+e x)^4}-\frac {3 b (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (d+e x)^5}+\frac {(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (d+e x)^6}-\frac {(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}-\frac {b^3 B}{3 e^5 (d+e x)^3} \]
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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)^8}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^7}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^6}+\frac {b^2 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^5}+\frac {b^3 B}{e^4 (d+e x)^4}\right ) \, dx \\ & = -\frac {(b d-a e)^3 (B d-A e)}{7 e^5 (d+e x)^7}+\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e)}{6 e^5 (d+e x)^6}-\frac {3 b (b d-a e) (2 b B d-A b e-a B e)}{5 e^5 (d+e x)^5}+\frac {b^2 (4 b B d-A b e-3 a B e)}{4 e^5 (d+e x)^4}-\frac {b^3 B}{3 e^5 (d+e x)^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {10 a^3 e^3 (6 A e+B (d+7 e x))+6 a^2 b e^2 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a b^2 e \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )}{420 e^5 (d+e x)^7} \]
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Time = 0.69 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.66
method | result | size |
risch | \(\frac {-\frac {b^{3} B \,x^{4}}{3 e}-\frac {b^{2} \left (3 A b e +9 B a e +4 B b d \right ) x^{3}}{12 e^{2}}-\frac {b \left (12 A a b \,e^{2}+3 A \,b^{2} d e +12 B \,a^{2} e^{2}+9 B a b d e +4 b^{2} B \,d^{2}\right ) x^{2}}{20 e^{3}}-\frac {\left (30 A \,a^{2} b \,e^{3}+12 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +10 B \,a^{3} e^{3}+12 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{60 e^{4}}-\frac {60 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}+12 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}+12 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{420 e^{5}}}{\left (e x +d \right )^{7}}\) | \(270\) |
default | \(-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{7 e^{5} \left (e x +d \right )^{7}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{3} B}{3 e^{5} \left (e x +d \right )^{3}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{4 e^{5} \left (e x +d \right )^{4}}\) | \(281\) |
gosper | \(-\frac {140 B \,x^{4} b^{3} e^{4}+105 A \,x^{3} b^{3} e^{4}+315 B \,x^{3} a \,b^{2} e^{4}+140 B \,x^{3} b^{3} d \,e^{3}+252 A \,x^{2} a \,b^{2} e^{4}+63 A \,x^{2} b^{3} d \,e^{3}+252 B \,x^{2} a^{2} b \,e^{4}+189 B \,x^{2} a \,b^{2} d \,e^{3}+84 B \,x^{2} b^{3} d^{2} e^{2}+210 A x \,a^{2} b \,e^{4}+84 A x a \,b^{2} d \,e^{3}+21 A x \,b^{3} d^{2} e^{2}+70 B x \,a^{3} e^{4}+84 B x \,a^{2} b d \,e^{3}+63 B x a \,b^{2} d^{2} e^{2}+28 B x \,b^{3} d^{3} e +60 a^{3} A \,e^{4}+30 A \,a^{2} b d \,e^{3}+12 A a \,b^{2} d^{2} e^{2}+3 A \,b^{3} d^{3} e +10 B \,a^{3} d \,e^{3}+12 B \,a^{2} b \,d^{2} e^{2}+9 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{420 e^{5} \left (e x +d \right )^{7}}\) | \(301\) |
norman | \(\frac {-\frac {b^{3} B \,x^{4}}{3 e}-\frac {\left (3 A \,b^{3} e^{3}+9 B a \,b^{2} e^{3}+4 b^{3} B d \,e^{2}\right ) x^{3}}{12 e^{4}}-\frac {\left (12 A a \,b^{2} e^{4}+3 A \,b^{3} d \,e^{3}+12 B \,a^{2} b \,e^{4}+9 B a \,b^{2} d \,e^{3}+4 b^{3} B \,d^{2} e^{2}\right ) x^{2}}{20 e^{5}}-\frac {\left (30 A \,a^{2} b \,e^{5}+12 A a \,b^{2} d \,e^{4}+3 A \,b^{3} d^{2} e^{3}+10 B \,a^{3} e^{5}+12 B \,a^{2} b d \,e^{4}+9 B a \,b^{2} d^{2} e^{3}+4 b^{3} B \,d^{3} e^{2}\right ) x}{60 e^{6}}-\frac {60 a^{3} A \,e^{6}+30 A \,a^{2} b d \,e^{5}+12 A a \,b^{2} d^{2} e^{4}+3 A \,b^{3} d^{3} e^{3}+10 B \,a^{3} d \,e^{5}+12 B \,a^{2} b \,d^{2} e^{4}+9 B a \,b^{2} d^{3} e^{3}+4 B \,b^{3} d^{4} e^{2}}{420 e^{7}}}{\left (e x +d \right )^{7}}\) | \(306\) |
parallelrisch | \(-\frac {140 b^{3} B \,x^{4} e^{6}+105 A \,b^{3} e^{6} x^{3}+315 B a \,b^{2} e^{6} x^{3}+140 B \,b^{3} d \,e^{5} x^{3}+252 A a \,b^{2} e^{6} x^{2}+63 A \,b^{3} d \,e^{5} x^{2}+252 B \,a^{2} b \,e^{6} x^{2}+189 B a \,b^{2} d \,e^{5} x^{2}+84 B \,b^{3} d^{2} e^{4} x^{2}+210 A \,a^{2} b \,e^{6} x +84 A a \,b^{2} d \,e^{5} x +21 A \,b^{3} d^{2} e^{4} x +70 B \,a^{3} e^{6} x +84 B \,a^{2} b d \,e^{5} x +63 B a \,b^{2} d^{2} e^{4} x +28 B \,b^{3} d^{3} e^{3} x +60 a^{3} A \,e^{6}+30 A \,a^{2} b d \,e^{5}+12 A a \,b^{2} d^{2} e^{4}+3 A \,b^{3} d^{3} e^{3}+10 B \,a^{3} d \,e^{5}+12 B \,a^{2} b \,d^{2} e^{4}+9 B a \,b^{2} d^{3} e^{3}+4 B \,b^{3} d^{4} e^{2}}{420 e^{7} \left (e x +d \right )^{7}}\) | \(310\) |
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Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (153) = 306\).
Time = 0.23 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \, {\left (4 \, B b^{3} d e^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \, {\left (4 \, B b^{3} d^{2} e^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \, {\left (4 \, B b^{3} d^{3} e + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (153) = 306\).
Time = 0.23 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \, {\left (4 \, B b^{3} d e^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \, {\left (4 \, B b^{3} d^{2} e^{2} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \, {\left (4 \, B b^{3} d^{3} e + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {140 \, B b^{3} e^{4} x^{4} + 140 \, B b^{3} d e^{3} x^{3} + 315 \, B a b^{2} e^{4} x^{3} + 105 \, A b^{3} e^{4} x^{3} + 84 \, B b^{3} d^{2} e^{2} x^{2} + 189 \, B a b^{2} d e^{3} x^{2} + 63 \, A b^{3} d e^{3} x^{2} + 252 \, B a^{2} b e^{4} x^{2} + 252 \, A a b^{2} e^{4} x^{2} + 28 \, B b^{3} d^{3} e x + 63 \, B a b^{2} d^{2} e^{2} x + 21 \, A b^{3} d^{2} e^{2} x + 84 \, B a^{2} b d e^{3} x + 84 \, A a b^{2} d e^{3} x + 70 \, B a^{3} e^{4} x + 210 \, A a^{2} b e^{4} x + 4 \, B b^{3} d^{4} + 9 \, B a b^{2} d^{3} e + 3 \, A b^{3} d^{3} e + 12 \, B a^{2} b d^{2} e^{2} + 12 \, A a b^{2} d^{2} e^{2} + 10 \, B a^{3} d e^{3} + 30 \, A a^{2} b d e^{3} + 60 \, A a^{3} e^{4}}{420 \, {\left (e x + d\right )}^{7} e^{5}} \]
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Time = 1.29 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^8} \, dx=-\frac {\frac {10\,B\,a^3\,d\,e^3+60\,A\,a^3\,e^4+12\,B\,a^2\,b\,d^2\,e^2+30\,A\,a^2\,b\,d\,e^3+9\,B\,a\,b^2\,d^3\,e+12\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+3\,A\,b^3\,d^3\,e}{420\,e^5}+\frac {x\,\left (10\,B\,a^3\,e^3+12\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+9\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{60\,e^4}+\frac {b^2\,x^3\,\left (3\,A\,b\,e+9\,B\,a\,e+4\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (12\,B\,a^2\,e^2+9\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{20\,e^3}+\frac {B\,b^3\,x^4}{3\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
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