Integrand size = 20, antiderivative size = 133 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^4}{60 e (b d-a e)^3 (d+e x)^4} \]
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Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=\frac {b (a+b x)^4 (-3 a B e+A b e+2 b B d)}{60 e (d+e x)^4 (b d-a e)^3}+\frac {(a+b x)^4 (-3 a B e+A b e+2 b B d)}{15 e (d+e x)^5 (b d-a e)^2}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) \int \frac {(a+b x)^3}{(d+e x)^6} \, dx}{3 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {(b (2 b B d+A b e-3 a B e)) \int \frac {(a+b x)^3}{(d+e x)^5} \, dx}{15 e (b d-a e)^2} \\ & = -\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^4}{60 e (b d-a e)^3 (d+e x)^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {2 a^3 e^3 (5 A e+B (d+6 e x))+3 a^2 b e^2 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+3 a b^2 e \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )}{60 e^5 (d+e x)^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs. \(2(127)=254\).
Time = 0.70 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {b^{2} \left (A b e +3 B a e +2 B b d \right ) x^{3}}{3 e^{2}}-\frac {b \left (3 A a b \,e^{2}+A \,b^{2} d e +3 B \,a^{2} e^{2}+3 B a b d e +2 b^{2} B \,d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (6 A \,a^{2} b \,e^{3}+3 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +2 B \,a^{3} e^{3}+3 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) x}{10 e^{4}}-\frac {10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5}}}{\left (e x +d \right )^{6}}\) | \(266\) |
default | \(-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{3 e^{5} \left (e x +d \right )^{3}}-\frac {b^{3} B}{2 e^{5} \left (e x +d \right )^{2}}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{4 e^{5} \left (e x +d \right )^{4}}\) | \(281\) |
norman | \(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {\left (A \,b^{3} e^{2}+3 B a \,b^{2} e^{2}+2 b^{3} B d e \right ) x^{3}}{3 e^{3}}-\frac {\left (3 A a \,b^{2} e^{3}+A \,b^{3} d \,e^{2}+3 B \,a^{2} b \,e^{3}+3 B a \,b^{2} d \,e^{2}+2 b^{3} B \,d^{2} e \right ) x^{2}}{4 e^{4}}-\frac {\left (6 A \,a^{2} b \,e^{4}+3 A a \,b^{2} d \,e^{3}+A \,b^{3} d^{2} e^{2}+2 B \,a^{3} e^{4}+3 B \,a^{2} b d \,e^{3}+3 B a \,b^{2} d^{2} e^{2}+2 b^{3} B \,d^{3} e \right ) x}{10 e^{5}}-\frac {10 a^{3} A \,e^{5}+6 A \,a^{2} b d \,e^{4}+3 A a \,b^{2} d^{2} e^{3}+A \,b^{3} d^{3} e^{2}+2 B \,a^{3} d \,e^{4}+3 B \,a^{2} b \,d^{2} e^{3}+3 B a \,b^{2} d^{3} e^{2}+2 b^{3} B \,d^{4} e}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(294\) |
gosper | \(-\frac {30 B \,x^{4} b^{3} e^{4}+20 A \,x^{3} b^{3} e^{4}+60 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+45 A \,x^{2} a \,b^{2} e^{4}+15 A \,x^{2} b^{3} d \,e^{3}+45 B \,x^{2} a^{2} b \,e^{4}+45 B \,x^{2} a \,b^{2} d \,e^{3}+30 B \,x^{2} b^{3} d^{2} e^{2}+36 A x \,a^{2} b \,e^{4}+18 A x a \,b^{2} d \,e^{3}+6 A x \,b^{3} d^{2} e^{2}+12 B x \,a^{3} e^{4}+18 B x \,a^{2} b d \,e^{3}+18 B x a \,b^{2} d^{2} e^{2}+12 B x \,b^{3} d^{3} e +10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\) | \(300\) |
parallelrisch | \(-\frac {30 b^{3} B \,x^{4} e^{5}+20 A \,b^{3} e^{5} x^{3}+60 B a \,b^{2} e^{5} x^{3}+40 B \,b^{3} d \,e^{4} x^{3}+45 A a \,b^{2} e^{5} x^{2}+15 A \,b^{3} d \,e^{4} x^{2}+45 B \,a^{2} b \,e^{5} x^{2}+45 B a \,b^{2} d \,e^{4} x^{2}+30 B \,b^{3} d^{2} e^{3} x^{2}+36 A \,a^{2} b \,e^{5} x +18 A a \,b^{2} d \,e^{4} x +6 A \,b^{3} d^{2} e^{3} x +12 B \,a^{3} e^{5} x +18 B \,a^{2} b d \,e^{4} x +18 B a \,b^{2} d^{2} e^{3} x +12 B \,b^{3} d^{3} e^{2} x +10 a^{3} A \,e^{5}+6 A \,a^{2} b d \,e^{4}+3 A a \,b^{2} d^{2} e^{3}+A \,b^{3} d^{3} e^{2}+2 B \,a^{3} d \,e^{4}+3 B \,a^{2} b \,d^{2} e^{3}+3 B a \,b^{2} d^{3} e^{2}+2 b^{3} B \,d^{4} e}{60 e^{6} \left (e x +d \right )^{6}}\) | \(307\) |
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Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (127) = 254\).
Time = 0.22 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (127) = 254\).
Time = 0.24 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (127) = 254\).
Time = 0.30 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 40 \, B b^{3} d e^{3} x^{3} + 60 \, B a b^{2} e^{4} x^{3} + 20 \, A b^{3} e^{4} x^{3} + 30 \, B b^{3} d^{2} e^{2} x^{2} + 45 \, B a b^{2} d e^{3} x^{2} + 15 \, A b^{3} d e^{3} x^{2} + 45 \, B a^{2} b e^{4} x^{2} + 45 \, A a b^{2} e^{4} x^{2} + 12 \, B b^{3} d^{3} e x + 18 \, B a b^{2} d^{2} e^{2} x + 6 \, A b^{3} d^{2} e^{2} x + 18 \, B a^{2} b d e^{3} x + 18 \, A a b^{2} d e^{3} x + 12 \, B a^{3} e^{4} x + 36 \, A a^{2} b e^{4} x + 2 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} + 2 \, B a^{3} d e^{3} + 6 \, A a^{2} b d e^{3} + 10 \, A a^{3} e^{4}}{60 \, {\left (e x + d\right )}^{6} e^{5}} \]
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Time = 1.44 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {\frac {2\,B\,a^3\,d\,e^3+10\,A\,a^3\,e^4+3\,B\,a^2\,b\,d^2\,e^2+6\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+3\,A\,a\,b^2\,d^2\,e^2+2\,B\,b^3\,d^4+A\,b^3\,d^3\,e}{60\,e^5}+\frac {x\,\left (2\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2+6\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+3\,A\,a\,b^2\,d\,e^2+2\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{10\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{3\,e^2}+\frac {b\,x^2\,\left (3\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+3\,A\,a\,b\,e^2+2\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{4\,e^3}+\frac {B\,b^3\,x^4}{2\,e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
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