Integrand size = 24, antiderivative size = 240 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=\frac {B c^2 x}{e^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{2 e^6 (d+e x)^2}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^6 (d+e x)}-\frac {c (5 B c d-2 b B e-A c e) \log (d+e x)}{e^6} \]
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Time = 0.16 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=\frac {2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac {d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac {c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac {B c^2 x}{e^5} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {B c^2}{e^5}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^4}+\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)^3}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^5 (d+e x)^2}+\frac {c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)}\right ) \, dx \\ & = \frac {B c^2 x}{e^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{2 e^6 (d+e x)^2}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^6 (d+e x)}-\frac {c (5 B c d-2 b B e-A c e) \log (d+e x)}{e^6} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=-\frac {A e \left (b^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 b c e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+B \left (3 b^2 e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-2 b c d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+12 c (5 B c d-2 b B e-A c e) (d+e x)^4 \log (d+e x)}{12 e^6 (d+e x)^4} \]
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Time = 0.24 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.20
method | result | size |
norman | \(\frac {\frac {B \,c^{2} x^{5}}{e}-\frac {d^{2} \left (A \,b^{2} e^{3}+6 A b c d \,e^{2}-25 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-50 B b c \,d^{2} e +125 B \,c^{2} d^{3}\right )}{12 e^{6}}-\frac {\left (2 A b c \,e^{2}-4 A \,c^{2} d e +B \,b^{2} e^{2}-8 B b c d e +20 B \,c^{2} d^{2}\right ) x^{3}}{e^{3}}-\frac {\left (A \,b^{2} e^{3}+6 A b c d \,e^{2}-18 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-36 B b c \,d^{2} e +90 B \,c^{2} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {d \left (A \,b^{2} e^{3}+6 A b c d \,e^{2}-22 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-44 B b c \,d^{2} e +110 B \,c^{2} d^{3}\right ) x}{3 e^{5}}}{\left (e x +d \right )^{4}}+\frac {c \left (A c e +2 B b e -5 B c d \right ) \ln \left (e x +d \right )}{e^{6}}\) | \(288\) |
default | \(\frac {B \,c^{2} x}{e^{5}}+\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{3 e^{6} \left (e x +d \right )^{3}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +B \,b^{2} e^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{e^{6} \left (e x +d \right )}+\frac {c \left (A c e +2 B b e -5 B c d \right ) \ln \left (e x +d \right )}{e^{6}}-\frac {d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{4 e^{6} \left (e x +d \right )^{4}}-\frac {A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{2 e^{6} \left (e x +d \right )^{2}}\) | \(298\) |
risch | \(\frac {B \,c^{2} x}{e^{5}}+\frac {\left (-2 A b c \,e^{4}+4 A \,c^{2} d \,e^{3}-B \,b^{2} e^{4}+8 B b c d \,e^{3}-10 B \,c^{2} d^{2} e^{2}\right ) x^{3}-\frac {e \left (A \,b^{2} e^{3}+6 A b c d \,e^{2}-18 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-36 B b c \,d^{2} e +50 B \,c^{2} d^{3}\right ) x^{2}}{2}-\frac {d \left (A \,b^{2} e^{3}+6 A b c d \,e^{2}-22 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-44 B b c \,d^{2} e +65 B \,c^{2} d^{3}\right ) x}{3}-\frac {d^{2} \left (A \,b^{2} e^{3}+6 A b c d \,e^{2}-25 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-50 B b c \,d^{2} e +77 B \,c^{2} d^{3}\right )}{12 e}}{e^{5} \left (e x +d \right )^{4}}+\frac {c^{2} \ln \left (e x +d \right ) A}{e^{5}}+\frac {2 c \ln \left (e x +d \right ) B b}{e^{5}}-\frac {5 c^{2} \ln \left (e x +d \right ) B d}{e^{6}}\) | \(306\) |
parallelrisch | \(\frac {-125 B \,c^{2} d^{5}-240 B \ln \left (e x +d \right ) x \,c^{2} d^{4} e +48 A \ln \left (e x +d \right ) x \,c^{2} d^{3} e^{2}+48 A \ln \left (e x +d \right ) x^{3} c^{2} d \,e^{4}-240 B \ln \left (e x +d \right ) x^{3} c^{2} d^{2} e^{3}+24 B \ln \left (e x +d \right ) x^{4} b c \,e^{5}-60 B \ln \left (e x +d \right ) x^{4} c^{2} d \,e^{4}+72 A \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{3}-360 B \ln \left (e x +d \right ) x^{2} c^{2} d^{3} e^{2}-60 B \ln \left (e x +d \right ) c^{2} d^{5}-12 B \,x^{3} b^{2} e^{5}-6 A \,x^{2} b^{2} e^{5}+12 B \,x^{5} c^{2} e^{5}+24 B \ln \left (e x +d \right ) b c \,d^{4} e +176 B x b c \,d^{3} e^{2}-24 A x b c \,d^{2} e^{3}+216 B \,x^{2} b c \,d^{2} e^{3}-36 A \,x^{2} b c d \,e^{4}+96 B \,x^{3} b c d \,e^{4}-A \,b^{2} d^{2} e^{3}+25 A \,c^{2} d^{4} e -3 B \,b^{2} d^{3} e^{2}+96 B \ln \left (e x +d \right ) x^{3} b c d \,e^{4}-6 A b c \,d^{3} e^{2}+50 B b c \,d^{4} e +12 A \ln \left (e x +d \right ) x^{4} c^{2} e^{5}+144 B \ln \left (e x +d \right ) x^{2} b c \,d^{2} e^{3}+96 B \ln \left (e x +d \right ) x b c \,d^{3} e^{2}-24 A \,x^{3} b c \,e^{5}+48 A \,x^{3} c^{2} d \,e^{4}-240 B \,x^{3} c^{2} d^{2} e^{3}+108 A \,x^{2} c^{2} d^{2} e^{3}-18 B \,x^{2} b^{2} d \,e^{4}-540 B \,x^{2} c^{2} d^{3} e^{2}-4 A x \,b^{2} d \,e^{4}+88 A x \,c^{2} d^{3} e^{2}-12 B x \,b^{2} d^{2} e^{3}-440 B x \,c^{2} d^{4} e +12 A \ln \left (e x +d \right ) c^{2} d^{4} e}{12 e^{6} \left (e x +d \right )^{4}}\) | \(579\) |
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Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (234) = 468\).
Time = 0.31 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.98 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=\frac {12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} + 25 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 12 \, {\left (4 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 6 \, {\left (42 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 4 \, {\left (62 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 12 \, {\left (5 \, B c^{2} d^{5} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (5 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B c^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} + 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (5 \, B c^{2} d^{4} e - {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]
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Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=\text {Timed out} \]
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none
Time = 0.20 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=-\frac {77 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} - 25 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 12 \, {\left (10 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 6 \, {\left (50 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (65 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac {B c^{2} x}{e^{5}} - \frac {{\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (234) = 468\).
Time = 0.27 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.98 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=\frac {{\left (e x + d\right )} B c^{2}}{e^{6}} + \frac {{\left (5 \, B c^{2} d - 2 \, B b c e - A c^{2} e\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{6}} - \frac {\frac {120 \, B c^{2} d^{2} e^{22}}{e x + d} - \frac {60 \, B c^{2} d^{3} e^{22}}{{\left (e x + d\right )}^{2}} + \frac {20 \, B c^{2} d^{4} e^{22}}{{\left (e x + d\right )}^{3}} - \frac {3 \, B c^{2} d^{5} e^{22}}{{\left (e x + d\right )}^{4}} - \frac {96 \, B b c d e^{23}}{e x + d} - \frac {48 \, A c^{2} d e^{23}}{e x + d} + \frac {72 \, B b c d^{2} e^{23}}{{\left (e x + d\right )}^{2}} + \frac {36 \, A c^{2} d^{2} e^{23}}{{\left (e x + d\right )}^{2}} - \frac {32 \, B b c d^{3} e^{23}}{{\left (e x + d\right )}^{3}} - \frac {16 \, A c^{2} d^{3} e^{23}}{{\left (e x + d\right )}^{3}} + \frac {6 \, B b c d^{4} e^{23}}{{\left (e x + d\right )}^{4}} + \frac {3 \, A c^{2} d^{4} e^{23}}{{\left (e x + d\right )}^{4}} + \frac {12 \, B b^{2} e^{24}}{e x + d} + \frac {24 \, A b c e^{24}}{e x + d} - \frac {18 \, B b^{2} d e^{24}}{{\left (e x + d\right )}^{2}} - \frac {36 \, A b c d e^{24}}{{\left (e x + d\right )}^{2}} + \frac {12 \, B b^{2} d^{2} e^{24}}{{\left (e x + d\right )}^{3}} + \frac {24 \, A b c d^{2} e^{24}}{{\left (e x + d\right )}^{3}} - \frac {3 \, B b^{2} d^{3} e^{24}}{{\left (e x + d\right )}^{4}} - \frac {6 \, A b c d^{3} e^{24}}{{\left (e x + d\right )}^{4}} + \frac {6 \, A b^{2} e^{25}}{{\left (e x + d\right )}^{2}} - \frac {8 \, A b^{2} d e^{25}}{{\left (e x + d\right )}^{3}} + \frac {3 \, A b^{2} d^{2} e^{25}}{{\left (e x + d\right )}^{4}}}{12 \, e^{28}} \]
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Time = 10.85 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (A\,c^2\,e-5\,B\,c^2\,d+2\,B\,b\,c\,e\right )}{e^6}-\frac {x^2\,\left (\frac {3\,B\,b^2\,d\,e^3}{2}+\frac {A\,b^2\,e^4}{2}-18\,B\,b\,c\,d^2\,e^2+3\,A\,b\,c\,d\,e^3+25\,B\,c^2\,d^3\,e-9\,A\,c^2\,d^2\,e^2\right )+\frac {3\,B\,b^2\,d^3\,e^2+A\,b^2\,d^2\,e^3-50\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2+77\,B\,c^2\,d^5-25\,A\,c^2\,d^4\,e}{12\,e}+x\,\left (B\,b^2\,d^2\,e^2+\frac {A\,b^2\,d\,e^3}{3}-\frac {44\,B\,b\,c\,d^3\,e}{3}+2\,A\,b\,c\,d^2\,e^2+\frac {65\,B\,c^2\,d^4}{3}-\frac {22\,A\,c^2\,d^3\,e}{3}\right )+x^3\,\left (B\,b^2\,e^4-8\,B\,b\,c\,d\,e^3+2\,A\,b\,c\,e^4+10\,B\,c^2\,d^2\,e^2-4\,A\,c^2\,d\,e^3\right )}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}+\frac {B\,c^2\,x}{e^5} \]
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