Integrand size = 20, antiderivative size = 86 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {(B d-A e) (a+b x)^4}{5 e (b d-a e) (d+e x)^5}+\frac {(4 b B d+A b e-5 a B e) (a+b x)^4}{20 e (b d-a e)^2 (d+e x)^4} \]
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Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=\frac {(a+b x)^4 (-5 a B e+A b e+4 b B d)}{20 e (d+e x)^4 (b d-a e)^2}-\frac {(a+b x)^4 (B d-A e)}{5 e (d+e x)^5 (b d-a e)} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {(B d-A e) (a+b x)^4}{5 e (b d-a e) (d+e x)^5}+\frac {(4 b B d+A b e-5 a B e) \int \frac {(a+b x)^3}{(d+e x)^5} \, dx}{5 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^4}{5 e (b d-a e) (d+e x)^5}+\frac {(4 b B d+A b e-5 a B e) (a+b x)^4}{20 e (b d-a e)^2 (d+e x)^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(86)=172\).
Time = 0.07 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.45 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {a^3 e^3 (4 A e+B (d+5 e x))+a^2 b e^2 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+a b^2 e \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )}{20 e^5 (d+e x)^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(82)=164\).
Time = 0.69 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.07
method | result | size |
risch | \(\frac {-\frac {b^{3} B \,x^{4}}{e}-\frac {b^{2} \left (A b e +3 B a e +4 B b d \right ) x^{3}}{2 e^{2}}-\frac {b \left (2 A a b \,e^{2}+A \,b^{2} d e +2 B \,a^{2} e^{2}+3 B a b d e +4 b^{2} B \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {\left (3 A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +B \,a^{3} e^{3}+2 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{4 e^{4}}-\frac {4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 e^{5}}}{\left (e x +d \right )^{5}}\) | \(264\) |
norman | \(\frac {-\frac {b^{3} B \,x^{4}}{e}-\frac {\left (A \,b^{3} e +3 B a \,b^{2} e +4 b^{3} B d \right ) x^{3}}{2 e^{2}}-\frac {\left (2 A a \,b^{2} e^{2}+A \,b^{3} d e +2 B \,a^{2} b \,e^{2}+3 B a \,b^{2} d e +4 b^{3} B \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {\left (3 A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +B \,a^{3} e^{3}+2 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +4 b^{3} B \,d^{3}\right ) x}{4 e^{4}}-\frac {4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 e^{5}}}{\left (e x +d \right )^{5}}\) | \(272\) |
default | \(-\frac {b^{3} B}{e^{5} \left (e x +d \right )}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{e^{5} \left (e x +d \right )^{3}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{2 e^{5} \left (e x +d \right )^{2}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{4 e^{5} \left (e x +d \right )^{4}}\) | \(281\) |
gosper | \(-\frac {20 B \,x^{4} b^{3} e^{4}+10 A \,x^{3} b^{3} e^{4}+30 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+20 A \,x^{2} a \,b^{2} e^{4}+10 A \,x^{2} b^{3} d \,e^{3}+20 B \,x^{2} a^{2} b \,e^{4}+30 B \,x^{2} a \,b^{2} d \,e^{3}+40 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+10 A x a \,b^{2} d \,e^{3}+5 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+10 B x \,a^{2} b d \,e^{3}+15 B x a \,b^{2} d^{2} e^{2}+20 B x \,b^{3} d^{3} e +4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 \left (e x +d \right )^{5} e^{5}}\) | \(299\) |
parallelrisch | \(-\frac {20 B \,x^{4} b^{3} e^{4}+10 A \,x^{3} b^{3} e^{4}+30 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+20 A \,x^{2} a \,b^{2} e^{4}+10 A \,x^{2} b^{3} d \,e^{3}+20 B \,x^{2} a^{2} b \,e^{4}+30 B \,x^{2} a \,b^{2} d \,e^{3}+40 B \,x^{2} b^{3} d^{2} e^{2}+15 A x \,a^{2} b \,e^{4}+10 A x a \,b^{2} d \,e^{3}+5 A x \,b^{3} d^{2} e^{2}+5 B x \,a^{3} e^{4}+10 B x \,a^{2} b d \,e^{3}+15 B x a \,b^{2} d^{2} e^{2}+20 B x \,b^{3} d^{3} e +4 a^{3} A \,e^{4}+3 A \,a^{2} b d \,e^{3}+2 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +B \,a^{3} d \,e^{3}+2 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +4 b^{3} B \,d^{4}}{20 \left (e x +d \right )^{5} e^{5}}\) | \(299\) |
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (82) = 164\).
Time = 0.22 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 4 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 10 \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 10 \, {\left (4 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 5 \, {\left (4 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{20 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (82) = 164\).
Time = 0.24 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 4 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 10 \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 10 \, {\left (4 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 5 \, {\left (4 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{20 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 298, normalized size of antiderivative = 3.47 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {20 \, B b^{3} e^{4} x^{4} + 40 \, B b^{3} d e^{3} x^{3} + 30 \, B a b^{2} e^{4} x^{3} + 10 \, A b^{3} e^{4} x^{3} + 40 \, B b^{3} d^{2} e^{2} x^{2} + 30 \, B a b^{2} d e^{3} x^{2} + 10 \, A b^{3} d e^{3} x^{2} + 20 \, B a^{2} b e^{4} x^{2} + 20 \, A a b^{2} e^{4} x^{2} + 20 \, B b^{3} d^{3} e x + 15 \, B a b^{2} d^{2} e^{2} x + 5 \, A b^{3} d^{2} e^{2} x + 10 \, B a^{2} b d e^{3} x + 10 \, A a b^{2} d e^{3} x + 5 \, B a^{3} e^{4} x + 15 \, A a^{2} b e^{4} x + 4 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 2 \, B a^{2} b d^{2} e^{2} + 2 \, A a b^{2} d^{2} e^{2} + B a^{3} d e^{3} + 3 \, A a^{2} b d e^{3} + 4 \, A a^{3} e^{4}}{20 \, {\left (e x + d\right )}^{5} e^{5}} \]
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Time = 0.18 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.57 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^6} \, dx=-\frac {\frac {B\,a^3\,d\,e^3+4\,A\,a^3\,e^4+2\,B\,a^2\,b\,d^2\,e^2+3\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+2\,A\,a\,b^2\,d^2\,e^2+4\,B\,b^3\,d^4+A\,b^3\,d^3\,e}{20\,e^5}+\frac {x\,\left (B\,a^3\,e^3+2\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+4\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+4\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+4\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{2\,e^3}+\frac {B\,b^3\,x^4}{e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]
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