Integrand size = 46, antiderivative size = 448 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {256 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g) \sqrt {d+e x}}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Time = 0.45 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {802, 670, 662} \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {256 \sqrt {d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rule 662
Rule 670
Rule 802
Rubi steps \begin{align*} \text {integral}& = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(7 c e f+13 c d g-10 b e g) \int \frac {(d+e x)^{11/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(8 (7 c e f+13 c d g-10 b e g)) \int \frac {(d+e x)^{9/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{21 c^2 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(16 (2 c d-b e) (7 c e f+13 c d g-10 b e g)) \int \frac {(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{35 c^3 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (64 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g)\right ) \int \frac {(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{105 c^4 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (128 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g)\right ) \int \frac {(d+e x)^{3/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{105 c^5 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {256 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g) \sqrt {d+e x}}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (7 e f+78 d g-15 e g x)-32 b^3 c^2 e^3 \left (953 d^2 g+2 d e (91 f-204 g x)+3 e^2 x (-14 f+5 g x)\right )+16 b^2 c^3 e^2 \left (2844 d^3 g+3 d^2 e (287 f-681 g x)+e^3 x^2 (21 f+5 g x)+6 d e^2 x (-77 f+29 g x)\right )+c^5 \left (9414 d^5 g+3 d^4 e (1687 f-4707 g x)+3 e^5 x^4 (7 f+5 g x)+2 d e^4 x^3 (98 f+57 g x)+2 d^2 e^3 x^2 (903 f+257 g x)+12 d^3 e^2 x (-637 f+292 g x)\right )-2 b c^4 e \left (16563 d^4 g+12 d^3 e (581 f-1482 g x)+e^4 x^3 (28 f+15 g x)+12 d e^3 x^2 (63 f+16 g x)+6 d^2 e^2 x (-1106 f+449 g x)\right )\right )}{105 c^6 e^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \]
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Time = 0.37 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 d^{4} f \,c^{5} e \right )}{105 \sqrt {e x +d}\, \left (x c e +b e -c d \right )^{2} c^{6} e^{2}}\) | \(529\) |
gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 d^{4} f \,c^{5} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{105 c^{6} e^{2} \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(535\) |
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Time = 0.46 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (15 \, c^{5} e^{5} g x^{5} + 3 \, {\left (7 \, c^{5} e^{5} f + 2 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 2 \, {\left (14 \, {\left (7 \, c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} f + {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 6 \, {\left (7 \, {\left (43 \, c^{5} d^{2} e^{3} - 36 \, b c^{4} d e^{4} + 8 \, b^{2} c^{3} e^{5}\right )} f + 2 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 7 \, {\left (723 \, c^{5} d^{4} e - 1992 \, b c^{4} d^{3} e^{2} + 1968 \, b^{2} c^{3} d^{2} e^{3} - 832 \, b^{3} c^{2} d e^{4} + 128 \, b^{4} c e^{5}\right )} f + 2 \, {\left (4707 \, c^{5} d^{5} - 16563 \, b c^{4} d^{4} e + 22752 \, b^{2} c^{3} d^{3} e^{2} - 15248 \, b^{3} c^{2} d^{2} e^{3} + 4992 \, b^{4} c d e^{4} - 640 \, b^{5} e^{5}\right )} g - 3 \, {\left (28 \, {\left (91 \, c^{5} d^{3} e^{2} - 158 \, b c^{4} d^{2} e^{3} + 88 \, b^{2} c^{3} d e^{4} - 16 \, b^{3} c^{2} e^{5}\right )} f + {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c^{8} e^{5} x^{3} + c^{8} d^{3} e^{2} - 2 \, b c^{7} d^{2} e^{3} + b^{2} c^{6} d e^{4} - {\left (c^{8} d e^{4} - 2 \, b c^{7} e^{5}\right )} x^{2} - {\left (c^{8} d^{2} e^{3} - b^{2} c^{6} e^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{4} e^{4} x^{4} + 723 \, c^{4} d^{4} - 1992 \, b c^{3} d^{3} e + 1968 \, b^{2} c^{2} d^{2} e^{2} - 832 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + 4 \, {\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} x^{3} + 6 \, {\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} x^{2} - 12 \, {\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} x\right )} f}{15 \, {\left (c^{6} e^{2} x - c^{6} d e + b c^{5} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (15 \, c^{5} e^{5} x^{5} + 9414 \, c^{5} d^{5} - 33126 \, b c^{4} d^{4} e + 45504 \, b^{2} c^{3} d^{3} e^{2} - 30496 \, b^{3} c^{2} d^{2} e^{3} + 9984 \, b^{4} c d e^{4} - 1280 \, b^{5} e^{5} + 6 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} x^{4} + 2 \, {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} x^{3} + 12 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} x^{2} - 3 \, {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} x\right )} g}{105 \, {\left (c^{7} e^{3} x - c^{7} d e^{2} + b c^{6} e^{3}\right )} \sqrt {-c e x + c d - b e}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1092 vs. \(2 (412) = 824\).
Time = 0.37 (sec) , antiderivative size = 1092, normalized size of antiderivative = 2.44 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
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Time = 12.41 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (-2560\,g\,b^5\,e^5+19968\,g\,b^4\,c\,d\,e^4+1792\,f\,b^4\,c\,e^5-60992\,g\,b^3\,c^2\,d^2\,e^3-11648\,f\,b^3\,c^2\,d\,e^4+91008\,g\,b^2\,c^3\,d^3\,e^2+27552\,f\,b^2\,c^3\,d^2\,e^3-66252\,g\,b\,c^4\,d^4\,e-27888\,f\,b\,c^4\,d^3\,e^2+18828\,g\,c^5\,d^5+10122\,f\,c^5\,d^4\,e\right )}{105\,c^8\,e^5}+\frac {2\,g\,x^5\,\sqrt {d+e\,x}}{7\,c^3}+\frac {4\,x^3\,\sqrt {d+e\,x}\,\left (40\,g\,b^2\,e^2-192\,g\,b\,c\,d\,e-28\,f\,b\,c\,e^2+257\,g\,c^2\,d^2+98\,f\,c^2\,d\,e\right )}{105\,c^5\,e^2}+\frac {2\,x^4\,\sqrt {d+e\,x}\,\left (38\,c\,d\,g-10\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^4\,e}-\frac {x\,\sqrt {d+e\,x}\,\left (3840\,g\,b^4\,c\,e^5-26112\,g\,b^3\,c^2\,d\,e^4-2688\,f\,b^3\,c^2\,e^5+65376\,g\,b^2\,c^3\,d^2\,e^3+14784\,f\,b^2\,c^3\,d\,e^4-71136\,g\,b\,c^4\,d^3\,e^2-26544\,f\,b\,c^4\,d^2\,e^3+28242\,g\,c^5\,d^4\,e+15288\,f\,c^5\,d^3\,e^2\right )}{105\,c^8\,e^5}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,g\,b^3\,c^2\,e^5+5568\,g\,b^2\,c^3\,d\,e^4+672\,f\,b^2\,c^3\,e^5-10776\,g\,b\,c^4\,d^2\,e^3-3024\,f\,b\,c^4\,d\,e^4+7008\,g\,c^5\,d^3\,e^2+3612\,f\,c^5\,d^2\,e^3\right )}{105\,c^8\,e^5}\right )}{x^3+\frac {x\,\left (105\,b^2\,c^6\,e^5-105\,c^8\,d^2\,e^3\right )}{105\,c^8\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \]
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