Integrand size = 20, antiderivative size = 86 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=-\frac {(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac {(3 b B d+A b e-4 a B e) (a+b x)^3}{12 e (b d-a e)^2 (d+e x)^3} \]
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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} \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=\frac {(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac {(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac {(3 b B d+A b e-4 a B e) \int \frac {(a+b x)^2}{(d+e x)^4} \, dx}{4 e (b d-a e)} \\ & = -\frac {(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac {(3 b B d+A b e-4 a B e) (a+b x)^3}{12 e (b d-a e)^2 (d+e x)^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=-\frac {a^2 e^2 (3 A e+B (d+4 e x))+2 a b e \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]
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Time = 0.68 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.77
method | result | size |
risch | \(\frac {-\frac {b^{2} B \,x^{3}}{e}-\frac {b \left (A b e +2 B a e +3 B b d \right ) x^{2}}{2 e^{2}}-\frac {\left (2 A a b \,e^{2}+A \,b^{2} d e +B \,a^{2} e^{2}+2 B a b d e +3 b^{2} B \,d^{2}\right ) x}{3 e^{3}}-\frac {3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(152\) |
norman | \(\frac {-\frac {b^{2} B \,x^{3}}{e}-\frac {\left (A \,b^{2} e +2 B a b e +3 b^{2} B d \right ) x^{2}}{2 e^{2}}-\frac {\left (2 A a b \,e^{2}+A \,b^{2} d e +B \,a^{2} e^{2}+2 B a b d e +3 b^{2} B \,d^{2}\right ) x}{3 e^{3}}-\frac {3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(156\) |
default | \(-\frac {b^{2} B}{e^{4} \left (e x +d \right )}-\frac {2 A a b \,e^{2}-2 A \,b^{2} d e +B \,a^{2} e^{2}-4 B a b d e +3 b^{2} B \,d^{2}}{3 e^{4} \left (e x +d \right )^{3}}-\frac {b \left (A b e +2 B a e -3 B b d \right )}{2 e^{4} \left (e x +d \right )^{2}}-\frac {a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}}{4 e^{4} \left (e x +d \right )^{4}}\) | \(166\) |
gosper | \(-\frac {12 b^{2} B \,x^{3} e^{3}+6 A \,x^{2} b^{2} e^{3}+12 B \,x^{2} a b \,e^{3}+18 B \,x^{2} b^{2} d \,e^{2}+8 A x a b \,e^{3}+4 A x \,b^{2} d \,e^{2}+4 B x \,a^{2} e^{3}+8 B x a b d \,e^{2}+12 B x \,b^{2} d^{2} e +3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{12 \left (e x +d \right )^{4} e^{4}}\) | \(167\) |
parallelrisch | \(-\frac {12 b^{2} B \,x^{3} e^{3}+6 A \,x^{2} b^{2} e^{3}+12 B \,x^{2} a b \,e^{3}+18 B \,x^{2} b^{2} d \,e^{2}+8 A x a b \,e^{3}+4 A x \,b^{2} d \,e^{2}+4 B x \,a^{2} e^{3}+8 B x a b d \,e^{2}+12 B x \,b^{2} d^{2} e +3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 b^{2} B \,d^{3}}{12 \left (e x +d \right )^{4} e^{4}}\) | \(167\) |
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Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (82) = 164\).
Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.17 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=-\frac {12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} + {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B b^{2} d^{2} e + {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (75) = 150\).
Time = 5.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.59 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=\frac {- 3 A a^{2} e^{3} - 2 A a b d e^{2} - A b^{2} d^{2} e - B a^{2} d e^{2} - 2 B a b d^{2} e - 3 B b^{2} d^{3} - 12 B b^{2} e^{3} x^{3} + x^{2} \left (- 6 A b^{2} e^{3} - 12 B a b e^{3} - 18 B b^{2} d e^{2}\right ) + x \left (- 8 A a b e^{3} - 4 A b^{2} d e^{2} - 4 B a^{2} e^{3} - 8 B a b d e^{2} - 12 B b^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (82) = 164\).
Time = 0.21 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.17 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=-\frac {12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} + {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B b^{2} d^{2} e + {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (82) = 164\).
Time = 0.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.70 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=-\frac {\frac {12 \, B a b}{{\left (e x + d\right )}^{2}} + \frac {6 \, A b^{2}}{{\left (e x + d\right )}^{2}} - \frac {16 \, B a b d}{{\left (e x + d\right )}^{3}} - \frac {8 \, A b^{2} d}{{\left (e x + d\right )}^{3}} + \frac {6 \, B a b d^{2}}{{\left (e x + d\right )}^{4}} + \frac {3 \, A b^{2} d^{2}}{{\left (e x + d\right )}^{4}} + \frac {12 \, B b^{2}}{{\left (e x + d\right )} e} - \frac {18 \, B b^{2} d}{{\left (e x + d\right )}^{2} e} + \frac {12 \, B b^{2} d^{2}}{{\left (e x + d\right )}^{3} e} - \frac {3 \, B b^{2} d^{3}}{{\left (e x + d\right )}^{4} e} + \frac {4 \, B a^{2} e}{{\left (e x + d\right )}^{3}} + \frac {8 \, A a b e}{{\left (e x + d\right )}^{3}} - \frac {3 \, B a^{2} d e}{{\left (e x + d\right )}^{4}} - \frac {6 \, A a b d e}{{\left (e x + d\right )}^{4}} + \frac {3 \, A a^{2} e^{2}}{{\left (e x + d\right )}^{4}}}{12 \, e^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.14 \[ \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx=-\frac {\frac {B\,a^2\,d\,e^2+3\,A\,a^2\,e^3+2\,B\,a\,b\,d^2\,e+2\,A\,a\,b\,d\,e^2+3\,B\,b^2\,d^3+A\,b^2\,d^2\,e}{12\,e^4}+\frac {x\,\left (B\,a^2\,e^2+2\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+3\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{3\,e^3}+\frac {b\,x^2\,\left (A\,b\,e+2\,B\,a\,e+3\,B\,b\,d\right )}{2\,e^2}+\frac {B\,b^2\,x^3}{e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]
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